From 7e5f427963138be95e83865d83baf38f779234d5 Mon Sep 17 00:00:00 2001 From: ccl-core Date: Fri, 21 Feb 2025 10:40:33 +0000 Subject: [PATCH 1/7] Update docs on CLA --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index eefedf459..410c39408 100644 --- a/README.md +++ b/README.md @@ -146,7 +146,7 @@ Here is an extremely simple example of the Croissant format, with comments showi - [Join](https://mlcommons.org/community/subscribe/) the mailing list - Attend Croissant meetings (please joint the list to automatically receive the invite) - [File issues for](https://github.com/mlcommons/croissant) bugs for feature requests -- [Contribute code](https://github.com/mlcommons/croissant) (please sign the MLCommons Association CLA first!) +- [Contribute to the code](https://github.com/mlcommons/croissant). To merge PRs, you will need to sign the MLCommons Association CLA at: https://mlcommons.org/community/subscribe/ ## Integrations From f9b7fc8f9f5ceb30a77fc228c4129a418d7dfd54 Mon Sep 17 00:00:00 2001 From: NevadaM <100001600+NevadaM@users.noreply.github.com> Date: Sat, 22 Feb 2025 16:42:27 +0000 Subject: [PATCH 2/7] [URGENT] pinning version of dependency for editor (#812) you might have noticed the HF space that holds the croissant editor was out of date and is now completely broken. I've been trying to figure out why - it's a dependency issue with streamlit-nested-layout, but it's impossible to replicate in local tests in the docker container or even in venvs. locally, version 0.1.2 of the library has a similar error with up-to-date streamlit versions, and I'm thinking that the HF space (when it creates the container) has some sort of caching that means it's loading 0.1.2 rather than 0.1.4 (which works as intended). So, this PR is just a pin in requirements.txt and an instruction to not cache in Dockerfile to maybe avoid that problem. Locally this works, but it seems it's a dice roll on whether it will work on HF. NOTE - commits to editor files here do not change the HF space. LONG TERM - should remove dependency on streamlit-nested-layout, which is serviced by two people and doesn't offer tons of value. I guess we might move away from streamlit anyway. --- editor/Dockerfile | 2 +- editor/requirements.txt | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/editor/Dockerfile b/editor/Dockerfile index 8b103f7b9..a352d9134 100644 --- a/editor/Dockerfile +++ b/editor/Dockerfile @@ -15,7 +15,7 @@ RUN apt-get update && apt-get install -y \ COPY ./ /app/ -RUN python3 -m pip install -r requirements.txt +RUN python3 -m pip install --no-cache-dir -r requirements.txt EXPOSE 8501 diff --git a/editor/requirements.txt b/editor/requirements.txt index a127b5746..36892aaf0 100644 --- a/editor/requirements.txt +++ b/editor/requirements.txt @@ -7,5 +7,5 @@ python-magic rdflib requests streamlit -streamlit-nested-layout +streamlit-nested-layout==0.1.4 twisted From 29f690cfa58b22ed52807ca437155b76a5cdade3 Mon Sep 17 00:00:00 2001 From: Pierre Marcenac Date: Mon, 24 Feb 2025 11:28:06 +0100 Subject: [PATCH 3/7] Fix dependencies in the editor and check that dependencies are correctly referenced by Hugging Face. (#814) contact Pierre Marcena or Neil Majithia for info --- editor/deploy_to_hf.sh | 6 ++++++ editor/requirements.txt | 2 +- 2 files changed, 7 insertions(+), 1 deletion(-) diff --git a/editor/deploy_to_hf.sh b/editor/deploy_to_hf.sh index 6cf38b95c..5be1dfa29 100755 --- a/editor/deploy_to_hf.sh +++ b/editor/deploy_to_hf.sh @@ -2,6 +2,12 @@ HF_REPO=/tmp/hf-croissant echo "Deleting $HF_REPO..." rm -rf ${HF_REPO} git clone git@hf.co:spaces/marcenacp/croissant-editor ${HF_REPO} +streamlit_version_in_hf=$(grep -oP 'sdk_version: \K[0-9]+\.[0-9]+\.[0-9]+' ${HF_REPO}/README.md) +streamlit_version_in_gh=$(grep -oP 'streamlit\=\=\K[0-9]+\.[0-9]+\.[0-9]+' requirements.txt) +if [ "$streamlit_version_in_hf" != "$streamlit_version_in_gh" ]; then + echo "ERROR: Versions of Streamlit in the requirements.txt and ${HF_REPO}/README.md should be the same." + exit 1 +fi echo "Copying files from $PWD to $HF_REPO..." rsync -aP --exclude="README.md" --exclude="*node_modules*" --exclude="cypress/*" --exclude="*__pycache__*" . ${HF_REPO} cd ${HF_REPO} diff --git a/editor/requirements.txt b/editor/requirements.txt index 36892aaf0..aa494670a 100644 --- a/editor/requirements.txt +++ b/editor/requirements.txt @@ -6,6 +6,6 @@ pytest python-magic rdflib requests -streamlit +streamlit==1.39.0 streamlit-nested-layout==0.1.4 twisted From 27d70f165213fd9febbfe26e53f3f75cdbea64c0 Mon Sep 17 00:00:00 2001 From: Pierre Marcenac Date: Mon, 24 Feb 2025 11:46:59 +0100 Subject: [PATCH 4/7] Add comment to understand why deps are fixed. (#815) --- editor/requirements.txt | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/editor/requirements.txt b/editor/requirements.txt index aa494670a..d7239ecd9 100644 --- a/editor/requirements.txt +++ b/editor/requirements.txt @@ -6,6 +6,6 @@ pytest python-magic rdflib requests -streamlit==1.39.0 +streamlit==1.39.0 # streamlit-nested-layout is only tested on 1.39.0 streamlit-nested-layout==0.1.4 twisted From 64e649f474c31635dcbd0b123bee34878a42ad9f Mon Sep 17 00:00:00 2001 From: ccl-core Date: Tue, 25 Feb 2025 14:51:21 +0000 Subject: [PATCH 5/7] New dataset with one-dim and multiple-dim arrays. --- .../metadata.json | 233 ++++++++++++++++++ .../output/default.jsonl | 2 + 2 files changed, 235 insertions(+) create mode 100644 datasets/1.1/huggingface-recipe_RL_data_roberta-base/metadata.json create mode 100644 datasets/1.1/huggingface-recipe_RL_data_roberta-base/output/default.jsonl diff --git a/datasets/1.1/huggingface-recipe_RL_data_roberta-base/metadata.json b/datasets/1.1/huggingface-recipe_RL_data_roberta-base/metadata.json new file mode 100644 index 000000000..01475b72b --- /dev/null +++ b/datasets/1.1/huggingface-recipe_RL_data_roberta-base/metadata.json @@ -0,0 +1,233 @@ +{ + "@context": { + "@language": "en", + "@vocab": "https://schema.org/", + "citeAs": "cr:citeAs", + "column": "cr:column", + "conformsTo": "dct:conformsTo", + "cr": "http://mlcommons.org/croissant/", + "data": { + "@id": "cr:data", + "@type": "@json" + }, + "dataBiases": "cr:dataBiases", + "dataCollection": "cr:dataCollection", + "dataType": { + "@id": "cr:dataType", + "@type": "@vocab" + }, + "dct": "http://purl.org/dc/terms/", + "extract": "cr:extract", + "field": "cr:field", + "fileProperty": "cr:fileProperty", + "fileObject": "cr:fileObject", + "fileSet": "cr:fileSet", + "format": "cr:format", + "includes": "cr:includes", + "isLiveDataset": "cr:isLiveDataset", + "jsonPath": "cr:jsonPath", + "key": "cr:key", + "md5": "cr:md5", + "parentField": "cr:parentField", + "path": "cr:path", + "personalSensitiveInformation": "cr:personalSensitiveInformation", + "recordSet": "cr:recordSet", + "references": "cr:references", + "regex": "cr:regex", + "repeated": "cr:repeated", + "replace": "cr:replace", + "sc": "https://schema.org/", + "separator": "cr:separator", + "source": "cr:source", + "subField": "cr:subField", + "transform": "cr:transform" + }, + "@type": "sc:Dataset", + "distribution": [ + { + "@type": "cr:FileObject", + "@id": "repo", + "name": "repo", + "description": "The Hugging Face git repository.", + "contentUrl": "https://huggingface.co/datasets/AnonymousSub/recipe_RL_data_roberta-base/tree/refs%2Fconvert%2Fparquet", + "encodingFormat": "git+https", + "sha256": "https://github.com/mlcommons/croissant/issues/80" + }, + { + "@type": "cr:FileSet", + "@id": "parquet-files-for-config-default", + "name": "parquet-files-for-config-default", + "description": "The underlying Parquet files as converted by Hugging Face (see: https://huggingface.co/docs/dataset-viewer/parquet).", + "containedIn": { + "@id": "repo" + }, + "encodingFormat": "application/x-parquet", + "includes": "default/*/*.parquet" + } + ], + "recordSet": [ + { + "@type": "cr:RecordSet", + "dataType": "cr:Split", + "key": { + "@id": "default_splits/split_name" + }, + "@id": "default_splits", + "name": "default_splits", + "description": "Splits for the default config.", + "field": [ + { + "@type": "cr:Field", + "@id": "default_splits/split_name", + "name": "split_name", + "description": "The name of the split.", + "dataType": "sc:Text" + } + ], + "data": [ + { + "default_splits/split_name": "train" + } + ] + }, + { + "@type": "cr:RecordSet", + "@id": "default", + "name": "default", + "description": "AnonymousSub/recipe_RL_data_roberta-base - 'default' subset", + "field": [ + { + "@type": "cr:Field", + "@id": "default/split", + "name": "default/split", + "description": "Split to which the example belongs to.", + "dataType": "sc:Text", + "source": { + "fileSet": { + "@id": "parquet-files-for-config-default" + }, + "extract": { + "fileProperty": "fullpath" + }, + "transform": { + "regex": "default/(?:partial-)?(train)/.+parquet$" + } + }, + "references": { + "field": { + "@id": "default_splits/split_name" + } + } + }, + { + "@type": "cr:Field", + "@id": "default/steps", + "name": "default/steps", + "description": "Column 'steps' from the Hugging Face parquet file.", + "dataType": "sc:Integer", + "isArray": true, + "arrayShape": "-1,-1", + "source": { + "fileSet": { + "@id": "parquet-files-for-config-default" + }, + "extract": { + "column": "steps" + } + } + }, + { + "@type": "cr:Field", + "@id": "default/step_attn_masks", + "name": "default/step_attn_masks", + "description": "Column 'step_attn_masks' from the Hugging Face parquet file.", + "dataType": "sc:Integer", + "isArray": true, + "arrayShape": "-1,-1", + "source": { + "fileSet": { + "@id": "parquet-files-for-config-default" + }, + "extract": { + "column": "step_attn_masks" + } + } + }, + { + "@type": "cr:Field", + "@id": "default/rewards", + "name": "default/rewards", + "description": "Column 'rewards' from the Hugging Face parquet file.", + "dataType": "sc:Float", + "isArray": true, + "source": { + "fileSet": { + "@id": "parquet-files-for-config-default" + }, + "extract": { + "column": "rewards" + } + } + }, + { + "@type": "cr:Field", + "@id": "default/actions", + "name": "default/actions", + "description": "Column 'actions' from the Hugging Face parquet file.", + "dataType": "sc:Integer", + "isArray": true, + "arrayShape": "-1,-1", + "source": { + "fileSet": { + "@id": "parquet-files-for-config-default" + }, + "extract": { + "column": "actions" + } + } + }, + { + "@type": "cr:Field", + "@id": "default/dones", + "name": "default/dones", + "description": "Column 'dones' from the Hugging Face parquet file.", + "dataType": "sc:Boolean", + "isArray": true, + "source": { + "fileSet": { + "@id": "parquet-files-for-config-default" + }, + "extract": { + "column": "dones" + } + } + } + ] + } + ], + "conformsTo": "http://mlcommons.org/croissant/1.0", + "name": "recipe_RL_data_roberta-base", + "description": "\n\t\n\t\t\n\t\n\t\n\t\tDataset Description\n\t\n\n\n\t\n\t\t\n\t\n\t\n\t\tStructure\n\t\n\n\nConsists of 5 fields\nEach row corresponds to a policy - sequence of actions, given an initial \u003CSTART\u003E state, and corresponding rewards at each step.\n\n\n\t\n\t\t\n\t\n\t\n\t\tFields\n\t\n\nsteps, step_attn_masks, rewards, actions, dones\n\n\t\n\t\t\n\t\n\t\n\t\tField descriptions\n\t\n\n\nsteps (List of lists of Ints) - tokenized step tokens of all the steps in the policy sequence (here we use the roberta-base tokenizer, as roberta-base would be used to encode each… See the full description on the dataset page: https://huggingface.co/datasets/AnonymousSub/recipe_RL_data_roberta-base.", + "alternateName": [ + "AnonymousSub/recipe_RL_data_roberta-base", + "recipe RL roberta base" + ], + "creator": { + "@type": "Person", + "name": "Anonymous", + "url": "https://huggingface.co/AnonymousSub" + }, + "keywords": [ + "monolingual", + "English", + "1M - 10M", + "parquet", + "Datasets", + "Dask", + "Croissant", + "Polars", + "🇺🇸 Region: US" + ], + "license": "https://choosealicense.com/licenses/undefined/", + "url": "https://huggingface.co/datasets/AnonymousSub/recipe_RL_data_roberta-base" +} diff --git a/datasets/1.1/huggingface-recipe_RL_data_roberta-base/output/default.jsonl b/datasets/1.1/huggingface-recipe_RL_data_roberta-base/output/default.jsonl new file mode 100644 index 000000000..56e939565 --- /dev/null +++ b/datasets/1.1/huggingface-recipe_RL_data_roberta-base/output/default.jsonl @@ -0,0 +1,2 @@ +{"default/split": "train", "default/steps": "[[0, 41552, 4014, 11328, 15698, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 44252, 209, 1964, 35, 112, 19821, 22126, 6, 32350, 8, 263, 548, 6158, 6, 112, 2125, 14099, 6, 290, 25119, 15039, 6, 112, 29387, 7636, 620, 13161, 6, 112, 21161, 2], [0, 10169, 13014, 5, 22126, 11, 14099, 10580, 13, 389, 728, 4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 42606, 833, 15039, 6, 7636, 620, 13161, 6, 8, 6740, 11, 10, 17793, 5749, 172, 3344, 157, 4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 20763, 5, 7689, 8, 5803, 14351, 454, 70, 5, 7075, 32, 157, 7664, 4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 44644, 10, 5730, 8, 9650, 12, 179, 5, 6836, 681, 4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 1779, 5, 681, 16, 2131, 615, 6, 10645, 10, 2125, 9, 22126, 11, 5, 15867, 36, 38457, 5, 7886, 13526, 43, 172, 1844, 12, 506, 1506, 454, 5, 3195, 9, 2], [0, 47583, 31, 5, 5730, 50, 25950, 254, 172, 317, 11, 10, 5299, 19, 2225, 21342, 36, 9226, 40, 15709, 7400, 681, 322, 2, 1, 1, 1, 1, 1, 1, 1, 1], [0, 104, 14477, 19, 4045, 8, 16933, 8929, 4, 1702, 8, 2254, 328, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]]", "default/step_attn_masks": "[[1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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16933, 8929, 4, 1702, 8, 2254, 328, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 1779, 5, 681, 16, 2131, 615, 6, 10645, 10, 2125, 9, 22126, 11, 5, 15867, 36, 38457, 5, 7886, 13526, 43, 172, 1844, 12, 506, 1506, 454, 5, 3195, 9, 2], [0, 42606, 833, 15039, 6, 7636, 620, 13161, 6, 8, 6740, 11, 10, 17793, 5749, 172, 3344, 157, 4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]]", "default/step_attn_masks": "[[1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]", "default/rewards": "[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]", "default/actions": "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]", "default/dones": "[False, False, False, False, False, False, False, False, False, True, True, True, True, True, True, True]"} From af614dee78a74f30655aeb72df3b056fe8f4718d Mon Sep 17 00:00:00 2001 From: ccl-core Date: Tue, 25 Feb 2025 15:07:37 +0000 Subject: [PATCH 6/7] Format new dataset --- .../metadata.json | 28 ++++++++----------- 1 file changed, 12 insertions(+), 16 deletions(-) diff --git a/datasets/1.1/huggingface-recipe_RL_data_roberta-base/metadata.json b/datasets/1.1/huggingface-recipe_RL_data_roberta-base/metadata.json index 01475b72b..ac13c425e 100644 --- a/datasets/1.1/huggingface-recipe_RL_data_roberta-base/metadata.json +++ b/datasets/1.1/huggingface-recipe_RL_data_roberta-base/metadata.json @@ -2,10 +2,12 @@ "@context": { "@language": "en", "@vocab": "https://schema.org/", + "arrayShape": "cr:arrayShape", "citeAs": "cr:citeAs", "column": "cr:column", "conformsTo": "dct:conformsTo", "cr": "http://mlcommons.org/croissant/", + "rai": "http://mlcommons.org/croissant/RAI/", "data": { "@id": "cr:data", "@type": "@json" @@ -17,6 +19,10 @@ "@type": "@vocab" }, "dct": "http://purl.org/dc/terms/", + "examples": { + "@id": "cr:examples", + "@type": "@json" + }, "extract": "cr:extract", "field": "cr:field", "fileProperty": "cr:fileProperty", @@ -24,6 +30,7 @@ "fileSet": "cr:fileSet", "format": "cr:format", "includes": "cr:includes", + "isArray": "cr:isArray", "isLiveDataset": "cr:isLiveDataset", "jsonPath": "cr:jsonPath", "key": "cr:key", @@ -84,11 +91,9 @@ "dataType": "sc:Text" } ], - "data": [ - { - "default_splits/split_name": "train" - } - ] + "data": { + "default_splits/split_name": "train" + } }, { "@type": "cr:RecordSet", @@ -141,9 +146,9 @@ "@id": "default/step_attn_masks", "name": "default/step_attn_masks", "description": "Column 'step_attn_masks' from the Hugging Face parquet file.", + "arrayShape": "-1,-1", "dataType": "sc:Integer", "isArray": true, - "arrayShape": "-1,-1", "source": { "fileSet": { "@id": "parquet-files-for-config-default" @@ -174,9 +179,9 @@ "@id": "default/actions", "name": "default/actions", "description": "Column 'actions' from the Hugging Face parquet file.", + "arrayShape": "-1,-1", "dataType": "sc:Integer", "isArray": true, - "arrayShape": "-1,-1", "source": { "fileSet": { "@id": "parquet-files-for-config-default" @@ -208,15 +213,6 @@ "conformsTo": "http://mlcommons.org/croissant/1.0", "name": "recipe_RL_data_roberta-base", "description": "\n\t\n\t\t\n\t\n\t\n\t\tDataset Description\n\t\n\n\n\t\n\t\t\n\t\n\t\n\t\tStructure\n\t\n\n\nConsists of 5 fields\nEach row corresponds to a policy - sequence of actions, given an initial \u003CSTART\u003E state, and corresponding rewards at each step.\n\n\n\t\n\t\t\n\t\n\t\n\t\tFields\n\t\n\nsteps, step_attn_masks, rewards, actions, dones\n\n\t\n\t\t\n\t\n\t\n\t\tField descriptions\n\t\n\n\nsteps (List of lists of Ints) - tokenized step tokens of all the steps in the policy sequence (here we use the roberta-base tokenizer, as roberta-base would be used to encode each… See the full description on the dataset page: https://huggingface.co/datasets/AnonymousSub/recipe_RL_data_roberta-base.", - "alternateName": [ - "AnonymousSub/recipe_RL_data_roberta-base", - "recipe RL roberta base" - ], - "creator": { - "@type": "Person", - "name": "Anonymous", - "url": "https://huggingface.co/AnonymousSub" - }, "keywords": [ "monolingual", "English", From 3f8ee88c7c6f8939bc9b4fc5726d246e0f760d87 Mon Sep 17 00:00:00 2001 From: ccl-core Date: Wed, 26 Feb 2025 10:29:19 +0000 Subject: [PATCH 7/7] add open r1 math raw dataset --- .../metadata.json | 349 ++++++++++++++++++ .../output/default.jsonl | 2 + 2 files changed, 351 insertions(+) create mode 100644 datasets/1.0/huggingface-open-r1-math-raw/metadata.json create mode 100644 datasets/1.0/huggingface-open-r1-math-raw/output/default.jsonl diff --git a/datasets/1.0/huggingface-open-r1-math-raw/metadata.json b/datasets/1.0/huggingface-open-r1-math-raw/metadata.json new file mode 100644 index 000000000..1f88c402b --- /dev/null +++ b/datasets/1.0/huggingface-open-r1-math-raw/metadata.json @@ -0,0 +1,349 @@ +{ + "@context": { + "@language": "en", + "@vocab": "https://schema.org/", + "citeAs": "cr:citeAs", + "column": "cr:column", + "conformsTo": "dct:conformsTo", + "cr": "http://mlcommons.org/croissant/", + "data": { + "@id": "cr:data", + "@type": "@json" + }, + "dataBiases": "cr:dataBiases", + "dataCollection": "cr:dataCollection", + "dataType": { + "@id": "cr:dataType", + "@type": "@vocab" + }, + "dct": "http://purl.org/dc/terms/", + "extract": "cr:extract", + "field": "cr:field", + "fileProperty": "cr:fileProperty", + "fileObject": "cr:fileObject", + "fileSet": "cr:fileSet", + "format": "cr:format", + "includes": "cr:includes", + "isLiveDataset": "cr:isLiveDataset", + "jsonPath": "cr:jsonPath", + "key": "cr:key", + "md5": "cr:md5", + "parentField": "cr:parentField", + "path": "cr:path", + "personalSensitiveInformation": "cr:personalSensitiveInformation", + "recordSet": "cr:recordSet", + "references": "cr:references", + "regex": "cr:regex", + "repeated": "cr:repeated", + "replace": "cr:replace", + "sc": "https://schema.org/", + "separator": "cr:separator", + "source": "cr:source", + "subField": "cr:subField", + "transform": "cr:transform" + }, + "@type": "sc:Dataset", + "distribution": [ + { + "@type": "cr:FileObject", + "@id": "repo", + "name": "repo", + "description": "The Hugging Face git repository.", + "contentUrl": "https://huggingface.co/datasets/open-r1/OpenR1-Math-Raw/tree/refs%2Fconvert%2Fparquet", + "encodingFormat": "git+https", + "sha256": "https://github.com/mlcommons/croissant/issues/80" + }, + { + "@type": "cr:FileSet", + "@id": "parquet-files-for-config-default", + "containedIn": { + "@id": "repo" + }, + "encodingFormat": "application/x-parquet", + "includes": "default/*/*.parquet" + } + ], + "recordSet": [ + { + "@type": "cr:RecordSet", + "dataType": "cr:Split", + "key": { + "@id": "default_splits/split_name" + }, + "@id": "default_splits", + "name": "default_splits", + "description": "Splits for the default config.", + "field": [ + { + "@type": "cr:Field", + "@id": "default_splits/split_name", + "dataType": "sc:Text" + } + ], + "data": [ + { + "default_splits/split_name": "train" + } + ] + }, + { + "@type": "cr:RecordSet", + "@id": "default", + "description": "open-r1/OpenR1-Math-Raw - 'default' subset", + "field": [ + { + "@type": "cr:Field", + "@id": "default/split", + "dataType": "sc:Text", + "source": { + "fileSet": { + "@id": "parquet-files-for-config-default" + }, + "extract": { + "fileProperty": "fullpath" + }, + "transform": { + "regex": "default/(?:partial-)?(train)/.+parquet$" + } + }, + "references": { + "field": { + "@id": "default_splits/split_name" + } + } + }, + { + "@type": "cr:Field", + "@id": "default/problem", + "dataType": "sc:Text", + "source": { + "fileSet": { + "@id": "parquet-files-for-config-default" + }, + "extract": { + "column": "problem" + } + } + }, + { + "@type": "cr:Field", + "@id": "default/solution", + "dataType": "sc:Text", + "source": { + "fileSet": { + "@id": "parquet-files-for-config-default" + }, + "extract": { + "column": "solution" + } + } + }, + { + "@type": "cr:Field", + "@id": "default/answer", + "dataType": "sc:Text", + "source": { + "fileSet": { + "@id": "parquet-files-for-config-default" + }, + "extract": { + "column": "answer" + } + } + }, + { + "@type": "cr:Field", + "@id": "default/problem_type", + "dataType": "sc:Text", + "source": { + "fileSet": { + "@id": "parquet-files-for-config-default" + }, + "extract": { + "column": "problem_type" + } + } 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It consists of 516k math problems sourced from AI-MO/NuminaMath-1.5 with 1 to 8 reasoning traces generated by DeepSeek R1. \nThe traces were verified using Math Verify and LLM-as-Judge based verifier (Llama-3.3-70B-Instruct)\nThe dataset contains:\n\n516,499 problems\n1,209,403 R1-generated solutions, with 2.3 solutions per problem on average\nre-parsed answers… See the full description on the dataset page: https://huggingface.co/datasets/open-r1/OpenR1-Math-Raw.", + "alternateName": [ + "open-r1/OpenR1-Math-Raw" + ], + "creator": { + "@type": "Organization", + "name": "Open R1", + "url": "https://huggingface.co/open-r1" + }, + "keywords": [ + "English", + "apache-2.0", + "100K - 1M", + "parquet", + "Text", + "Datasets", + "Dask", + "Croissant", + "Polars", + "🇺🇸 Region: US" + ], + "license": "https://choosealicense.com/licenses/apache-2.0/", + "url": "https://huggingface.co/datasets/open-r1/OpenR1-Math-Raw" +} diff --git a/datasets/1.0/huggingface-open-r1-math-raw/output/default.jsonl b/datasets/1.0/huggingface-open-r1-math-raw/output/default.jsonl new file mode 100644 index 000000000..39f16cedd --- /dev/null +++ b/datasets/1.0/huggingface-open-r1-math-raw/output/default.jsonl @@ -0,0 +1,2 @@ +{"default/split": "train", "default/problem": "\nProblem 1. Find all prime numbers $p$ for which there exist positive integers $x, y$ and $z$ such that the number\n\n$$\nx^{p}+y^{p}+z^{p}-x-y-z\n$$\n\nis a product of exactly three distinct prime numbers.\n", "default/solution": "\nSolution. Let $A=x^{p}+y^{p}+z^{p}-x-y-z$. For $p=2$, we take $x=y=4$ and $z=3$. Then $A=30=2 \\cdot 3 \\cdot 5$. For $p=3$ we can take $x=3$ and $y=2$ and $z=1$. Then again $A=30=2 \\cdot 3 \\cdot 5$. For $p=5$ we can take $x=2$ and $y=1$ and $z=1$. Again $A=30=2 \\cdot 3 \\cdot 5$.\n\nAssume now that $p \\geqslant 7$. Working modulo 2 and modulo 3 we see that $A$ is divisible by both 2 and 3. Moreover, by Fermat's Little Theorem, we have\n\n$$\nx^{p}+y^{p}+z^{p}-x-y-z \\equiv x+y+z-x-y-z=0 \\bmod p \\text {. }\n$$\n\nTherefore, by the given condition, we have to solve the equation\n\n$$\nx^{p}+y^{p}+z^{p}-x-y-z=6 p\n$$\n\nIf one of the numbers $x, y$ and $z$ is bigger than or equal to 2 , let's say $x \\geqslant 2$, then\n\n$$\n6 p \\geqslant x^{p}-x=x\\left(x^{p-1}-1\\right) \\geqslant 2\\left(2^{p-1}-1\\right)=2^{p}-2\n$$\n\nIt is easy to check by induction that $2^{n}-2>6 n$ for all natural numbers $n \\geqslant 6$. This contradiction shows that there are no more values of $p$ which satisfy the required property.\n\nRemark. There are a couple of other ways to prove that $2^{p}-2>6 p$ for $p \\geqslant 7$. For example, we can use the Binomial Theorem as follows:\n\n$$\n2^{p}-2 \\geqslant 1+p+\\frac{p(p-1)}{2}+\\frac{p(p-1)(p-2)}{6}-2 \\geqslant 1+p+3 p+5 p-2>6 p\n$$\n\nWe can also use Bernoulli's Inequality as follows:\n\n$$\n2^{p}-2=8(1+1)^{p-3}-2 \\geqslant 8(1+(p-3))-2=8 p-18>6 p\n$$\n\nThe last inequality is true for $p \\geqslant 11$. For $p=7$ we can see directly that $2^{p}-2>6 p$.\n\nOne can also use calculus to show that $f(x)=2^{x}-6 x$ is increasing for $x \\geqslant 5$.\n", "default/answer": "proof", "default/problem_type": "Number Theory", "default/question_type": "math-word-problem", "default/problem_is_valid": "Yes", "default/solution_is_valid": "Yes", "default/source": "olympiads", "default/synthetic": false, "default/generations": "[b'\\nOkay, so I need to find all prime numbers p where there exist positive integers x, y, z such that the expression x^p + y^p + z^p - x - y - z is a product of exactly three distinct prime numbers. Hmm, let me break this down step by step.\\n\\nFirst, since p is a prime number, possible values for p are 2, 3, 5, 7, etc. Maybe I should start by testing small primes like 2 and 3 first because they\\'re simpler and might give me some pattern or rule.\\n\\nLet\\'s start with p=2. Then the expression becomes x\\xc2\\xb2 + y\\xc2\\xb2 + z\\xc2\\xb2 - x - y - z. I need this to be a product of three distinct primes. Let me try small values for x, y, z. Since they are positive integers, the smallest values are 1, 1, 1.\\n\\nPlugging in x=1, y=1, z=1: 1 + 1 + 1 - 1 - 1 - 1 = 0. That\\'s not helpful. So maybe 1,1,2?\\n\\nx=1, y=1, z=2: 1 + 1 + 4 -1 -1 -2 = 2. 2 is a prime, but we need a product of three distinct primes. So that\\'s not it.\\n\\nHow about x=2, y=2, z=2: 4 + 4 + 4 - 2 - 2 -2 = 6. 6 factors into 2*3, which is two primes. Not enough.\\n\\nMaybe try different combinations. Let\\'s see x=2, y=1, z=1: 4 + 1 +1 -2 -1 -1 = 2. Still prime.\\n\\nx=3, y=1, z=1: 9 +1 +1 -3 -1 -1=6. Again 2*3. Hmm.\\n\\nWhat if x=2, y=2, z=3: 4 +4 +9 -2 -2 -3=10. 10=2*5. Still two primes.\\n\\nWait, maybe larger numbers. Let\\'s try x=3, y=2, z=2: 9 +4 +4 -3 -2 -2=10. Same as above.\\n\\nx=3, y=3, z=3: 9 +9 +9 -3 -3 -3=18. 18=2*3*3. Three primes but two are the same. So not distinct.\\n\\nHmm. Maybe x=4, y=1, z=1: 16 +1 +1 -4 -1 -1=12. 12=2*2*3. Again duplicates.\\n\\nx=2, y=3, z=4: 4 +9 +16 -2 -3 -4=20. 20=2*2*5. Nope.\\n\\nWait, maybe higher numbers. Let\\'s try x=5, y=2, z=1: 25 +4 +1 -5 -2 -1=22. 22=2*11. Two primes.\\n\\nx=4, y=3, z=2: 16 +9 +4 -4 -3 -2=20. Still 20. Hmm.\\n\\nThis is getting tedious. Maybe there\\'s a pattern here. For p=2, the expression is x\\xc2\\xb2 + y\\xc2\\xb2 + z\\xc2\\xb2 -x -y -z. Let\\'s see if we can represent this in another way. Maybe complete the square for each variable.\\n\\nFor each variable, say x\\xc2\\xb2 -x. Completing the square: x\\xc2\\xb2 -x = x\\xc2\\xb2 -x + 1/4 -1/4 = (x - 1/2)\\xc2\\xb2 -1/4. But since x is an integer, (x - 1/2)\\xc2\\xb2 is a square of a half-integer. Not sure if that helps.\\n\\nAlternatively, note that x\\xc2\\xb2 -x = x(x -1). Similarly for y and z. So the entire expression is x(x -1) + y(y -1) + z(z -1). So x(x -1) is the product of two consecutive integers, so it\\'s always even. Similarly for y and z. Therefore, the entire expression is even + even + even = even. So the result is even, meaning it\\'s divisible by 2. So the product of three distinct primes must include 2. Therefore, the expression is 2 times two other distinct primes. So the total would be 2*q*r where q and r are distinct primes different from 2.\\n\\nSo maybe we can set x(x -1) + y(y -1) + z(z -1) = 2*q*r.\\n\\nBut how do we find x, y, z such that this is true? Let\\'s note that x(x -1) is 0 when x=1, 2 when x=2, 6 when x=3, 12 when x=4, etc.\\n\\nSo the possible contributions from each variable:\\n\\nIf x=1: 0\\n\\nx=2: 2\\n\\nx=3: 6\\n\\nx=4: 12\\n\\nx=5: 20\\n\\nx=6: 30\\n\\nSimilarly for y and z.\\n\\nSo let\\'s consider possible combinations of x, y, z where the sum of their x(x-1) terms equals a number which is 2*q*r. Let\\'s see.\\n\\nLet me try x=2, y=2, z=3: 2 + 2 + 6 = 10. 10=2*5. But we need three primes. Wait, 10 is 2*5, which is two primes. Not enough.\\n\\nx=2, y=3, z=3: 2 +6 +6=14. 14=2*7. Again two primes.\\n\\nx=2, y=3, z=4: 2 +6 +12=20. 20=2*2*5. Three primes but duplicates.\\n\\nx=3, y=3, z=3: 6 +6 +6=18. 18=2*3*3. Again duplicates.\\n\\nx=3, y=3, z=4: 6 +6 +12=24. 24=2*2*2*3. Too many duplicates.\\n\\nx=2, y=4, z=5: 2 +12 +20=34. 34=2*17. Two primes.\\n\\nx=3, y=4, z=5: 6 +12 +20=38. 38=2*19. Two primes.\\n\\nx=4, y=4, z=4: 12 +12 +12=36. 36=2*2*3*3. Nope.\\n\\nWait, perhaps if one of the variables is 1. Let\\'s try x=1, y=2, z=3: 0 +2 +6=8. 8=2*2*2. All duplicates.\\n\\nx=1, y=3, z=4: 0 +6 +12=18. 18=2*3*3. Again duplicates.\\n\\nx=1, y=4, z=5: 0 +12 +20=32. 32=2^5. Not enough.\\n\\nx=1, y=5, z=6: 0 +20 +30=50. 50=2*5*5. Still duplicates.\\n\\nHmm. Maybe we need one term to be larger. Let\\'s see x=5: 20, so x=5, y=2, z=2: 20 +2 +2=24. Same as before.\\n\\nx=5, y=3, z=2: 20 +6 +2=28. 28=2*2*7. Duplicates.\\n\\nx=5, y=5, z=5: 20 +20 +20=60. 60=2*2*3*5. Duplicates.\\n\\nx=6, y=2, z=3: 30 +2 +6=38=2*19. Two primes.\\n\\nx=6, y=6, z=6: 30 +30 +30=90=2*3*3*5. Duplicates.\\n\\nWait, maybe if two variables are 1? Let\\'s try x=1, y=1, z=4: 0 +0 +12=12=2*2*3. Duplicates.\\n\\nx=1, y=1, z=5: 0 +0 +20=20=2*2*5. Duplicates.\\n\\nx=1, y=1, z=6: 0 +0 +30=30=2*3*5. Wait, 30 is 2*3*5. That\\'s three distinct primes! So x=1, y=1, z=6: the expression is 30, which is the product of three distinct primes. So that works for p=2.\\n\\nWait, let\\'s verify that. If p=2, x=1, y=1, z=6:\\n\\nx\\xc2\\xb2 + y\\xc2\\xb2 + z\\xc2\\xb2 -x - y - z = 1 + 1 + 36 -1 -1 -6 = 30. 30=2*3*5. Exactly three distinct primes. So p=2 is possible.\\n\\nIs this the only case? Let\\'s check other possibilities where two variables are 1. Let\\'s try x=1, y=1, z=7: 0 +0 +42=42=2*3*7. Also three distinct primes. So 42. So that would also work. Wait, but z=7: x^2 + y^2 + z^2 -x -y -z = 1 +1 +49 -1 -1 -7=42. 42=2*3*7. So that\\'s another example. So there are multiple solutions for p=2. So p=2 is possible.\\n\\nWait, but the problem says \"Find all prime numbers p\". So if p=2 works, maybe other primes work as well. Let me check p=3.\\n\\nFor p=3, the expression is x\\xc2\\xb3 + y\\xc2\\xb3 + z\\xc2\\xb3 -x - y - z. Again, we need this to be a product of exactly three distinct primes. Let\\'s try small values.\\n\\nStarting with x=1, y=1, z=1: 1 +1 +1 -1 -1 -1=0. Not helpful.\\n\\nx=1, y=1, z=2: 1 +1 +8 -1 -1 -2=6. 6=2*3. Two primes.\\n\\nx=1, y=1, z=3: 1 +1 +27 -1 -1 -3=24. 24=2*2*2*3. Too many duplicates.\\n\\nx=1, y=1, z=4: 1 +1 +64 -1 -1 -4=60. 60=2*2*3*5. Duplicates.\\n\\nx=1, y=1, z=5: 1 +1 +125 -1 -1 -5=120. 120=2*2*2*3*5. Nope.\\n\\nx=1, y=2, z=2: 1 +8 +8 -1 -2 -2=12. 12=2*2*3. Duplicates.\\n\\nx=2, y=2, z=2: 8 +8 +8 -2 -2 -2=18. 18=2*3*3. Duplicates.\\n\\nx=1, y=2, z=3: 1 +8 +27 -1 -2 -3=30. 30=2*3*5. Three distinct primes! So here, x=1, y=2, z=3 for p=3 gives 30, which is 2*3*5. So p=3 is also possible.\\n\\nWait, let me check: 1\\xc2\\xb3 + 2\\xc2\\xb3 + 3\\xc2\\xb3 -1 -2 -3 = 1 +8 +27 -1 -2 -3 = 36 -6 = 30. Yes, correct.\\n\\nSo p=3 also works. Interesting. So both p=2 and p=3 are possible. Let\\'s check p=5.\\n\\nFor p=5, the expression is x\\xe2\\x81\\xb5 + y\\xe2\\x81\\xb5 + z\\xe2\\x81\\xb5 -x - y - z. Let\\'s try small numbers.\\n\\nx=1, y=1, z=1: 1 +1 +1 -1 -1 -1=0. No good.\\n\\nx=1, y=1, z=2: 1 +1 +32 -1 -1 -2=30. 30=2*3*5. Three distinct primes. Wait, so x=1, y=1, z=2: 1\\xe2\\x81\\xb5 +1\\xe2\\x81\\xb5 +2\\xe2\\x81\\xb5 -1 -1 -2=1 +1 +32 -1 -1 -2=30. So that\\'s 2*3*5. Exactly three distinct primes. So p=5 also works.\\n\\nWait, so p=5 also works. Hmm. Maybe p=7?\\n\\nLet me check p=7. x=1, y=1, z=2: 1 +1 +128 -1 -1 -2=126. 126=2*3*3*7. Duplicates. Not good.\\n\\nx=1, y=1, z=3: 1 +1 +2187 -1 -1 -3=2183. Hmm, 2183 is a prime? Wait, 2187 is 3^7, so 2187 -3=2184. Wait, wait, x=1, y=1, z=3: 1^7 +1^7 +3^7 -1 -1 -3=1 +1 +2187 -1 -1 -3=2187 -3=2184. 2184 divided by 2 is 1092. Divided by 2 again is 546. Divided by 2 again is 273. 273=3*91=3*7*13. So 2184=2^3*3*7*13. That\\'s four prime factors, with duplicates. So not three distinct.\\n\\nWhat if x=1, y=2, z=2: 1 +128 +128 -1 -2 -2=252. 252=2^2*3^2*7. Duplicates.\\n\\nx=1, y=2, z=3: 1 +128 +2187 -1 -2 -3=2187 +128 -6=2309. Is 2309 a prime? Let me check. 2309 divided by primes: 2309 \\xc3\\xb72=1154.5. Not integer. \\xc3\\xb73: 2309/3\\xe2\\x89\\x88769.666. \\xc3\\xb75: no. \\xc3\\xb77: 2309/7\\xe2\\x89\\x88329.857. Not integer. \\xc3\\xb711: 2309/11\\xe2\\x89\\x88209.909. Hmm, not sure. Maybe it\\'s prime. If so, then 2309 is a prime, so the expression would be a single prime, which is not a product of three. But actually, wait: x=1, y=2, z=3 for p=7: 1 + 128 + 2187 -1 -2 -3= (1 -1) + (128 -2) + (2187 -3)=0 + 126 + 2184=2310. Wait, I think I miscalculated earlier. Let me recheck.\\n\\nWait: 1^7 is 1, 2^7 is 128, 3^7 is 2187. So sum is 1 +128 +2187=2316. Then subtract x + y + z =1 +2 +3=6. So 2316 -6=2310. 2310. 2310 is a well-known number; it\\'s 2*3*5*7*11. So that\\'s five distinct primes. So 2310 is a product of five distinct primes. Therefore, that\\'s more than three, so it doesn\\'t fit.\\n\\nBut maybe if we take smaller z? Let\\'s try x=1, y=1, z=2: 1 +1 +128 -1 -1 -2=126. Which is 2*3^2*7. Still duplicates.\\n\\nx=1, y=1, z=4: 1 +1 +16384 -1 -1 -4=16384 -4=16380. 16380 factors into 2^2*3^2*5*7*13. Too many primes.\\n\\nHow about x=2, y=2, z=2: 128 +128 +128 -2 -2 -2= 384 -6=378. 378=2*3^3*7. Duplicates.\\n\\nx=1, y=3, z=3: 1 + 2187 +2187 -1 -3 -3=4374 -7=4367. Not sure what that factors into. Let\\'s see: 4367 \\xc3\\xb7 13=335.923. Not integer. \\xc3\\xb77=623.857. Not. \\xc3\\xb711=396.09. No. Maybe prime? If so, then it\\'s a single prime, not a product.\\n\\nAlternatively, maybe I need to look for other combinations. For p=7, is there any x, y, z such that x^7 + y^7 + z^7 -x -y -z is a product of three distinct primes?\\n\\nAlternatively, maybe try x=1, y=1, z=1: 0, as before. x=1, y=1, z=2: 126, which is 2*3^2*7. Not good.\\n\\nx=1, y=2, z=2: 1 +128 +128 -1 -2 -2=252=2^2*3^2*7. Duplicates.\\n\\nx=1, y=2, z=4: 1 +128 +16384 -1 -2 -4=16384 +128 -6=16506. 16506 divided by 2=8253. 8253 divided by 3=2751. Divided by 3 again=917. 917 \\xc3\\xb77=131, which is prime. So 16506=2*3^2*7*131. Four distinct primes, but with duplicates. Not good.\\n\\nHmm, maybe p=7 doesn\\'t work. What about p=5? Wait, earlier with p=5, x=1, y=1, z=2 gives 30=2*3*5. So that works. What about p=5 with other numbers?\\n\\nx=1, y=1, z=3: 1 +1 +243 -1 -1 -3=240. 240=2^4*3*5. Too many factors.\\n\\nx=1, y=2, z=2: 1 +32 +32 -1 -2 -2=60. 60=2^2*3*5. Duplicates.\\n\\nx=2, y=2, z=2:32 +32 +32 -2 -2 -2=96 -6=90. 90=2*3^2*5. Duplicates.\\n\\nx=1, y=2, z=3:1 +32 +243 -1 -2 -3=243 +32 -6=269. 269 is a prime. So that\\'s a single prime, not a product.\\n\\nWait, so p=5 can only give us three distinct primes in some specific cases. We already saw that x=1, y=1, z=2 gives 30=2*3*5. Let me see if there are other combinations for p=5.\\n\\nx=1, y=1, z=4:1 +1 +1024 -1 -1 -4=1020. 1020=2^2*3*5*17. Four primes with duplicates.\\n\\nx=1, y=2, z=4:1 +32 +1024 -1 -2 -4=1050. 1050=2*3*5^2*7. Four primes with duplicates.\\n\\nx=2, y=2, z=3:32 +32 +243 -2 -2 -3=243 +64 -7=300. 300=2^2*3*5^2. Duplicates.\\n\\nx=1, y=3, z=3:1 +243 +243 -1 -3 -3=486 -7=479. 479 is a prime.\\n\\nSo seems like for p=5, the only way to get a product of three distinct primes is when x=1, y=1, z=2, giving 30. Or maybe other combinations?\\n\\nWait, x=1, y=1, z=2 is unique? Let\\'s check x=2, y=1, z=1: same as above, 30. Similarly permutated variables. So the only case for p=5 is when two variables are 1 and the third is 2. So gives 30.\\n\\nSo p=5 works. What about p=7? Doesn\\'t seem to get a case with exactly three primes. Maybe higher primes?\\n\\nWait, let\\'s check p=7 again. Suppose we take x=1, y=1, z=2: as before, 126=2*3^2*7. Which is three primes but with duplicates. Not allowed. If we take x=1, y=1, z=3: 1 +1 +2187 -1 -1 -3=2183. Is 2183 a prime?\\n\\n2183 \\xc3\\xb713=167.923, not integer. Let\\'s check 2183 \\xc3\\xb77=311.857, not. \\xc3\\xb73=727.666, nope. Let\\'s check if 2183 is prime. Let me see. 2183. The square root is around 46.7. So check primes up to 47.\\n\\n2183 \\xc3\\xb75=436.6, not. \\xc3\\xb77=311.857, \\xc3\\xb711=198.454, \\xc3\\xb713=167.923, \\xc3\\xb717=128.411, \\xc3\\xb719=114.894, \\xc3\\xb723=94.913, \\xc3\\xb729=75.275, \\xc3\\xb731=70.419, \\xc3\\xb737=59.0, \\xc3\\xb737=59.0 (exact?), 37*59=2183? 37*50=1850, 37*9=333, total 1850+333=2183. Yes! So 2183=37*59. Both primes. So 2183=37*59. So that\\'s two primes. So 2183 is a product of two primes, so not three. So that doesn\\'t work.\\n\\nAlternatively, x=1, y=2, z=4:1 +128 +16384 -1 -2 -4=16384 +128 -6=16506. As before, which factors into 2*3^2*7*131. Four distinct primes but with duplicates.\\n\\nSo maybe p=7 doesn\\'t work. Let\\'s check p=7 with x=1, y=3, z=3:1 +2187 +2187 -1 -3 -3=4371. 4371. Let\\'s factor this. \\xc3\\xb73: 1457. 1457 \\xc3\\xb713=112.07. \\xc3\\xb77=208.14. \\xc3\\xb711=132.45. Not obvious. Maybe 1457 \\xc3\\xb717=85.7. Maybe prime? If 1457 is prime, then 4371=3*1457. But 1457 \\xc3\\xb731=47.0. 31*47=1457? 31*40=1240, 31*7=217, total 1240+217=1457. Yes! So 1457=31*47. Thus, 4371=3*31*47. Three distinct primes! So 3*31*47. So that\\'s three distinct primes. So p=7, x=1, y=3, z=3 gives 4371=3*31*47. So three distinct primes. Therefore, p=7 also works.\\n\\nWait, let\\'s verify that. x=1, y=3, z=3. For p=7:\\n\\n1^7 +3^7 +3^7 -1 -3 -3 =1 +2187 +2187 -1 -3 -3= (1 -1) + (2187 +2187) - (3 +3)=0 +4374 -6=4368. Wait, wait, that conflicts with earlier calculation. Wait, 1 + 2187 +2187 = 4375. Then subtract 1 +3 +3=7. So 4375 -7=4368. Wait, so 4368. Let me factor 4368.\\n\\nDivide by 2: 2184. Again by 2: 1092. Again by 2: 546. Again by 2: 273. 273=3*91=3*7*13. So 4368=2^4*3*7*13. That\\'s four distinct primes with exponents. So that\\'s 2,3,7,13. So more than three. So my mistake earlier was miscalculating the total.\\n\\nWait, maybe I did the arithmetic wrong. Let\\'s compute again: 1^7=1, 3^7=2187. So 1 +2187 +2187=4375. Then subtract x+y+z=1+3+3=7. So 4375 -7=4368. 4368 factors into 2^4 *3 *7 *13. So four distinct primes. Therefore, not three. So my initial thought was wrong. So that doesn\\'t work.\\n\\nHmm. So maybe p=7 doesn\\'t have a solution. Let me try another combination. x=1, y=2, z=5:1 +128 + 78125 -1 -2 -5=78125 +128 -8=78245. 78245. Let\\'s divide by 5: 78245 \\xc3\\xb75=15649. Check if 15649 is prime. Let\\'s see, sqrt(15649)\\xe2\\x89\\x88125.1. Check primes up to 127. \\xc3\\xb77=2235.571, \\xc3\\xb711=1422.636, \\xc3\\xb713=1203.769, \\xc3\\xb717=920.529, \\xc3\\xb719=823.63, \\xc3\\xb723=680.391, \\xc3\\xb729=539.62, \\xc3\\xb731=504.8, \\xc3\\xb737=422.946, etc. Not obvious. Maybe it\\'s prime. If so, 78245=5*15649. If 15649 is prime, then product of two primes. Not three. So that\\'s not helpful.\\n\\nHow about x=2, y=2, z=3:128 +128 +2187 -2 -2 -3=2187 +256 -7=2436. 2436 divided by 2=1218, \\xc3\\xb72=609, \\xc3\\xb73=203, \\xc3\\xb77=29. So 2436=2^2*3*7*29. Four distinct primes. Not good.\\n\\nx=2, y=3, z=3:128 +2187 +2187 -2 -3 -3=2187*2 +128 -8=4374 +120=4494. 4494 \\xc3\\xb72=2247. \\xc3\\xb73=749. Check if 749 is prime: \\xc3\\xb77=107. So 749=7*107. Thus, 4494=2*3*7*107. Four distinct primes. So again, more than three.\\n\\nHmm. So maybe p=7 doesn\\'t have a solution. Let\\'s check p=5 again. Wait, we had x=1, y=1, z=2 gives 30=2*3*5. What if x=1, y=1, z=3:1 +1 +243 -1 -1 -3=240=2^4*3*5. So duplicates. Not acceptable. So p=5 seems to only have that one case.\\n\\nSimilarly, for p=3, we had x=1, y=2, z=3 gives 30. Are there other cases for p=3?\\n\\nx=1, y=1, z=2:1 +1 +8 -1 -1 -2=6=2*3. Two primes.\\n\\nx=1, y=1, z=4:1 +1 +64 -1 -1 -4=60=2^2*3*5. Duplicates.\\n\\nx=1, y=3, z=3:1 +27 +27 -1 -3 -3=54 -7=47. Prime. Single prime.\\n\\nx=2, y=2, z=2:8 +8 +8 -2 -2 -2=18=2*3*3. Duplicates.\\n\\nx=2, y=2, z=3:8 +8 +27 -2 -2 -3=36=2^2*3^2. Duplicates.\\n\\nx=2, y=3, z=4:8 +27 +64 -2 -3 -4=94=2*47. Two primes.\\n\\nSo for p=3, only the case x=1, y=2, z=3 gives 30=2*3*5. So p=3 works.\\n\\nSo far, p=2,3,5 have solutions. Wait, what about p=7? Maybe there\\'s another combination. Let me think.\\n\\nAlternatively, try x=1, y=2, z=2:1 +128 +128 -1 -2 -2=252=2^2*3^2*7. Not good.\\n\\nx=1, y=4, z=4:1 +16384 +16384 -1 -4 -4=32768 -9=32759. Is 32759 a prime? Not sure. Probably difficult to factor, but even if it\\'s prime, that\\'s only one prime. Not helpful.\\n\\nAlternatively, x=3, y=3, z=3:2187*3 -9=6561 -9=6552. 6552=8*819=8*3*273=8*3*3*91=8*3*3*7*13. So 2^3*3^2*7*13. Four distinct primes with duplicates.\\n\\nSo maybe p=7 doesn\\'t work. Let\\'s check p=11. Maybe even higher primes, but likely numbers get too big. Let\\'s try p=11 with x=1, y=1, z=2:1 +1 +2048 -1 -1 -2=2046. 2046=2*3*11*31. Four distinct primes. So more than three. Not good.\\n\\nx=1, y=1, z=3:1 +1 +177147 -1 -1 -3=177143. Not sure, but likely a prime or a product of few primes. 177143 \\xc3\\xb73=59047.666, \\xc3\\xb77=25306.142, \\xc3\\xb711=16103.909. Maybe prime. If so, product of one prime. Not helpful.\\n\\nSo seems like higher primes either give numbers with more than three prime factors or duplicates.\\n\\nWait, let\\'s check p=7 with x=2, y=2, z=2:128*3 -6=384-6=378=2*3^3*7. Duplicates.\\n\\nAlternatively, p=7 and x=1, y=2, z=7:1 +128 +823543 -1 -2 -7=823543 +128 -10=823661. That\\'s a huge number. Probably not factoring into three primes.\\n\\nAlternatively, maybe some other combination? It\\'s getting too time-consuming. Let\\'s recap:\\n\\nFor p=2: possible, example x=1, y=1, z=6 gives 30=2*3*5.\\n\\nFor p=3: possible, example x=1, y=2, z=3 gives 30=2*3*5.\\n\\nFor p=5: possible, example x=1, y=1, z=2 gives 30=2*3*5.\\n\\nFor p=7: Tried several combinations, but couldn\\'t find any that result in exactly three distinct primes. Seems like p=7 doesn\\'t work.\\n\\nWait, but for p=5, if we take x=1, y=1, z=2, we get 30=2*3*5. Similarly for p=2 and p=3. So maybe all primes p can do this by taking x=1, y=1, z=2 or something similar. Wait, let\\'s check for p=7: x=1, y=1, z=2 gives 1 +1 +128 -1 -1 -2=126=2*3^2*7. That\\'s three primes but with duplicates. So not acceptable.\\n\\nBut for other primes, like p=7, if we take x=1, y=1, z=k, what do we get? 1 +1 +k^7 -1 -1 -k =k^7 -k. So k^7 -k. We need k^7 -k to be a product of three distinct primes.\\n\\nk=2:128 -2=126=2*3^2*7.\\n\\nk=3:2187 -3=2184=2^3*3*7*13.\\n\\nk=4:16384 -4=16380=2^2*3^2*5*7*13.\\n\\nSo seems like as k increases, the number of prime factors increases. So likely, for p\\xe2\\x89\\xa52, if we take x=1, y=1, z=k, then the expression is k^p -k. So we can factor this as k(k^{p-1} -1). For k=2: 2(2^{p-1} -1). For this to be a product of three distinct primes, 2 must be one prime, and (2^{p-1} -1) must be a product of two distinct primes.\\n\\nSo for example, when p=2: 2^{1} -1=1. So 2*1=2. Not three primes.\\n\\nWait, but earlier we saw for p=2, taking x=1, y=1, z=6 gives 6\\xc2\\xb2 -6=30=2*3*5. So in that case, z=6: 6\\xc2\\xb2 -6=30. So 6*5=30. So factorization is 6*5=2*3*5.\\n\\nSo generalizing, if we take x=1, y=1, z=k, then the expression is k^p -k. For this to be a product of three distinct primes, k^p -k must equal q*r*s where q, r, s are distinct primes.\\n\\nAlternatively, if k is composite, k^p -k may factor into more components. For example, k=6, p=2: 6^2 -6=30=2*3*5. So three primes. For k=6 and p=2: 6=2*3, so k^p -k=6*5=2*3*5. So this works.\\n\\nSimilarly, for other primes p, can we find k such that k^p -k is the product of three distinct primes?\\n\\nFor p=3: Let\\'s take k=2: 8 -2=6=2*3. Two primes.\\n\\nk=3:27 -3=24=2^3*3. Duplicates.\\n\\nk=4:64 -4=60=2^2*3*5. Duplicates.\\n\\nk=5:125 -5=120=2^3*3*5. Duplicates.\\n\\nk=6:216 -6=210=2*3*5*7. Four primes. Too many.\\n\\nk=7:343 -7=336=2^4*3*7. Duplicates.\\n\\nk=5: already checked.\\n\\nWait, but for p=3, we had a different solution: x=1, y=2, z=3. Let\\'s see what that gives:1 +8 +27 -1 -2 -3=30=2*3*5. So this is 30. But how does this relate to k? It\\'s not of the form k^p -k. Instead, it\\'s x^3 + y^3 + z^3 -x -y -z. So not the same as the previous case. So there might be multiple ways to achieve the product of three primes for different primes p.\\n\\nSimilarly, for p=5, we had x=1, y=1, z=2:1 +1 +32 -1 -1 -2=30=2*3*5.\\n\\nSo in this case, k=2: 2^5 -2=30. Which factors into 2*3*5. So here, k=2 for p=5 gives a product of three primes.\\n\\nSimilarly, if we can find for a prime p, a k such that k^p -k is a product of three distinct primes, then p is a possible solution.\\n\\nSo let\\'s formalize this. For prime p, if there exists a positive integer k such that k^p -k is a product of three distinct primes, then p is a solution. Additionally, there might be solutions where not all variables are 1 except one, but those might be harder to find.\\n\\nSo let\\'s analyze k^p -k.\\n\\nFactor k^p -k =k(k^{p-1} -1). For this to be a product of three distinct primes, k must be either:\\n\\n1) k is a prime, and (k^{p-1} -1) is a product of two distinct primes.\\n\\nOr\\n\\n2) k is a product of two primes, and (k^{p-1} -1) is a prime.\\n\\nBut since k is a positive integer greater than 1 (as k=1 gives 1 -1=0). Let\\'s check case 1: k is prime.\\n\\nCase 1: k is prime. Then k^p -k =k*(k^{p-1} -1). We need this to be a product of three distinct primes. So since k is prime, k^{p-1} -1 must be a product of two distinct primes. Note that k and the primes in k^{p-1} -1 must be distinct.\\n\\nFor example, take p=5, k=2:2^5 -2=30=2*3*5. Here, k=2, and k^{p-1}-1=2^{4} -1=16-1=15=3*5. So 3 and 5. So 2*15=2*3*5. So yes, three distinct primes. Similarly, for p=3 and k=2:2^3 -2=6=2*3. Only two primes. Doesn\\'t work, but earlier solution for p=3 was different.\\n\\nWait, but in p=3, the solution was not of the form k^p -k. So maybe the k^p -k approach is only one way, but other combinations can also give solutions. However, for primes p=2,3,5, we can find solutions either through k^p -k or other variable combinations. But for higher primes, like p=7, even the k^p -k approach doesn\\'t yield a product of three primes.\\n\\nLet me check for p=7, k=2:2^7 -2=128 -2=126=2*3^2*7. Three primes but with duplicates. So not allowed.\\n\\nk=3:3^7 -3=2187 -3=2184=2^3*3*7*13. Four primes.\\n\\nk=5:5^7 -5=78125 -5=78120=2^3*3*5*7*11*17. Way too many primes.\\n\\nk=7:7^7 -7=823543 -7=823536=2^4*3^3*7^4. Lots of duplicates.\\n\\nk=4:4^7 -4=16384 -4=16380=2^2*3^2*5*7*13. Five primes.\\n\\nk=6:6^7 -6=279936 -6=279930=2*3*5*7*31*43. Six primes.\\n\\nSo even for p=7, k from 2 to 7 don\\'t give products of three distinct primes. So perhaps p=7 is not possible. Similarly, higher primes would lead to even larger numbers with more prime factors.\\n\\nTherefore, the possible primes p are 2,3,5. Let me confirm for p=5:\\n\\nWe saw that k=2 gives 30=2*3*5, which works. Also, maybe other k for p=5? For example, k=3:3^5 -3=243 -3=240=2^4*3*5. Duplicates. k=4:4^5 -4=1024 -4=1020=2^2*3*5*17. Four primes. So only k=2 works for p=5.\\n\\nSimilarly, for p=3, k=2 gives 6=2*3. Only two primes, but we had another solution with x=1, y=2, z=3, which gives 30. So even if k^p -k doesn\\'t work, other combinations might. But in this case, even for p=3, we needed a different combination.\\n\\nSimilarly, for p=2, the solution came from k=6:6\\xc2\\xb2 -6=30. So 6 is not prime. But 6=2*3, so k=6, a composite, gives k^2 -k=30. Let\\'s analyze this case.\\n\\nFor composite k, k=ab, then k^p -k=ab(ab)^{p-1} -ab=ab[(ab)^{p-1} -1]. For example, k=6=2*3, p=2:6\\xc2\\xb2 -6=30=2*3*5. Here, 6=2*3, and [(6)^{1} -1]=5. So 6*5=2*3*5. So in this case, for composite k=6 and p=2, we get three distinct primes.\\n\\nSimilarly, maybe other composite k and primes p.\\n\\nSo perhaps the possible primes p are those where there exists either a prime k or composite k such that k^p -k factors into three distinct primes.\\n\\nFor p=2, k=6 gives 30=2*3*5.\\n\\nFor p=3, x=1, y=2, z=3 gives 30=2*3*5.\\n\\nFor p=5, k=2 gives 30=2*3*5.\\n\\nInterestingly, all three cases result in 30. Maybe 30 is the magic number here.\\n\\nIs there another number which is the product of three distinct primes that can be formed for other primes p?\\n\\nAlternatively, maybe all primes p can form 30 in some way.\\n\\nWait, let\\'s check:\\n\\nFor p=2: x=1, y=1, z=6:1 +1 +36 -1 -1 -6=30.\\n\\nFor p=3: x=1, y=2, z=3:1 +8 +27 -1 -2 -3=30.\\n\\nFor p=5: x=1, y=1, z=2:1 +1 +32 -1 -1 -2=30.\\n\\nSo for p=2,3,5, we can get 30, which is 2*3*5. For p=7, can we get 30? Let\\'s try:\\n\\nx^7 + y^7 + z^7 -x -y -z=30.\\n\\nBut 30 +x + y + z =x^7 + y^7 + z^7. Let\\'s see if there are small x,y,z such that x^7 + y^7 + z^7=30 +x + y + z.\\n\\nThe smallest possible values:\\n\\nx=1:1 + y^7 + z^7=30 +1 + y + z \\xe2\\x86\\x92 y^7 + z^7=29 + y + z.\\n\\nTry y=1:1 + z^7=29 +1 + z \\xe2\\x86\\x92z^7 -z=28.\\n\\nz^7 -z=28. For z=2:128 -2=126\\xe2\\x89\\xa028. z=1:0. z=3:2187 -3=2184\\xe2\\x89\\xa028. So no solution.\\n\\ny=2:128 + z^7=29 +2 + z \\xe2\\x86\\x92z^7= -97 +z. Not possible since z\\xe2\\x89\\xa51.\\n\\nSo no solutions for p=7. Hence, p=7 cannot give 30, and other numbers also don\\'t seem to work.\\n\\nTherefore, the only primes p for which there exist x,y,z such that x^p + y^p + z^p -x -y -z is a product of three distinct primes are p=2,3,5.\\n\\nTo confirm that there are no others, we need to ensure that for p\\xe2\\x89\\xa57, no possible x,y,z can make the expression a product of three distinct primes. Given that for p=7, attempts to find such x,y,z have failed and numbers tend to get too large with multiple prime factors, it\\'s reasonable to conjecture that p=2,3,5 are the only solutions.\\n\\n**Final Answer**\\nThe primes are \\\\boxed{2}, \\\\boxed{3}, and \\\\boxed{5}.\\n\\n\\nTo find all prime numbers \\\\( p \\\\) for which there exist positive integers \\\\( x, y, \\\\) and \\\\( z \\\\) such that the expression \\\\( x^p + y^p + z^p - x - y - z \\\\) is a product of exactly three distinct prime numbers, we analyze different values of \\\\( p \\\\).\\n\\n1. **Case \\\\( p = 2 \\\\)**:\\n - The expression becomes \\\\( x^2 + y^2 + z^2 - x - y - z \\\\).\\n - For \\\\( x = 1 \\\\), \\\\( y = 1 \\\\), \\\\( z = 6 \\\\), we get:\\n \\\\[\\n 1^2 + 1^2 + 6^2 - 1 - 1 - 6 = 1 + 1 + 36 - 1 - 1 - 6 = 30\\n \\\\]\\n - \\\\( 30 = 2 \\\\times 3 \\\\times 5 \\\\), which is a product of three distinct primes. Thus, \\\\( p = 2 \\\\) works.\\n\\n2. **Case \\\\( p = 3 \\\\)**:\\n - The expression becomes \\\\( x^3 + y^3 + z^3 - x - y - z \\\\).\\n - For \\\\( x = 1 \\\\), \\\\( y = 2 \\\\), \\\\( z = 3 \\\\), we get:\\n \\\\[\\n 1^3 + 2^3 + 3^3 - 1 - 2 - 3 = 1 + 8 + 27 - 1 - 2 - 3 = 30\\n \\\\]\\n - \\\\( 30 = 2 \\\\times 3 \\\\times 5 \\\\), which is a product of three distinct primes. Thus, \\\\( p = 3 \\\\) works.\\n\\n3. **Case \\\\( p = 5 \\\\)**:\\n - The expression becomes \\\\( x^5 + y^5 + z^5 - x - y - z \\\\).\\n - For \\\\( x = 1 \\\\), \\\\( y = 1 \\\\), \\\\( z = 2 \\\\), we get:\\n \\\\[\\n 1^5 + 1^5 + 2^5 - 1 - 1 - 2 = 1 + 1 + 32 - 1 - 1 - 2 = 30\\n \\\\]\\n - \\\\( 30 = 2 \\\\times 3 \\\\times 5 \\\\), which is a product of three distinct primes. Thus, \\\\( p = 5 \\\\) works.\\n\\n4. **Case \\\\( p \\\\geq 7 \\\\)**:\\n - Testing various combinations of \\\\( x, y, z \\\\) for primes \\\\( p = 7 \\\\) and higher results in numbers with more than three prime factors or duplicates. For example, \\\\( p = 7 \\\\) with \\\\( x = 1 \\\\), \\\\( y = 1 \\\\), \\\\( z = 2 \\\\) gives \\\\( 126 = 2 \\\\times 3^2 \\\\times 7 \\\\), which has duplicate primes. No valid combinations were found for higher primes.\\n\\nThus, the primes \\\\( p \\\\) for which the expression can be a product of exactly three distinct prime numbers are \\\\( \\\\boxed{2} \\\\), \\\\( \\\\boxed{3} \\\\), and \\\\( \\\\boxed{5} \\\\).', b'\\nOkay, so I need to find all prime numbers p such that there exist positive integers x, y, z where the expression x^p + y^p + z^p - x - y - z is a product of exactly three distinct prime numbers. Hmm, let\\'s break this down step by step.\\n\\nFirst, since p is a prime number, possible values are 2, 3, 5, 7, etc. The problem is asking for primes p where such x, y, z exist. So maybe I can check for small primes first and see if there\\'s a pattern or a possible restriction.\\n\\nLet me start with p=2. Then the expression becomes x\\xc2\\xb2 + y\\xc2\\xb2 + z\\xc2\\xb2 - x - y - z. I need to find x, y, z positive integers such that this expression is the product of three distinct primes. Let\\'s see if this is possible.\\n\\nMaybe try small values for x, y, z. Let\\'s take x, y, z as 1, 1, 1. Then the expression would be 1 + 1 + 1 - 1 - 1 - 1 = 0. That\\'s not helpful. How about x=2, y=2, z=2? Then it\\'s 4 + 4 + 4 - 2 - 2 - 2 = 12 - 6 = 6. 6 factors into 2*3, which is two primes, not three. So that\\'s not enough.\\n\\nWhat if we mix the numbers? Let\\'s try x=2, y=3, z=4. Then compute 4 + 9 + 16 - 2 - 3 - 4 = 29 - 9 = 20. 20 is 2\\xc2\\xb2*5, which is not three distinct primes. Hmm.\\n\\nWait, maybe x=1, y=2, z=3. Then 1 + 4 + 9 - 1 - 2 - 3 = 14 - 6 = 8. 8 is 2\\xc2\\xb3, not three primes. Not good.\\n\\nHow about x=3, y=3, z=3? 9 + 9 + 9 - 3 - 3 - 3 = 27 - 9 = 18. 18 is 2*3\\xc2\\xb2, again only two distinct primes.\\n\\nWait, maybe we need different values where the result is a product of three distinct primes. Let\\'s think. Let me try x=2, y=3, z=5. Then 4 + 9 + 25 - 2 - 3 - 5 = 38 - 10 = 28. 28 is 2\\xc2\\xb2*7. Not three distinct primes.\\n\\nWait, maybe x=1, y=1, z=2. Then 1 + 1 + 4 - 1 - 1 - 2 = 6 - 4 = 2. 2 is a prime, but we need three distinct primes multiplied. Not enough.\\n\\nHmm. Maybe larger numbers. Let\\'s try x=3, y=4, z=5. Then 9 + 16 + 25 - 3 - 4 - 5 = 50 - 12 = 38. 38 is 2*19. Still two primes.\\n\\nWhat if x=4, y=5, z=6. Then 16 + 25 + 36 - 4 - 5 - 6 = 77 - 15 = 62. 62 is 2*31. Again two primes.\\n\\nWait, maybe if the result is 30. 30 is 2*3*5, three distinct primes. Let\\'s see if we can get 30. So, x\\xc2\\xb2 + y\\xc2\\xb2 + z\\xc2\\xb2 - x - y - z = 30. Let\\'s solve this equation.\\n\\nSo x\\xc2\\xb2 - x + y\\xc2\\xb2 - y + z\\xc2\\xb2 - z = 30. Each term x\\xc2\\xb2 - x is equal to x(x - 1). Since x is a positive integer, x(x - 1) is at least 0. For x=1, x\\xc2\\xb2 -x = 0. For x=2, it\\'s 2. x=3, 6. x=4, 12. x=5, 20. x=6, 30. Hmm, so maybe if x, y, z are small, their contributions could add up to 30.\\n\\nLet me try combinations. Let\\'s see if x=3, y=3, z=4. Then x\\xc2\\xb2 -x = 6, y\\xc2\\xb2 - y=6, z\\xc2\\xb2 - z=12. Total 6+6+12=24. Not enough.\\n\\nx=4, y=4, z=4: 12+12+12=36. That\\'s over 30. Maybe x=4, y=3, z=3: 12+6+6=24. Still no. x=5, y=2, z=2: 20 + 2 + 2=24. Hmm.\\n\\nHow about x=5, y=3, z=2: 20 + 6 + 2=28. Still 2 short. Then x=5, y=4, z=2: 20 + 12 + 2=34. Too much.\\n\\nWait, perhaps x=4, y=5, z=3: 12 + 20 + 6=38. No. Maybe x=3, y=4, z=5: 6 + 12 + 20=38. Still 38. Hmm.\\n\\nAlternatively, maybe x=6, y=2, z=2: 30 + 2 + 2=34. Still over. Hmm.\\n\\nWait, maybe x=2, y=5, z=5: 2 + 20 + 20=42. No.\\n\\nAlternatively, perhaps the individual terms can be different. For example, x=4 (12), y=3 (6), z=5 (20). Sum 12+6+20=38. Not 30.\\n\\nAlternatively, x=5 (20), y=2 (2), z=3 (6). 20+2+6=28. Close. Maybe x=5, y=3, z=3: 20 + 6 + 6=32. Hmm.\\n\\nWait, is there a combination where x\\xc2\\xb2 -x + y\\xc2\\xb2 - y + z\\xc2\\xb2 - z =30? Maybe x=5, y=2, z=2: 20 + 2 + 2=24. x=5, y=3, z=1: 20 +6 +0=26. x=5, y=4, z=1:20 +12 +0=32. x=5, y=5, z=1:20 +20 +0=40. x=4, y=4, z=3:12+12+6=30. Wait, x=4, y=4, z=3. Then x=4, y=4, z=3. Let\\'s check: 4\\xc2\\xb2 +4\\xc2\\xb2 +3\\xc2\\xb2 -4 -4 -3 = 16+16+9 -11= 41 -11=30. Yes! So 30 is 2*3*5, three distinct primes. So for p=2, there exists x=4, y=4, z=3 such that the expression is 30, which is a product of three distinct primes. Therefore, p=2 is a possible candidate.\\n\\nOkay, so p=2 works. Now check if there are other primes. Let\\'s try p=3. The expression becomes x\\xc2\\xb3 + y\\xc2\\xb3 + z\\xc2\\xb3 - x - y - z. We need this to be a product of three distinct primes.\\n\\nAgain, maybe try small x, y, z. Let\\'s start with x=1, y=1, z=1: 1 +1 +1 -1 -1 -1=0. Not good. x=2, y=2, z=2: 8 +8 +8 -2 -2 -2=24 -6=18=2*3\\xc2\\xb2. Not three distinct primes.\\n\\nx=1, y=1, z=2: 1 +1 +8 -1 -1 -2=6. 6=2*3. Two primes.\\n\\nx=1, y=2, z=3: 1 +8 +27 -1 -2 -3=36 -6=30=2*3*5. Hey, 30 is three distinct primes. Wait, so if x=1, y=2, z=3 with p=3, the expression is 1\\xc2\\xb3 +2\\xc2\\xb3 +3\\xc2\\xb3 -1 -2 -3=1 +8 +27 -6=30=2*3*5. So that\\'s a product of exactly three distinct primes. Therefore, p=3 is also possible.\\n\\nWait, so p=3 works. Let me verify that again. 1\\xc2\\xb3 +2\\xc2\\xb3 +3\\xc2\\xb3 =1 +8 +27=36. Subtract x+y+z=1+2+3=6. 36-6=30. Yep, that\\'s correct. 30 factors into 2*3*5. So p=3 works.\\n\\nNow, check p=5. Let\\'s see. The expression is x\\xe2\\x81\\xb5 + y\\xe2\\x81\\xb5 + z\\xe2\\x81\\xb5 -x -y -z. Need this to be a product of three distinct primes. Let\\'s try small x, y, z. Let\\'s try x=1, y=1, z=1: 1+1+1 -1-1-1=0. Not good. x=1, y=1, z=2:1 +1 +32 -1 -1 -2=32 -4=28=2\\xc2\\xb2*7. Not three primes.\\n\\nx=1, y=2, z=2:1 +32 +32 -1 -2 -2=65 -5=60=2\\xc2\\xb2*3*5. That\\'s four prime factors, but with repetition, so not three distinct primes.\\n\\nx=1, y=2, z=3:1 +32 +243 -1 -2 -3=276 -6=270. 270=2*3\\xc2\\xb3*5. Again multiple factors. Not three distinct.\\n\\nx=2, y=2, z=2:32*3 -6=96-6=90=2*3\\xc2\\xb2*5. Not three distinct.\\n\\nHow about x=1, y=2, z=4:1 +32 +1024 -1 -2 -4=1057 -7=1050. 1050=2*3*5\\xc2\\xb2*7. That\\'s four distinct primes with repetition. Not three.\\n\\nx=2, y=3, z=4:32 +243 +1024 -2 -3 -4=1299 -9=1290. 1290=2*3*5*43. Four distinct primes. So that\\'s four, not three.\\n\\nHmm. Let\\'s try x=1, y=1, z=3:1 +1 +243 -1 -1 -3=244 -5=239. 239 is a prime. So that\\'s a single prime, not a product of three.\\n\\nx=1, y=3, z=3:1 +243 +243 -1 -3 -3=487 -7=480. 480=2^5*3*5. Three distinct primes but with exponents higher than 1. But the problem says \"a product of exactly three distinct prime numbers\", so exponents don\\'t matter as long as each prime is present once. Wait, no. Wait, if it\\'s a product of exactly three distinct primes, then it should be equal to p1*p2*p3 where p1, p2, p3 are distinct primes. So exponents all must be 1. So 480 is not acceptable because it\\'s 2^5*3*5. So even though it has three distinct primes, the exponents are higher. Therefore, 480 is out.\\n\\nSo maybe p=5 is not possible. Let me try x=2, y=3, z=5. 32 +243 +3125 -2 -3 -5=3400 -10=3390. 3390 factors into 2*3*5*113. So four distinct primes. Hmm.\\n\\nAlternatively, x=1, y=1, z=5:1 +1 +3125 -1 -1 -5=3126 -7=3119. 3119 is a prime? Let me check. 3119 divided by 2: no. 3: 3+1+1+9=14, not divisible by 3. 5: ends with 9, no. 7: 7*445=3115, 3119-3115=4. Not divisible. 11: 11*283=3113, 3119-3113=6. Not. 13: 13*239=3107, 3119-3107=12. Not divisible. 17: 17*183=3111, 3119-3111=8. Not. 19: 19*164=3116, 3119-3116=3. Not. 23: 23*135=3105, 3119-3105=14. Not divisible. So 3119 might be a prime. Therefore, that\\'s a single prime, which isn\\'t the product of three.\\n\\nAlternatively, try x=1, y=3, z=2:1 +243 +32 -1 -3 -2=276 -6=270. As before, 2*3^3*5. Not three distinct.\\n\\nWait, maybe x=2, y=2, z=3:32 +32 +243 -2 -2 -3=307 -7=300. 300=2\\xc2\\xb2*3*5\\xc2\\xb2. Again multiple exponents. So even though three distinct primes, the exponents are higher than one, so not a product of exactly three distinct primes. The problem specifies \"a product of exactly three distinct prime numbers\", which I think means that the total is a semiprime with three distinct primes, each to the first power. So exponents must be 1.\\n\\nSo for p=5, seems challenging to get such a product. Maybe possible with larger numbers? Let me try x=2, y=2, z=5:32 +32 +3125 -2 -2 -5=3189 -9=3180. 3180=2\\xc2\\xb2*3*5*53. Four distinct primes. Not good.\\n\\nx=3, y=3, z=3:243*3 -9=729 -9=720=2^4*3\\xc2\\xb2*5. Nope.\\n\\nx=2, y=3, z=3:32 +243 +243 -2 -3 -3=518 -8=510=2*3*5*17. Four distinct primes.\\n\\nHmm. Maybe it\\'s not possible for p=5. Let me check another way. Let\\'s suppose that x, y, z are small. For example, x=1, y=2, z=5. Then 1 +32 +3125 -1 -2 -5=3126 -8=3118. 3118=2*1559. 1559 is a prime. So 2*1559, which is two primes. Not three.\\n\\nAlternatively, x=1, y=4, z=5:1 +1024 +3125 -1 -4 -5=4150 -10=4140. 4140=2\\xc2\\xb2*3\\xc2\\xb2*5*23. Four distinct primes. Hmm.\\n\\nMaybe trying x=2, y=4, z=5:32 +1024 +3125 -2 -4 -5=4181 -11=4170. 4170=2*3*5*139. Four distinct primes.\\n\\nAlternatively, x=3, y=4, z=5:243 + 1024 + 3125 -3 -4 -5=4392 -12=4380. 4380=2\\xc2\\xb2*3*5*73. Again four distinct primes.\\n\\nThis seems tough. Maybe p=5 is not possible. Let me check if there\\'s any possible combination. Alternatively, perhaps x, y, z are larger. For example, x=2, y=1, z=1:32 +1 +1 -2 -1 -1=34 -4=30. Wait, 30 is 2*3*5. So x=2, y=1, z=1, p=5. Let\\'s compute x\\xe2\\x81\\xb5 + y\\xe2\\x81\\xb5 + z\\xe2\\x81\\xb5 -x -y -z. 32 +1 +1 -2 -1 -1=32+1+1=34; 34-4=30. Yes! 30 is 2*3*5. So that\\'s three distinct primes. Therefore, p=5 works.\\n\\nWait, really? Let me check again. x=2, y=1, z=1. Then 2\\xe2\\x81\\xb5 +1\\xe2\\x81\\xb5 +1\\xe2\\x81\\xb5 -2 -1 -1=32 +1 +1 -2 -1 -1=34 -4=30. Yes! 30 is indeed 2*3*5. So p=5 works as well. So, that\\'s surprising. So for p=5, x=2, y=1, z=1 gives 30. So p=5 is possible.\\n\\nHmm. So then p=5 is possible. Then maybe p=7? Let\\'s check. The expression is x\\xe2\\x81\\xb7 + y\\xe2\\x81\\xb7 + z\\xe2\\x81\\xb7 -x -y -z. Let\\'s try small x, y, z.\\n\\nTake x=1, y=1, z=1:1 +1 +1 -1 -1 -1=0. Not good. x=2, y=1, z=1:128 +1 +1 -2 -1 -1=130 -4=126. 126=2*3\\xc2\\xb2*7. Not three distinct primes.\\n\\nx=2, y=2, z=1:128 +128 +1 -2 -2 -1=257 -5=252. 252=2\\xc2\\xb2*3\\xc2\\xb2*7. Not three distinct.\\n\\nx=2, y=1, z=2: same as above. 128 +1 +128 -2 -1 -2=257 -5=252.\\n\\nx=1, y=2, z=3:1 +128 +2187 -1 -2 -3=2316 -6=2310. 2310=2*3*5*7*11. That\\'s five distinct primes. Not three.\\n\\nx=1, y=1, z=2:1 +1 +128 -1 -1 -2=128 -3=125=5\\xc2\\xb3. Not three.\\n\\nx=1, y=1, z=3:1 +1 +2187 -1 -1 -3=2187 -4=2183. Let\\'s check if 2183 is a prime. Divided by 3: 2+1+8+3=14, not divisible. Divided by 7: 7*311=2177, 2183-2177=6. Not. 11: 11*198=2178, 2183-2178=5. Not. 13: 13*167=2171, 2183-2171=12, divisible by 13? 12 isn\\'t. 17: 17*128=2176, 2183-2176=7. Not. 19: 19*114=2166, 2183-2166=17. Not. 23: 23*94=2162, 2183-2162=21. Not. 29: 29*75=2175, 2183-2175=8. Not. 31: 31*70=2170, 2183-2170=13. Not. So 2183 might be prime. If so, then it\\'s a single prime, not three.\\n\\nAlternatively, x=2, y=3, z=4:128 + 2187 + 16384 -2 -3 -4=187, 128+2187=2315; 2315+16384=187, 2315+16384=187, wait, 128+2187=2315, 2315+16384=187, that can\\'t be. Wait, 128+2187=2315, 2315+16384=187, that\\'s incorrect. Wait, 128+2187=2315, then 2315+16384=2315+16384=187, no. 2315+16384=187, that\\'s not possible. Wait, 16384 + 2315=187, no, 16384 + 2000=18384, no. Wait, 16384 + 2315. Let\\'s compute 16384 + 2000=18384, then +315=187, 18384+315=187, no, that\\'s 187, but 16384 + 2315=187, which is impossible. Wait, my mistake. 16384 + 2315=16384 + 2000=18384, 18384 + 315=187, 18384 +315=187, no. 18384 +315=187, 18384+315=187, that\\'s incorrect. Wait, 16384 + 2315 is 16384 + 2000=18384, then 18384 + 315=187, no. Wait, 16384 + 2315. Let\\'s compute 16384 + 2000=18384, then add 315: 18384 + 315=187, no. Wait, 16384 +2315: 16000 +2300=18300, 384 +15=399, so total 18300 +399=18699. Then subtract 2+3+4=9. So total 18699 -9=18690. 18690=1869*10=10*1869. 1869 divided by 3: 1+8+6+9=24, divisible by 3. 1869/3=623. 623: 6+2+3=11, not divisible by 3. Divided by 7: 7*89=623. 89 is prime. So 18690=2*3*5*7*89. Five distinct primes. Not three.\\n\\nAlternatively, x=2, y=1, z=1:128 +1 +1 -2 -1 -1=130 -4=126=2*3\\xc2\\xb2*7. Not three distinct primes.\\n\\nWait, but maybe x=3, y=1, z=1:3\\xe2\\x81\\xb7=2187. So 2187 +1 +1 -3 -1 -1=2189 -5=2184. 2184=2\\xc2\\xb3*3*7*13. Four distinct primes.\\n\\nHmm. Maybe x=1, y=2, z=2:1 +128 +128 -1 -2 -2=257 -5=252=2\\xc2\\xb2*3\\xc2\\xb2*7. Not three.\\n\\nWait, what about x=1, y=1, z=4:1 +1 +16384 -1 -1 -4=16385 -6=16379. Check if 16379 is a prime. Divided by 2: no. 3:1+6+3+7+9=26, not divisible. 5: ends with 9. 7: 7*2340=16380, 16379-16380=-1. Not divisible. 11: 11*1489=16379? 11*1400=15400, 11*89=979, total 15400+979=16379. So 11*1489=16379. Is 1489 a prime? Let\\'s check. 1489 divided by 13: 13*114=1482, 1489-1482=7. Not. 17: 17*87=1479, 1489-1479=10. Not. 19: 19*78=1482, 1489-1482=7. Not. 23: 23*64=1472, 1489-1472=17. Not. So 1489 is prime. Therefore, 16379=11*1489, two primes. Not three.\\n\\nAlternatively, x=1, y=3, z=1:1 +2187 +1 -1 -3 -1=2189 -5=2184 as before.\\n\\nAlternatively, x=2, y=2, z=2:3*128 -6=384 -6=378=2*3\\xc2\\xb3*7. Not three distinct.\\n\\nHmm. Maybe p=7 is not possible. Let\\'s try another approach. Let\\'s see if we can get 30 again, but with p=7. 30=2*3*5. So can x\\xe2\\x81\\xb7 + y\\xe2\\x81\\xb7 + z\\xe2\\x81\\xb7 -x -y -z=30? Let\\'s see. Since x, y, z are positive integers. Let\\'s see:\\n\\nIf x=2, y=1, z=1:128 +1 +1 -2 -1 -1=126. 126 is larger than 30. x=1, y=1, z=2: same. 126.\\n\\nx=1, y=1, z=1:0. Not. x=1, y=1, z=3:2187 +1 +1 -1 -1 -3=2185. Way larger.\\n\\nx=1, y=2, z=1:1 +128 +1 -1 -2 -1=126. So 126. So even the minimal values give 126. So can\\'t get 30. So seems like p=7 cannot produce 30. So maybe p=7 is not possible.\\n\\nAlternatively, let\\'s see if we can get another number that\\'s a product of three distinct primes. Let\\'s try for p=7, x=2, y=3, z=1:128 +2187 +1 -2 -3 -1=2316 -6=2310=2*3*5*7*11. Five primes. Not good. x=2, y=3, z=4: as before, 18690=2*3*5*7*89. Five primes.\\n\\nx=3, y=3, z=3:3*2187 -9=6561 -9=6552. 6552 factors into... let\\'s see: 6552 divided by 8=819. 819=3*273=3*3*91=3*3*7*13. So 6552=8*3*3*7*13=2\\xc2\\xb3*3\\xc2\\xb2*7*13. Four distinct primes.\\n\\nx=2, y=4, z=5:128 + 16384 + 78125 -2 -4 -5=128+16384=16512; 16512+78125=94637; 94637 -11=94626. That\\'s a big number. Let\\'s factorize: 94626 divided by 2=47313. 47313 divided by 3=15771. Divided by 3=5257. Check if 5257 is prime. 5+2+5+7=19, not divisible by 3. Divided by 5: ends with 7. 7: 7*751=5257. 7*700=4900, 7*51=357, total 4900+357=5257. So 5257=7*751. 751 is prime. So 94626=2*3\\xc2\\xb2*7*751. Four distinct primes. Not good.\\n\\nHmm. So for p=7, seems challenging. Let me check x=1, y=2, z=5. 1 +128 + 78125 -1 -2 -5=78254 -8=78246. 78246 divided by 2=39123. Divided by 3=13041. Divided by 3=4347. Divided by 3=1449. Divided by 3=483. Divided by 3=161. Which is 7*23. So 78246=2*3^5*7*23. Multiple primes. Not three.\\n\\nAlternatively, x=1, y=3, z=2:1 +2187 +128 -1 -3 -2=2316 -6=2310 as before.\\n\\nSo seems like for p=7, any small x, y, z lead to either too many prime factors or primes with exponents. Maybe p=7 is impossible. Let me check if there\\'s a larger x, y, z combination that could result in three distinct primes. But this is getting cumbersome. Let\\'s think differently.\\n\\nNote that for p=2,3,5, we found solutions. For p=7, it\\'s unclear. Let\\'s check if there\\'s a mathematical reason why higher primes might not work. For example, perhaps for primes p >=7, x^p grows too quickly, so the expression x^p + y^p + z^p -x - y - z is too large to be a product of three small primes. But even so, the product of three primes can be large if the primes are large. However, the problem requires the product to be exactly three distinct primes, not necessarily small. However, as p increases, x^p becomes massive even for small x, making it difficult to control the total to be a product of exactly three primes.\\n\\nAlternatively, maybe only small primes work. Let\\'s check p=7 with x=1, y=1, z= some larger number. Let\\'s say z=2. Then x=1, y=1, z=2:1 +1 +128 -1 -1 -2=126. As before. 126=2*3\\xc2\\xb2*7. Not three distinct.\\n\\nz=3:1 +1 +2187 -1 -1 -3=2183. Which we thought might be prime. If it\\'s prime, then single prime. If composite, maybe product of two primes. If it\\'s product of two primes, then that\\'s not three. If product of three primes, but that\\'s unlikely. 2183: we tried dividing by primes up to 31 and it wasn\\'t divisible, so perhaps it\\'s a prime. So 2183 is a prime. Thus, no.\\n\\nSimilarly, x=1, y=1, z=4:16384 -4=16380. 16380=2\\xc2\\xb2*3\\xc2\\xb2*5*7*13. Five distinct primes.\\n\\nx=1, y=1, z=5:1 +1 + 161051 -1 -1 -5=161051 -6=161045. 161045 divided by 5=32209. Check if 32209 is prime. 32209 divided by 7: 7*4601=32207, remainder 2. 11: 11*2928=32208, remainder 1. 13: 13*2477=32201, remainder 8. 17: 17*1894=322, 17*1894=322, 17*100=1700, 17*894=15198, total 1700+15198=16898. Not. Maybe 32209 is prime. So 161045=5*32209. Two primes. Not three.\\n\\nTherefore, even with larger z, it\\'s hard to get three primes. So perhaps p=7 is not possible. Let\\'s check p=7, x=2, y=1, z=1:128 +1 +1 -2 -1 -1=126. As before, not good.\\n\\nAlternatively, x=3, y=1, z=1:2187 +1 +1 -3 -1 -1=2185. Check if 2185 is a product of three primes. 2185 divided by 5=437. 437: divided by 19=23. 19*23=437. So 2185=5*19*23. Three distinct primes! Wait, hold on. 2185=5*19*23. So that is a product of exactly three distinct primes. So, if x=3, y=1, z=1 with p=7, the expression is 3\\xe2\\x81\\xb7 +1\\xe2\\x81\\xb7 +1\\xe2\\x81\\xb7 -3 -1 -1=2187 +1 +1 -5=2184. Wait, no, 2187 +1 +1=2189. 2189 -3 -1 -1=2189 -5=2184. Wait, but 2184=2189-5=2184. Wait, earlier I thought x=3, y=1, z=1 gives 2184. But 2184 factors into 2\\xc2\\xb3*3*7*13. Four distinct primes. But wait, earlier, when I thought x=3, y=1, z=1:3\\xe2\\x81\\xb7=2187, then 2187 +1 +1=2189. Then subtract 3+1+1=5. So 2189 -5=2184. So 2184=2\\xc2\\xb3*3*7*13. Four distinct primes. So my mistake earlier, 2185 is different. Wait, how did I get 2185? Maybe a miscalculation. Let me recalculate:\\n\\nIf x=3, y=1, z=1 for p=7:3\\xe2\\x81\\xb7 +1\\xe2\\x81\\xb7 +1\\xe2\\x81\\xb7 -3 -1 -1=2187 +1 +1 -3 -1 -1=2189 -5=2184. So 2184, not 2185. So my previous thought about 2185 was incorrect. Therefore, even x=3, y=1, z=1 gives 2184=2\\xc2\\xb3*3*7*13. Still four distinct primes.\\n\\nSo maybe p=7 doesn\\'t work. Then, what about p=11? Wait, the problem says \"Find all prime numbers p\". So maybe p=2,3,5 work. Let\\'s check p=7 again. Wait, maybe there\\'s another combination. Let\\'s try x=4, y=1, z=1:4\\xe2\\x81\\xb7=16384. So 16384 +1 +1 -4 -1 -1=16386 -6=16380=2\\xc2\\xb2*3\\xc2\\xb2*5*7*13. Five distinct primes. Not good.\\n\\nWait, perhaps x=5, y=1, z=1:5\\xe2\\x81\\xb7=78125. 78125 +1 +1 -5 -1 -1=78127 -7=78120. 78120=2\\xc2\\xb3*3*5*7*11*17. Six distinct primes. No.\\n\\nAlternatively, x=2, y=3, z=5:128 +2187 + 161051 -2 -3 -5=163366 -10=163356. Factorizing that would be tedious, but likely multiple primes.\\n\\nAlternatively, x=3, y=3, z=1:2187*2 +1 -3*2 -1=4374 +1 -6 -1=4368. 4368=16*273=16*3*91=16*3*7*13=2^4*3*7*13. Four distinct primes.\\n\\nHmm. It seems that for p=7, no combination of x, y, z gives a product of exactly three distinct primes. Therefore, perhaps p=2,3,5 are possible, and primes larger than 5 are not.\\n\\nWait, but p=5 worked with x=2, y=1, z=1. Let\\'s check p=7, x=2, y=1, z=1:128 +1 +1 -2 -1 -1=126=2*3\\xc2\\xb2*7. Not three distinct primes.\\n\\nWait, is there another approach? Let\\'s note that for p=2,3,5, we can get 30, which is 2*3*5. For p=2, x=4, y=4, z=3; p=3, x=1, y=2, z=3; p=5, x=2, y=1, z=1. So in each case, we can set the expression to 30. For higher primes, like p=7, is 30 achievable? Let\\'s check. For p=7, x^7 + y^7 + z^7 -x -y -z=30. Let\\'s see if there are x, y, z such that this holds.\\n\\nBut x, y, z are positive integers. The smallest x^7 for x=2 is 128, which is already larger than 30. So the only possible x, y, z are 1. But x=y=z=1 gives 0. If two variables are 1 and one is 2: 1 +1 +128 -1 -1 -2=126. Not 30. So p=7 cannot reach 30. So 30 is too small for p=7. For higher primes, even more so.\\n\\nTherefore, maybe the only primes are 2,3,5. Let me check p=7 again. Suppose we look for another product of three distinct primes, not necessarily 30. Let\\'s say the product is 105=3*5*7. Let\\'s see if x^7 + y^7 + z^7 -x -y -z=105.\\n\\nLet\\'s try x=2, y=2, z=2:3*128 -6=378. Too big. x=2, y=2, z=1:128 +128 +1 -2 -2 -1=257 -5=252. 252=2\\xc2\\xb2*3\\xc2\\xb2*7. Not 105.\\n\\nx=1, y=2, z=3:1 +128 +2187 -1 -2 -3=2316 -6=2310. 2310=2*3*5*7*11. Five primes. Not 105.\\n\\nx=1, y=1, z=3:1 +1 +2187 -1 -1 -3=2184. As before.\\n\\nx=2, y=1, z=3:128 +1 +2187 -2 -1 -3=2316 -6=2310. Again five primes.\\n\\nAlternatively, x=1, y=3, z=1: same as x=1, y=1, z=3. 2184. Not 105.\\n\\nWait, maybe x=1, y=2, z=2:1 +128 +128 -1 -2 -2=257 -5=252. Not 105.\\n\\nx=1, y=4, z=1:1 +16384 +1 -1 -4 -1=16385 -6=16379=11*1489. Two primes.\\n\\nHmm. Alternatively, target a larger product of three primes. Let\\'s say 385=5*7*11. Can we get 385?\\n\\nTry x=3, y=1, z=1:2187 +1 +1 -3 -1 -1=2184. Not 385.\\n\\nx=2, y=3, z=1:128 +2187 +1 -2 -3 -1=2316 -6=2310. 2310=2*3*5*7*11. That\\'s five primes. Close, but not 385.\\n\\nAlternatively, x=2, y=4, z=1:128 +16384 +1 -2 -4 -1=16513 -7=16506. 16506=2*3*3*7*131. Five primes.\\n\\nAlternatively, x=3, y=2, z=1:2187 +128 +1 -3 -2 -1=2316 -6=2310 again.\\n\\nAlternatively, x=1, y=3, z=2:1 +2187 +128 -1 -3 -2=2316 -6=2310.\\n\\nHmm. It seems that for p=7, it\\'s difficult to get a product of three distinct primes. All attempts either result in too many primes or exponents higher than one. Therefore, perhaps p=7 and higher primes do not work. Thus, the primes are p=2,3,5.\\n\\nBut let\\'s check p=5 again. We had x=2, y=1, z=1 giving 30. Any other examples for p=5? For example, x=3, y=1, z=1:243 +1 +1 -3 -1 -1=244 -5=239. Prime. Not three. x=3, y=2, z=1:243 +32 +1 -3 -2 -1=276 -6=270=2*3\\xc2\\xb3*5. Not three distinct.\\n\\nx=2, y=2, z=1:32 +32 +1 -2 -2 -1=65 -5=60=2\\xc2\\xb2*3*5. Three distinct primes but with exponents. So not acceptable.\\n\\nWait, the problem says \"a product of exactly three distinct prime numbers\". So even if exponents are higher, but the primes are three distinct, does it count? For example, 60=2\\xc2\\xb2*3*5. It has three distinct primes but with exponents. The problem says \"a product of exactly three distinct prime numbers\". The wording is ambiguous. If \"product\" is in the sense of multiplying three distinct primes each appearing once, then exponents must be 1. If it just requires that the total number of distinct prime factors is three, regardless of exponents, then 60 would qualify. But the problem says \"a product of exactly three distinct prime numbers\", which probably means that the expression factors as p1*p2*p3, where p1, p2, p3 are distinct primes. Therefore, exponents must be 1. Hence, for example, 60 wouldn\\'t qualify, but 30=2*3*5 would. So for p=5, x=2, y=1, z=1 gives 30, which works. For p=5, x=3, y=2, z=1 gives 270=2*3\\xc2\\xb3*5, which is not three distinct primes multiplied once. So only x=2, y=1, z=1 works for p=5.\\n\\nSimilarly, for p=2 and p=3, we have examples that give 30. Are there other examples for p=2 and p=3 that result in other products of three distinct primes?\\n\\nFor p=2, x=4, y=4, z=3 gives 30. Let me check x=5, y=5, z=6:25 +25 +36 -5 -5 -6=86 -16=70=2*5*7. Three distinct primes. So yes, another example. So 70 is 2*5*7. So p=2 allows other products as well. Similarly, x=5, y=5, z=6 gives x\\xc2\\xb2 + y\\xc2\\xb2 + z\\xc2\\xb2 -x -y -z=70. So p=2 is possible.\\n\\nSimilarly, for p=3, x=1, y=2, z=3 gives 30. Are there other possibilities? For example, x=2, y=3, z=4:8 +27 +64 -2 -3 -4=99 -9=90=2*3\\xc2\\xb2*5. Not three distinct. x=1, y=3, z=4:1 +27 +64 -1 -3 -4=92 -8=84=2\\xc2\\xb2*3*7. Not three distinct. x=1, y=1, z=4:1 +1 +64 -1 -1 -4=66 -6=60=2\\xc2\\xb2*3*5. Not three distinct. x=2, y=2, z=3:8 +8 +27 -2 -2 -3=43 -7=36=2\\xc2\\xb2*3\\xc2\\xb2. Not three. x=1, y=2, z=4:1 +8 +64 -1 -2 -4=73 -7=66=2*3*11. Three distinct primes. So 66=2*3*11. Therefore, for p=3, x=1, y=2, z=4 gives 66, which is a product of three distinct primes. So p=3 works here as well.\\n\\nSo p=2,3,5 can generate such products, while p=7 and higher primes likely cannot. Therefore, the answer should be p=2,3,5.\\n\\n**Final Answer**\\nThe prime numbers are \\\\boxed{2}, \\\\boxed{3}, and \\\\boxed{5}.\\n\\n\\nTo find all prime numbers \\\\( p \\\\) for which there exist positive integers \\\\( x, y, \\\\) and \\\\( z \\\\) such that the expression \\\\( x^p + y^p + z^p - x - y - z \\\\) is a product of exactly three distinct prime numbers, we analyze the problem for small prime values and check if such combinations exist.\\n\\n1. **For \\\\( p = 2 \\\\)**:\\n - The expression becomes \\\\( x^2 + y^2 + z^2 - x - y - z \\\\).\\n - Trying \\\\( x = 4 \\\\), \\\\( y = 4 \\\\), \\\\( z = 3 \\\\):\\n \\\\[\\n 4^2 + 4^2 + 3^2 - 4 - 4 - 3 = 16 + 16 + 9 - 4 - 4 - 3 = 30\\n \\\\]\\n - 30 factors into \\\\( 2 \\\\times 3 \\\\times 5 \\\\), which are three distinct primes.\\n - Other examples, such as \\\\( x = 5 \\\\), \\\\( y = 5 \\\\), \\\\( z = 6 \\\\), yield 70, which factors into \\\\( 2 \\\\times 5 \\\\times 7 \\\\).\\n\\n2. **For \\\\( p = 3 \\\\)**:\\n - The expression becomes \\\\( x^3 + y^3 + z^3 - x - y - z \\\\).\\n - Trying \\\\( x = 1 \\\\), \\\\( y = 2 \\\\), \\\\( z = 3 \\\\):\\n \\\\[\\n 1^3 + 2^3 + 3^3 - 1 - 2 - 3 = 1 + 8 + 27 - 1 - 2 - 3 = 30\\n \\\\]\\n - 30 factors into \\\\( 2 \\\\times 3 \\\\times 5 \\\\), which are three distinct primes.\\n - Another example, \\\\( x = 1 \\\\), \\\\( y = 2 \\\\), \\\\( z = 4 \\\\), yields 66, which factors into \\\\( 2 \\\\times 3 \\\\times 11 \\\\).\\n\\n3. **For \\\\( p = 5 \\\\)**:\\n - The expression becomes \\\\( x^5 + y^5 + z^5 - x - y - z \\\\).\\n - Trying \\\\( x = 2 \\\\), \\\\( y = 1 \\\\), \\\\( z = 1 \\\\):\\n \\\\[\\n 2^5 + 1^5 + 1^5 - 2 - 1 - 1 = 32 + 1 + 1 - 2 - 1 - 1 = 30\\n \\\\]\\n - 30 factors into \\\\( 2 \\\\times 3 \\\\times 5 \\\\), which are three distinct primes.\\n\\n4. **For primes \\\\( p \\\\geq 7 \\\\)**:\\n - Testing small values and combinations shows that the expression grows too quickly to be a product of exactly three distinct primes. For example, \\\\( x = 2 \\\\), \\\\( y = 1 \\\\), \\\\( z = 1 \\\\) for \\\\( p = 7 \\\\) results in 126, which factors into \\\\( 2 \\\\times 3^2 \\\\times 7 \\\\), not three distinct primes.\\n\\nThus, the prime numbers \\\\( p \\\\) that satisfy the condition are \\\\(\\\\boxed{2}\\\\), \\\\(\\\\boxed{3}\\\\), and \\\\(\\\\boxed{5}\\\\).']", "default/generations_count": 2, "default/correctness": {"default/correctness/llama_verification": "[True, True, True]", "default/correctness/math_verify_answer": "[True, True, True]", "default/correctness/math_verify_reparsed_answer": "[True, True, True]"}, "default/reparsed_answers": "None"} +{"default/split": "train", "default/problem": "\nProblem 2. Let $a, b$ be two distinct real numbers and let $c$ be a positive real number such that\n\n$$\na^{4}-2019 a=b^{4}-2019 b=c .\n$$\n\nProve that $-\\sqrt{c}\\nOkay, so I need to prove that if two distinct real numbers a and b, and a positive real number c satisfy the equations a\\xe2\\x81\\xb4 - 2019a = b\\xe2\\x81\\xb4 - 2019b = c, then the product ab is between -\\xe2\\x88\\x9ac and 0. Hmm, let me start by understanding what the problem is saying.\\n\\nFirst, both a and b are distinct real numbers, and c is positive. The equation given is a fourth-degree polynomial in terms of a and b, each set equal to c. So, both a and b are roots of the equation x\\xe2\\x81\\xb4 - 2019x - c = 0. But since a and b are distinct, this polynomial must have at least two real roots. Wait, but fourth-degree polynomials can have up to four real roots. However, given that c is positive, maybe the structure of the polynomial will restrict the number of real roots?\\n\\nLet me consider the function f(x) = x\\xe2\\x81\\xb4 - 2019x. The problem states that f(a) = f(b) = c, and c is positive. So, we need to analyze the behavior of this function to understand where the roots a and b could be.\\n\\nFirst, let me sketch or imagine the graph of f(x) = x\\xe2\\x81\\xb4 - 2019x. The leading term is x\\xe2\\x81\\xb4, which as x approaches positive or negative infinity, dominates, so the function tends to positive infinity. The derivative f\\xe2\\x80\\x99(x) = 4x\\xc2\\xb3 - 2019. Setting this equal to zero to find critical points:\\n\\n4x\\xc2\\xb3 - 2019 = 0 => x\\xc2\\xb3 = 2019/4 => x = cube root(2019/4). Let\\'s calculate that approximately. 2019 divided by 4 is 504.75. The cube root of 504.75 is roughly around 7.96, since 8\\xc2\\xb3 is 512, which is close. So, approximately 7.96. Let\\'s denote this critical point as x\\xe2\\x82\\x80 \\xe2\\x89\\x88 7.96. \\n\\nSo, the function f(x) has a critical point at x\\xe2\\x82\\x80. Since the coefficient of x\\xe2\\x81\\xb4 is positive, the function decreases to the left of x\\xe2\\x82\\x80 and increases to the right of x\\xe2\\x82\\x80. Therefore, the function has a minimum at x\\xe2\\x82\\x80. Let\\'s compute f(x\\xe2\\x82\\x80) to find the minimum value:\\n\\nf(x\\xe2\\x82\\x80) = (x\\xe2\\x82\\x80)\\xe2\\x81\\xb4 - 2019x\\xe2\\x82\\x80. Since x\\xe2\\x82\\x80\\xc2\\xb3 = 2019/4, so (x\\xe2\\x82\\x80)\\xe2\\x81\\xb4 = x\\xe2\\x82\\x80 * x\\xe2\\x82\\x80\\xc2\\xb3 = x\\xe2\\x82\\x80 * (2019/4). Therefore, f(x\\xe2\\x82\\x80) = (2019/4)x\\xe2\\x82\\x80 - 2019x\\xe2\\x82\\x80 = (2019/4 - 2019)x\\xe2\\x82\\x80 = (- (2019 * 3)/4)x\\xe2\\x82\\x80. That is negative because x\\xe2\\x82\\x80 is positive and multiplied by a negative coefficient. Therefore, the minimum value of f(x) is negative. \\n\\nBut the problem states that c is positive, so f(a) = f(b) = c > 0. Therefore, the points a and b are such that they lie where the function f(x) is equal to c, which is above the minimum. Since the function tends to infinity as x approaches both positive and negative infinity, there should be two real roots for f(x) = c when c is greater than the minimum value. Wait, but since the function has a minimum at x\\xe2\\x82\\x80, if c is greater than f(x\\xe2\\x82\\x80), then the equation f(x) = c will have two real roots: one to the left of x\\xe2\\x82\\x80 and another to the right of x\\xe2\\x82\\x80. But here, we have two distinct real numbers a and b such that f(a) = f(b) = c, so they must be these two roots. \\n\\nWait, but wait. Let me check. If the function is decreasing to the left of x\\xe2\\x82\\x80 and increasing to the right, then if c is greater than the minimum value f(x\\xe2\\x82\\x80), the equation f(x) = c will have two real roots: one on the left side (x < x\\xe2\\x82\\x80) and one on the right side (x > x\\xe2\\x82\\x80). Therefore, a and b must be these two roots. So, one of them is to the left of the minimum point, and the other is to the right. Therefore, a and b are on opposite sides of x\\xe2\\x82\\x80. Since x\\xe2\\x82\\x80 is positive (approx 7.96), this means one of them is less than x\\xe2\\x82\\x80 and the other is greater than x\\xe2\\x82\\x80? Wait, but x\\xe2\\x82\\x80 is positive, so the left side is negative?\\n\\nWait, actually, x\\xe2\\x82\\x80 is positive. So, left of x\\xe2\\x82\\x80 is smaller positive numbers, and right of x\\xe2\\x82\\x80 is larger positive numbers. But wait, the function\\'s behavior for negative x: since x\\xe2\\x81\\xb4 is positive, and -2019x is positive when x is negative. So, for negative x, f(x) = x\\xe2\\x81\\xb4 - 2019x is positive. Let\\'s check the value at x = 0: f(0) = 0 - 0 = 0. So, at x = 0, f(x) is zero. Then, as x approaches negative infinity, x\\xe2\\x81\\xb4 dominates, so f(x) tends to positive infinity. So, the graph of f(x) for x negative is a curve starting from positive infinity, decreasing to f(0) = 0, then as x increases from 0 to x\\xe2\\x82\\x80, f(x) decreases from 0 to the minimum f(x\\xe2\\x82\\x80), which is negative, and then increases back to positive infinity as x goes to positive infinity. Wait, but that contradicts the earlier analysis.\\n\\nWait, hold on. Wait, if x\\xe2\\x82\\x80 is the critical point, which is positive, then for x < x\\xe2\\x82\\x80, the function is decreasing, but for x < x\\xe2\\x82\\x80, when x is negative, how does the function behave?\\n\\nWait, let\\'s recast the derivative. The derivative f\\xe2\\x80\\x99(x) = 4x\\xc2\\xb3 - 2019. So, when x is negative, 4x\\xc2\\xb3 is negative, so f\\xe2\\x80\\x99(x) is negative (since -2019 is also negative). Wait, no: f\\xe2\\x80\\x99(x) = 4x\\xc2\\xb3 - 2019. If x is negative, then 4x\\xc2\\xb3 is negative, so f\\xe2\\x80\\x99(x) = negative - 2019, which is more negative. So, for x < x\\xe2\\x82\\x80, but x\\xe2\\x82\\x80 is positive. Wait, let me think again.\\n\\nWait, critical points occur where f\\xe2\\x80\\x99(x) = 0, which is x = cube root(2019/4) \\xe2\\x89\\x88 7.96, as before. So, for x < x\\xe2\\x82\\x80, the derivative f\\xe2\\x80\\x99(x) = 4x\\xc2\\xb3 - 2019. If x < x\\xe2\\x82\\x80, but x could be positive or negative. Let me check the sign of f\\xe2\\x80\\x99(x) in different intervals.\\n\\nFor x < x\\xe2\\x82\\x80, since x\\xe2\\x82\\x80 is the only critical point (as the derivative is a cubic function with only one real root, because 4x\\xc2\\xb3 - 2019 is a cubic equation with one real root and two complex roots). Therefore, for x < x\\xe2\\x82\\x80, the derivative f\\xe2\\x80\\x99(x) is negative (since for x < x\\xe2\\x82\\x80, 4x\\xc2\\xb3 < 2019, so 4x\\xc2\\xb3 - 2019 < 0), and for x > x\\xe2\\x82\\x80, f\\xe2\\x80\\x99(x) is positive. Therefore, the function f(x) is decreasing on (-\\xe2\\x88\\x9e, x\\xe2\\x82\\x80) and increasing on (x\\xe2\\x82\\x80, \\xe2\\x88\\x9e). \\n\\nBut then, at x = 0, f(0) = 0\\xe2\\x81\\xb4 - 2019*0 = 0. For x negative, f(x) = x\\xe2\\x81\\xb4 - 2019x. Since x is negative, -2019x is positive, so f(x) is positive for all x < 0. As x approaches negative infinity, x\\xe2\\x81\\xb4 dominates, so f(x) approaches positive infinity. So, the graph for x < 0 is a curve that starts at positive infinity, decreases to f(0) = 0. Then, for x between 0 and x\\xe2\\x82\\x80 (\\xe2\\x89\\x887.96), the function is decreasing, so from f(0)=0, it decreases to f(x\\xe2\\x82\\x80), which is negative (as we calculated before). Then, for x > x\\xe2\\x82\\x80, the function increases back to positive infinity. \\n\\nSo, the function f(x) = x\\xe2\\x81\\xb4 - 2019x has the following behavior:\\n- For x < 0: f(x) is positive and decreasing (since derivative is negative).\\n- At x = 0: f(x) = 0.\\n- For 0 < x < x\\xe2\\x82\\x80: f(x) is decreasing from 0 to f(x\\xe2\\x82\\x80) (negative).\\n- For x > x\\xe2\\x82\\x80: f(x) is increasing from f(x\\xe2\\x82\\x80) to infinity.\\n\\nTherefore, if we set c > 0, the equation f(x) = c will have two real roots: one in the interval (-\\xe2\\x88\\x9e, 0) and another in the interval (x\\xe2\\x82\\x80, \\xe2\\x88\\x9e). Wait, but when x < 0, the function is decreasing from infinity to 0, so for c > 0, there\\'s exactly one real root in (-\\xe2\\x88\\x9e, 0). Then, for x > x\\xe2\\x82\\x80, the function is increasing from f(x\\xe2\\x82\\x80) to infinity, so if c > 0, which is greater than f(x\\xe2\\x82\\x80) (since f(x\\xe2\\x82\\x80) is negative), there will be another real root in (x\\xe2\\x82\\x80, \\xe2\\x88\\x9e). Therefore, the equation f(x) = c will have two real roots: one negative and one positive. \\n\\nBut in the problem statement, a and b are two distinct real numbers. So, perhaps a is the negative root and b is the positive root (or vice versa). Since they are distinct, and the function f(x) = c has exactly two real roots when c > 0: one negative and one positive. Therefore, without loss of generality, let\\'s assume a < 0 and b > x\\xe2\\x82\\x80. \\n\\nWait, but the problem says \"a and b are two distinct real numbers\", so perhaps they can be both positive or one positive and one negative? But according to the previous analysis, for c > 0, there are two real roots: one negative and one positive (greater than x\\xe2\\x82\\x80). So, the two distinct roots must be one negative and one positive. Therefore, a and b are of opposite signs. Therefore, their product ab is negative. Hence, ab < 0, which is part of what we need to prove. So, ab is negative, so the upper bound is 0, and we need to show that ab > -\\xe2\\x88\\x9ac.\\n\\nTherefore, the problem reduces to showing that ab is greater than -\\xe2\\x88\\x9ac. So, the product of the two roots (one negative and one positive) is greater than -\\xe2\\x88\\x9ac. \\n\\nSo, given that a and b satisfy a\\xe2\\x81\\xb4 - 2019a = c and b\\xe2\\x81\\xb4 - 2019b = c, with a < 0 and b > x\\xe2\\x82\\x80, then we need to show that ab > -\\xe2\\x88\\x9ac. \\n\\nLet me denote a as the negative root and b as the positive root. So, let\\'s set a = -p, where p > 0. Then, we can write the equation for a:\\n\\n(-p)^4 - 2019*(-p) = p\\xe2\\x81\\xb4 + 2019p = c.\\n\\nSimilarly, for b, which is positive and greater than x\\xe2\\x82\\x80, the equation is:\\n\\nb\\xe2\\x81\\xb4 - 2019b = c.\\n\\nSo, we have p\\xe2\\x81\\xb4 + 2019p = c and b\\xe2\\x81\\xb4 - 2019b = c. Therefore, p\\xe2\\x81\\xb4 + 2019p = b\\xe2\\x81\\xb4 - 2019b. \\n\\nBut since a and b are roots of the equation x\\xe2\\x81\\xb4 - 2019x = c, but a is negative, so we substitute a = -p. Then, the equation becomes (as above) p\\xe2\\x81\\xb4 + 2019p = c. \\n\\nBut perhaps instead of substituting variables, we can consider the relationship between a and b. Since a and b are roots of the equation x\\xe2\\x81\\xb4 - 2019x - c = 0, perhaps we can use Vieta\\'s formula. But this is a quartic equation, but we only have two roots? Wait, no. The quartic equation x\\xe2\\x81\\xb4 - 2019x - c = 0 will have four roots. However, given that c is positive, as per the previous analysis, there are two real roots: one negative and one positive. The other two roots must be complex. Therefore, if we consider the quartic equation, it has two real roots (a and b) and two complex roots. But Vieta\\'s formulas apply to all roots, real and complex. \\n\\nWait, but since complex roots come in conjugate pairs, if there are two complex roots, they must be conjugates. However, Vieta\\'s formulas for quartic equations state that the sum of the roots is equal to the coefficient of x\\xc2\\xb3 (which is 0 in this case, since the equation is x\\xe2\\x81\\xb4 - 2019x - c = 0; coefficients of x\\xc2\\xb2, x\\xc2\\xb3 are zero). So, sum of roots: a + b + c1 + c2 = 0 (where c1 and c2 are complex conjugates). Similarly, the sum of products two at a time is 0 (since the coefficient of x\\xc2\\xb2 is 0), sum of products three at a time is -2019, and the product of all four roots is -c (since the constant term is -c). \\n\\nBut since we don\\'t know the complex roots, maybe this approach is complicated. Alternatively, perhaps considering that a and b are roots, we can treat the quartic as having factors (x - a)(x - b) and a quadratic factor for the complex roots. But maybe this is too involved.\\n\\nAlternatively, let\\'s consider that since a and b satisfy a\\xe2\\x81\\xb4 - 2019a = c and b\\xe2\\x81\\xb4 - 2019b = c, we can set these equal to each other:\\n\\na\\xe2\\x81\\xb4 - 2019a = b\\xe2\\x81\\xb4 - 2019b.\\n\\nTherefore, a\\xe2\\x81\\xb4 - b\\xe2\\x81\\xb4 = 2019(a - b).\\n\\nFactorizing the left-hand side:\\n\\n(a\\xc2\\xb2 - b\\xc2\\xb2)(a\\xc2\\xb2 + b\\xc2\\xb2) = 2019(a - b).\\n\\nWhich can be written as (a - b)(a + b)(a\\xc2\\xb2 + b\\xc2\\xb2) = 2019(a - b).\\n\\nSince a \\xe2\\x89\\xa0 b (they are distinct), we can divide both sides by (a - b):\\n\\n(a + b)(a\\xc2\\xb2 + b\\xc2\\xb2) = 2019.\\n\\nSo, this gives us a relationship between a and b: (a + b)(a\\xc2\\xb2 + b\\xc2\\xb2) = 2019. \\n\\nThis seems like a key equation. Let\\'s denote S = a + b and P = ab. Then, note that a\\xc2\\xb2 + b\\xc2\\xb2 = S\\xc2\\xb2 - 2P. Therefore, substituting into the equation:\\n\\nS(S\\xc2\\xb2 - 2P) = 2019.\\n\\nSo, S\\xc2\\xb3 - 2S P = 2019.\\n\\nTherefore, we can write:\\n\\nS\\xc2\\xb3 - 2S P = 2019. (1)\\n\\nOur goal is to relate P (which is ab) to c. Since we need to prove that -\\xe2\\x88\\x9ac < ab < 0, which is equivalent to -\\xe2\\x88\\x9ac < P < 0. \\n\\nWe already know that ab is negative because one root is negative and the other is positive. So, we need to show that P > -\\xe2\\x88\\x9ac. \\n\\nWe need another equation to relate S and P. Let\\'s recall that a and b each satisfy their own equations:\\n\\nFor a: a\\xe2\\x81\\xb4 - 2019a = c.\\n\\nFor b: b\\xe2\\x81\\xb4 - 2019b = c.\\n\\nLet me subtract these two equations:\\n\\na\\xe2\\x81\\xb4 - b\\xe2\\x81\\xb4 - 2019(a - b) = 0.\\n\\nWhich is the same as the previous equation leading to (a + b)(a\\xc2\\xb2 + b\\xc2\\xb2) = 2019. So, that gives us nothing new.\\n\\nAlternatively, perhaps adding the equations:\\n\\na\\xe2\\x81\\xb4 + b\\xe2\\x81\\xb4 - 2019(a + b) = 2c.\\n\\nSo, a\\xe2\\x81\\xb4 + b\\xe2\\x81\\xb4 = 2019(a + b) + 2c. (2)\\n\\nNow, let\\'s express a\\xe2\\x81\\xb4 + b\\xe2\\x81\\xb4 in terms of S and P. We know that:\\n\\na\\xe2\\x81\\xb4 + b\\xe2\\x81\\xb4 = (a\\xc2\\xb2 + b\\xc2\\xb2)\\xc2\\xb2 - 2a\\xc2\\xb2b\\xc2\\xb2.\\n\\nBut a\\xc2\\xb2 + b\\xc2\\xb2 = S\\xc2\\xb2 - 2P, so:\\n\\na\\xe2\\x81\\xb4 + b\\xe2\\x81\\xb4 = (S\\xc2\\xb2 - 2P)\\xc2\\xb2 - 2(ab)^2 = (S\\xc2\\xb2 - 2P)\\xc2\\xb2 - 2P\\xc2\\xb2.\\n\\nSubstituting into equation (2):\\n\\n(S\\xc2\\xb2 - 2P)\\xc2\\xb2 - 2P\\xc2\\xb2 = 2019S + 2c.\\n\\nExpanding (S\\xc2\\xb2 - 2P)\\xc2\\xb2:\\n\\n= S\\xe2\\x81\\xb4 - 4S\\xc2\\xb2 P + 4P\\xc2\\xb2 - 2P\\xc2\\xb2\\n\\n= S\\xe2\\x81\\xb4 - 4S\\xc2\\xb2 P + 2P\\xc2\\xb2.\\n\\nTherefore, equation (2) becomes:\\n\\nS\\xe2\\x81\\xb4 - 4S\\xc2\\xb2 P + 2P\\xc2\\xb2 = 2019S + 2c. (3)\\n\\nNow, from equation (1):\\n\\nS\\xc2\\xb3 - 2S P = 2019.\\n\\nWe can solve for 2019:\\n\\n2019 = S\\xc2\\xb3 - 2S P. \\n\\nSubstituting this into equation (3):\\n\\nS\\xe2\\x81\\xb4 - 4S\\xc2\\xb2 P + 2P\\xc2\\xb2 = S(S\\xc2\\xb3 - 2S P) + 2c.\\n\\nLet me compute the right-hand side:\\n\\n= S\\xe2\\x81\\xb4 - 2S\\xc2\\xb2 P + 2c.\\n\\nTherefore, equation (3) becomes:\\n\\nLeft-hand side: S\\xe2\\x81\\xb4 - 4S\\xc2\\xb2 P + 2P\\xc2\\xb2\\n\\nRight-hand side: S\\xe2\\x81\\xb4 - 2S\\xc2\\xb2 P + 2c\\n\\nSubtracting right-hand side from left-hand side:\\n\\n(S\\xe2\\x81\\xb4 - 4S\\xc2\\xb2 P + 2P\\xc2\\xb2) - (S\\xe2\\x81\\xb4 - 2S\\xc2\\xb2 P + 2c) = 0\\n\\nSimplify:\\n\\n-4S\\xc2\\xb2 P + 2P\\xc2\\xb2 + 2S\\xc2\\xb2 P - 2c = 0\\n\\nCombine like terms:\\n\\n(-4S\\xc2\\xb2 P + 2S\\xc2\\xb2 P) + 2P\\xc2\\xb2 - 2c = 0\\n\\nWhich is:\\n\\n-2S\\xc2\\xb2 P + 2P\\xc2\\xb2 - 2c = 0\\n\\nDivide both sides by 2:\\n\\n- S\\xc2\\xb2 P + P\\xc2\\xb2 - c = 0\\n\\nRearranged:\\n\\nP\\xc2\\xb2 - S\\xc2\\xb2 P - c = 0. (4)\\n\\nSo, now we have equation (4): P\\xc2\\xb2 - S\\xc2\\xb2 P - c = 0.\\n\\nOur variables are S and P, related by equations (1) and (4). Let\\'s recall equation (1):\\n\\nS\\xc2\\xb3 - 2S P = 2019. (1)\\n\\nEquation (4):\\n\\nP\\xc2\\xb2 - S\\xc2\\xb2 P - c = 0. (4)\\n\\nWe need to solve these two equations to relate P and c.\\n\\nLet me try to express S in terms of P from equation (1). However, equation (1) is cubic in S, which might be complicated. Alternatively, perhaps express S\\xc2\\xb2 from equation (4). Let\\'s see:\\n\\nFrom equation (4):\\n\\nP\\xc2\\xb2 - S\\xc2\\xb2 P = c\\n\\n=> S\\xc2\\xb2 P = P\\xc2\\xb2 - c\\n\\nAssuming P \\xe2\\x89\\xa0 0 (which is true since ab = P < 0, and c > 0, so P\\xc2\\xb2 - c could be positive or negative, but S\\xc2\\xb2 is positive). Therefore:\\n\\nS\\xc2\\xb2 = (P\\xc2\\xb2 - c)/P\\n\\nBut S\\xc2\\xb2 must be positive, so (P\\xc2\\xb2 - c)/P > 0. Since P = ab < 0 (as established before), so P is negative. Therefore, the denominator is negative. Therefore, for (P\\xc2\\xb2 - c)/P > 0, the numerator must be negative as well (since denominator is negative, so numerator must be negative to have positive overall). Therefore:\\n\\nP\\xc2\\xb2 - c < 0\\n\\n=> P\\xc2\\xb2 < c\\n\\n=> |P| < \\xe2\\x88\\x9ac\\n\\nBut since P is negative (P = ab < 0), this is equivalent to -\\xe2\\x88\\x9ac < P < 0. Which is exactly what we need to prove! \\n\\nWait, hold on. Let me check that again. \\n\\nWe have S\\xc2\\xb2 = (P\\xc2\\xb2 - c)/P. Since S\\xc2\\xb2 is a square, it must be non-negative. Therefore:\\n\\n(P\\xc2\\xb2 - c)/P \\xe2\\x89\\xa5 0.\\n\\nGiven that P = ab is negative (since a and b are of opposite signs), then denominator P is negative. Therefore, the numerator must also be negative for the fraction to be non-negative (since negative divided by negative is positive). Therefore:\\n\\nP\\xc2\\xb2 - c < 0.\\n\\nWhich implies:\\n\\nP\\xc2\\xb2 < c.\\n\\nTaking square roots:\\n\\n|P| < \\xe2\\x88\\x9ac.\\n\\nSince P is negative, this is equivalent to:\\n\\n-\\xe2\\x88\\x9ac < P < 0.\\n\\nWhich is exactly the inequality we needed to prove: -\\xe2\\x88\\x9ac < ab < 0.\\n\\nTherefore, this completes the proof.\\n\\nBut let me recap to ensure there are no gaps. So, by deriving equation (4) and using equation (1), we found that S\\xc2\\xb2 must equal (P\\xc2\\xb2 - c)/P. Since S\\xc2\\xb2 is non-negative, the expression (P\\xc2\\xb2 - c)/P must be non-negative. Given that P is negative (as a and b have opposite signs), this requires that the numerator (P\\xc2\\xb2 - c) must also be negative, leading to P\\xc2\\xb2 < c, hence |P| < \\xe2\\x88\\x9ac. Since P is negative, this gives -\\xe2\\x88\\x9ac < P < 0, which is the desired result. Therefore, the product ab is indeed between -\\xe2\\x88\\x9ac and 0.\\n\\n**Final Answer**\\n\\\\boxed{-\\\\sqrt{c} < ab < 0}\\n\\n\\nGiven two distinct real numbers \\\\(a\\\\) and \\\\(b\\\\) and a positive real number \\\\(c\\\\) such that \\\\(a^4 - 2019a = b^4 - 2019b = c\\\\), we need to prove that \\\\(-\\\\sqrt{c} < ab < 0\\\\).\\n\\n1. **Function Analysis**:\\n - Consider the function \\\\(f(x) = x^4 - 2019x\\\\).\\n - The critical points of \\\\(f(x)\\\\) are found by setting the derivative \\\\(f\\'(x) = 4x^3 - 2019\\\\) to zero, yielding \\\\(x = \\\\sqrt[3]{2019/4} \\\\approx 7.96\\\\).\\n - The function \\\\(f(x)\\\\) has a minimum at this critical point \\\\(x_0 \\\\approx 7.96\\\\) and is decreasing for \\\\(x < x_0\\\\) and increasing for \\\\(x > x_0\\\\).\\n\\n2. **Roots of the Equation**:\\n - Since \\\\(c > 0\\\\), the equation \\\\(f(x) = c\\\\) has two real roots: one negative and one positive (greater than \\\\(x_0\\\\)).\\n - Therefore, \\\\(a\\\\) and \\\\(b\\\\) must be these roots, with one negative and one positive.\\n\\n3. **Product of Roots**:\\n - Since \\\\(a\\\\) and \\\\(b\\\\) are of opposite signs, their product \\\\(ab\\\\) is negative, i.e., \\\\(ab < 0\\\\).\\n\\n4. **Key Equations**:\\n - Using the given equations \\\\(a^4 - 2019a = c\\\\) and \\\\(b^4 - 2019b = c\\\\), we derive:\\n \\\\[\\n (a + b)(a^2 + b^2) = 2019\\n \\\\]\\n - Let \\\\(S = a + b\\\\) and \\\\(P = ab\\\\). Then \\\\(a^2 + b^2 = S^2 - 2P\\\\), leading to:\\n \\\\[\\n S(S^2 - 2P) = 2019\\n \\\\]\\n - Another key equation derived is:\\n \\\\[\\n P^2 - S^2P - c = 0\\n \\\\]\\n\\n5. **Solving for \\\\(P\\\\)**:\\n - From the equation \\\\(P^2 - S^2P - c = 0\\\\), we express \\\\(S^2\\\\) as:\\n \\\\[\\n S^2 = \\\\frac{P^2 - c}{P}\\n \\\\]\\n - Since \\\\(S^2 \\\\geq 0\\\\), the right-hand side must be non-negative. Given \\\\(P < 0\\\\), the numerator \\\\(P^2 - c\\\\) must be negative:\\n \\\\[\\n P^2 < c \\\\implies |P| < \\\\sqrt{c}\\n \\\\]\\n - Since \\\\(P = ab\\\\) is negative, this translates to:\\n \\\\[\\n -\\\\sqrt{c} < ab < 0\\n \\\\]\\n\\nThus, we conclude that \\\\(-\\\\sqrt{c} < ab < 0\\\\), and the final answer is \\\\(\\\\boxed{-\\\\sqrt{c} < ab < 0}\\\\).', b\"\\nOkay, so I have this problem here where there are two distinct real numbers, a and b, and a positive real number c. The equations given are a\\xe2\\x81\\xb4 - 2019a = c and b\\xe2\\x81\\xb4 - 2019b = c. I need to prove that -\\xe2\\x88\\x9ac is less than ab, which is less than 0. Hmm, interesting. Let me try to unpack this step by step.\\n\\nFirst, since a and b are distinct real numbers and both satisfy the equation x\\xe2\\x81\\xb4 - 2019x = c, that means they are different roots of the equation x\\xe2\\x81\\xb4 - 2019x - c = 0. So, maybe I can consider the function f(x) = x\\xe2\\x81\\xb4 - 2019x - c and analyze its roots. Since c is positive, the equation is x\\xe2\\x81\\xb4 - 2019x = c. Let's see.\\n\\nLet me think about the graph of f(x) = x\\xe2\\x81\\xb4 - 2019x. Since it's a quartic function with a positive leading coefficient, as x approaches positive or negative infinity, f(x) tends to positive infinity. The derivative f\\xe2\\x80\\x99(x) = 4x\\xc2\\xb3 - 2019. Setting this equal to zero to find critical points: 4x\\xc2\\xb3 - 2019 = 0 \\xe2\\x86\\x92 x\\xc2\\xb3 = 2019/4 \\xe2\\x86\\x92 x = (2019/4)^(1/3). Let me compute that value numerically to get a sense of where the critical points are.\\n\\nCalculating (2019/4)^(1/3): 2019 divided by 4 is approximately 504.75. The cube root of 504.75 is around 7.96, so approximately 8. So, the function f(x) has a critical point at x \\xe2\\x89\\x88 8. So, the function is decreasing from negative infinity up to x \\xe2\\x89\\x88 8, and then increasing from x \\xe2\\x89\\x88 8 to infinity. Therefore, the graph of f(x) = x\\xe2\\x81\\xb4 - 2019x is U-shaped with a minimum at x \\xe2\\x89\\x88 8.\\n\\nBut since c is positive, the equation x\\xe2\\x81\\xb4 - 2019x = c would have two real roots if the horizontal line y = c intersects the graph of f(x) at two points. Wait, but since the function tends to infinity on both ends and has a minimum at x \\xe2\\x89\\x88 8, the equation x\\xe2\\x81\\xb4 - 2019x = c will have two real roots when c is greater than the minimum value of f(x). But in the problem, c is given as positive, but is it necessarily greater than the minimum? Wait, hold on. The problem states that a and b are distinct real numbers, so there must be two real roots. Therefore, c must be greater than or equal to the minimum value of f(x). But the problem says c is positive. Hmm. So, the minimum value of f(x) is f((2019/4)^(1/3)). Let's compute that.\\n\\nCompute f at x = (2019/4)^(1/3):\\n\\nLet\\xe2\\x80\\x99s denote k = (2019/4)^(1/3). Then f(k) = k\\xe2\\x81\\xb4 - 2019k. But since k\\xc2\\xb3 = 2019/4, we can write k\\xe2\\x81\\xb4 = k * k\\xc2\\xb3 = k*(2019/4). Therefore, f(k) = (2019/4)k - 2019k = (2019/4 - 2019)k = (-2019*(3/4))k. So, f(k) = - (6057/4)k. Since k is positive (because 2019/4 is positive), f(k) is negative. Therefore, the minimum value of f(x) is negative. Therefore, the equation x\\xe2\\x81\\xb4 - 2019x = c will have two real roots when c is greater than the minimum value, which is negative. Since c is positive, it's definitely above the minimum, so there are two real roots. Wait, but the problem says a and b are two distinct real numbers, so c must be such that there are two distinct roots. So, given that c is positive, the horizontal line y = c intersects the graph at two points. So, there are two real roots, a and b.\\n\\nBut in that case, how are the roots a and b positioned? Since the function is decreasing to the left of x \\xe2\\x89\\x88 8 and increasing to the right, one root is to the left of the minimum point, and the other is to the right. Wait, but if the function is U-shaped, then actually for c > f(k), which is the minimum value, the equation x\\xe2\\x81\\xb4 - 2019x = c will have two real roots: one to the left of the minimum and one to the right. But since the function is decreasing until x \\xe2\\x89\\x88 8 and increasing after that, then the left root would be in the decreasing part, and the right root would be in the increasing part. Wait, but when c is positive, since the minimum value is negative, the horizontal line y = c is above the x-axis, so the equation x\\xe2\\x81\\xb4 - 2019x = c will have two real roots: one negative and one positive? Wait, let's check.\\n\\nWait, if x is very large positive, then x\\xe2\\x81\\xb4 dominates, so f(x) is positive. If x is very large negative, x\\xe2\\x81\\xb4 is still positive, so f(x) is positive. The minimum at x \\xe2\\x89\\x88 8 is negative. Therefore, the graph crosses the x-axis somewhere. Wait, actually, the equation x\\xe2\\x81\\xb4 - 2019x = 0 would have roots at x=0 and the real roots of x\\xc2\\xb3 = 2019. So, x=0 and x= (2019)^(1/3) \\xe2\\x89\\x88 12.6. So, the graph crosses the x-axis at x=0 and x\\xe2\\x89\\x8812.6. Therefore, the function f(x) = x\\xe2\\x81\\xb4 - 2019x is negative between x=0 and x\\xe2\\x89\\x8812.6? Wait, no. Wait, x\\xe2\\x81\\xb4 - 2019x = 0 at x=0 and x= (2019)^(1/3). Let me compute f(x) at x=0: f(0)=0 -0=0. At x= (2019)^(1/3)\\xe2\\x89\\x8812.6, f(x)= (12.6)^4 - 2019*(12.6). Let me compute that. 12.6^4 is 12.6*12.6=158.76, then squared is \\xe2\\x89\\x8825204. But 2019*12.6 \\xe2\\x89\\x882019*10=20190 + 2019*2.6\\xe2\\x89\\x882019*2=4038 + 2019*0.6\\xe2\\x89\\x881211.4 \\xe2\\x86\\x92 20190 + 4038 + 1211.4\\xe2\\x89\\x8825439.4. So, 25204 -25439.4\\xe2\\x89\\x88-235.4. Wait, so at x\\xe2\\x89\\x8812.6, f(x)=0? Wait, no, that was solving x\\xc2\\xb3=2019. Wait, but f(x)=x\\xe2\\x81\\xb4 -2019x. If x= (2019)^(1/3), then x\\xc2\\xb3=2019, so x\\xe2\\x81\\xb4= x* x\\xc2\\xb3= x*2019. Therefore, f(x)=2019x -2019x=0. So, yes, x=0 and x= (2019)^(1/3) are roots of f(x)=0.\\n\\nBut the minimum of f(x) is at x=(2019/4)^(1/3)\\xe2\\x89\\x888, and f(8)=8^4 -2019*8=4096 -16152= -12056. That seems too big. Wait, maybe my previous calculation was wrong. Let me recalculate.\\n\\nWait, (2019/4) is 504.75. So, the cube root of 504.75 is approximately 7.96, so let's take x\\xe2\\x89\\x888. Then, x\\xc2\\xb3=512, which is a bit larger than 504.75, so x\\xe2\\x89\\x887.96. Let's take x=7.96. Then, x^4 = (7.96)^4. Let's compute:\\n\\nFirst, 7.96 squared: 7.96*7.96. 8*8=64, subtract 0.04*8 + 0.04*8 -0.04^2 = 64 - 0.64 -0.64 +0.0016=64 -1.28 +0.0016=62.7216. Then, square that: 62.7216 squared. Let's approximate:\\n\\n60^2=3600, 2.7216^2\\xe2\\x89\\x887.407, and cross term 2*60*2.7216\\xe2\\x89\\x88326.592. So total\\xe2\\x89\\x883600 + 326.592 +7.407\\xe2\\x89\\x883934. So, x^4\\xe2\\x89\\x883934. Then 2019x\\xe2\\x89\\x882019*7.96\\xe2\\x89\\x882019*8 -2019*0.04\\xe2\\x89\\x8816152 -80.76\\xe2\\x89\\x8816071.24. Then f(x)=x^4 -2019x\\xe2\\x89\\x883934 -16071\\xe2\\x89\\x88-12137. So, the minimum value is around -12137. So, f(x) reaches a minimum of about -12000 something at x\\xe2\\x89\\x888. So, the graph of f(x) is positive at x approaching \\xc2\\xb1infinity, crosses the x-axis at x=0 and x\\xe2\\x89\\x8812.6, and has a minimum at x\\xe2\\x89\\x888 of about -12000. So, if we set y = c, which is positive, then the equation x\\xe2\\x81\\xb4 -2019x = c will have two real roots: one left of x=0 (negative) and one right of x\\xe2\\x89\\x8812.6 (positive). Wait, is that possible?\\n\\nWait, when x is negative, x\\xe2\\x81\\xb4 is positive, and -2019x is positive because x is negative. So, f(x) = x\\xe2\\x81\\xb4 -2019x is positive for x negative. Therefore, when x is negative, f(x) is positive. Then, as x increases from negative infinity to 0, f(x) decreases from positive infinity to f(0)=0. Then, from x=0 to x\\xe2\\x89\\x888, f(x) decreases to the minimum at x\\xe2\\x89\\x888 of -12000. Then, from x\\xe2\\x89\\x888 onwards, f(x) increases back to positive infinity. Therefore, the equation x\\xe2\\x81\\xb4 -2019x = c (which is positive) will have two real roots: one negative (left of x=0) and one positive (right of x\\xe2\\x89\\x8812.6). Wait, but at x\\xe2\\x89\\x8812.6, f(x)=0. So, if we set c positive, then the equation x\\xe2\\x81\\xb4 -2019x = c will have two roots: one negative (since for x <0, f(x) is positive and decreasing from +infty to 0 as x goes from -infty to 0), so the negative root is somewhere left of 0. Then, the other root is to the right of x\\xe2\\x89\\x8812.6 where f(x) increases from -12000 to +infty, crossing y=c >0 at some x >12.6.\\n\\nWait, but if that's the case, then a and b would be one negative and one positive. So, their product ab would be negative. Which aligns with the upper bound of 0 in the problem statement. So, ab <0. That makes sense. So, the problem wants to show that ab is between -\\xe2\\x88\\x9ac and 0. So, ab is negative, and its absolute value is less than \\xe2\\x88\\x9ac.\\n\\nSo, first, let's confirm that a and b are of opposite signs. Since they are roots of x\\xe2\\x81\\xb4 -2019x -c =0, and c>0. If x is negative, then x\\xe2\\x81\\xb4 is positive, -2019x is positive (since x is negative), so x\\xe2\\x81\\xb4 -2019x is positive, so to equal c (positive), x must be such that x\\xe2\\x81\\xb4 -2019x is positive, which is true for all x negative. So, there is one negative root. Then, the other root is positive, where x\\xe2\\x81\\xb4 -2019x is positive. But when x is positive, x\\xe2\\x81\\xb4 -2019x. For x >0, x\\xe2\\x81\\xb4 grows faster than 2019x, but at x=0, it's 0. Then, as x increases, x\\xe2\\x81\\xb4 -2019x is negative until x is sufficiently large. Wait, at x=8, it's -12000. At x=12.6, it's 0. So, for x between 0 and 12.6, x\\xe2\\x81\\xb4 -2019x is negative. So, to get x\\xe2\\x81\\xb4 -2019x = c >0, the positive root must be greater than 12.6. Therefore, yes, the two roots are one negative (left of 0) and one positive (right of 12.6). Therefore, their product ab is negative, so ab <0. So, that's the upper bound. Now, we need to show that ab > -\\xe2\\x88\\x9ac.\\n\\nSo, how can we relate ab to c? Let's think. Since a and b are roots of x\\xe2\\x81\\xb4 -2019x -c =0, but since there are two roots, but the equation is quartic, which might have more roots, but in this problem, we are only considering two distinct roots. Wait, but actually, the equation x\\xe2\\x81\\xb4 -2019x -c =0 has four roots in total (counting multiplicities). But given that c is positive, as we saw, there are two real roots: one negative, one positive greater than 12.6. The other two roots must be complex, since the function tends to infinity on both ends. Wait, but maybe not. Wait, actually, if the function f(x) = x\\xe2\\x81\\xb4 -2019x -c, then for c >0, as x approaches -infty, f(x) approaches +infty, at x=0, f(0) = -c <0, then decreases to a minimum at x\\xe2\\x89\\x888, then increases to +infty. Wait, hold on, maybe I miscalculated before.\\n\\nWait, if we take the function f(x) = x\\xe2\\x81\\xb4 -2019x -c. So, the original function was x\\xe2\\x81\\xb4 -2019x, but here it's x\\xe2\\x81\\xb4 -2019x -c. So, when c is positive, this is shifting the original function down by c. So, if the original function f(x) = x\\xe2\\x81\\xb4 -2019x has a minimum at x\\xe2\\x89\\x888 of -12000, then f(x) -c would have a minimum at -12000 -c. But since c is positive, the minimum is even lower. Wait, but the problem states that there are two distinct real roots a and b. But for f(x) = x\\xe2\\x81\\xb4 -2019x -c, with c>0, how many real roots does it have?\\n\\nWait, when x is very negative, x\\xe2\\x81\\xb4 dominates, so f(x) is positive. At x=0, f(x) = -c (negative). Then, as x increases from 0 to the minimum at x\\xe2\\x89\\x888, f(x) decreases from -c to -12000 -c. Then, increases to infinity. So, in that case, the graph crosses the x-axis once on the left (negative side) and once on the right (after the minimum). Wait, but if the minimum is -12000 -c, which is very negative, then the graph crosses the x-axis only twice: once on the left of the minimum (negative x) and once on the right (positive x). Therefore, yes, two real roots, as given in the problem. So, a and b are these two roots. So, a is negative, b is positive.\\n\\nSo, a <0 and b >0. Therefore, ab <0. That gives the upper bound. Now, the lower bound is -\\xe2\\x88\\x9ac. So, we need to show that ab > -\\xe2\\x88\\x9ac. So, how can we relate ab to c?\\n\\nSince a and b satisfy the equation x\\xe2\\x81\\xb4 -2019x = c. So, let's write the equations:\\n\\nFor a: a\\xe2\\x81\\xb4 -2019a = c\\n\\nFor b: b\\xe2\\x81\\xb4 -2019b = c\\n\\nSo, a\\xe2\\x81\\xb4 -2019a = b\\xe2\\x81\\xb4 -2019b\\n\\nSo, a\\xe2\\x81\\xb4 - b\\xe2\\x81\\xb4 = 2019(a - b)\\n\\nFactor the left side: (a\\xc2\\xb2 - b\\xc2\\xb2)(a\\xc2\\xb2 + b\\xc2\\xb2) = 2019(a - b)\\n\\nThen, (a - b)(a + b)(a\\xc2\\xb2 + b\\xc2\\xb2) = 2019(a - b)\\n\\nSince a \\xe2\\x89\\xa0 b (they are distinct), we can divide both sides by (a - b):\\n\\n(a + b)(a\\xc2\\xb2 + b\\xc2\\xb2) = 2019\\n\\nHmm, that's a useful equation. So, (a + b)(a\\xc2\\xb2 + b\\xc2\\xb2) = 2019.\\n\\nAlso, from the original equations, a\\xe2\\x81\\xb4 -2019a = c and b\\xe2\\x81\\xb4 -2019b = c. So, both a and b satisfy x\\xe2\\x81\\xb4 -2019x = c.\\n\\nNow, maybe we can consider variables a and b, with a <0 and b>0, such that (a + b)(a\\xc2\\xb2 + b\\xc2\\xb2) =2019, and we need to relate ab to c.\\n\\nWe need to show that -\\xe2\\x88\\x9ac < ab <0. The upper bound is already established (since a <0, b>0). Now, the lower bound: ab > -\\xe2\\x88\\x9ac.\\n\\nSo, perhaps we can express c in terms of a and b, and then relate ab to c.\\n\\nGiven that a\\xe2\\x81\\xb4 -2019a = c and b\\xe2\\x81\\xb4 -2019b = c.\\n\\nSo, c = a\\xe2\\x81\\xb4 -2019a = b\\xe2\\x81\\xb4 -2019b.\\n\\nTherefore, a\\xe2\\x81\\xb4 - b\\xe2\\x81\\xb4 =2019(a - b), which we already used to get (a + b)(a\\xc2\\xb2 + b\\xc2\\xb2)=2019.\\n\\nLet me see. Let's denote S = a + b and P = ab. Maybe we can express things in terms of S and P.\\n\\nNote that a and b are roots of the equation x\\xe2\\x81\\xb4 -2019x -c =0. However, since it's a quartic equation, but we are only considering two roots, it's a bit tricky. Alternatively, perhaps we can relate S and P to c.\\n\\nWe have S = a + b\\n\\nWe have P = ab\\n\\nWe need to relate S and P to c.\\n\\nWe also have the equation (a + b)(a\\xc2\\xb2 + b\\xc2\\xb2)=2019.\\n\\nExpress a\\xc2\\xb2 + b\\xc2\\xb2 in terms of S and P: a\\xc2\\xb2 + b\\xc2\\xb2 = (a + b)^2 - 2ab = S\\xc2\\xb2 - 2P.\\n\\nTherefore, the equation becomes S*(S\\xc2\\xb2 - 2P) = 2019.\\n\\nSo, S^3 - 2S P =2019.\\n\\nThat's one equation relating S and P.\\n\\nNow, we need another equation. Let's see. Since a and b satisfy a\\xe2\\x81\\xb4 -2019a = c and b\\xe2\\x81\\xb4 -2019b =c. Let's compute a\\xe2\\x81\\xb4 + b\\xe2\\x81\\xb4 in terms of c. a\\xe2\\x81\\xb4 + b\\xe2\\x81\\xb4 = 2c +2019(a + b) = 2c +2019 S.\\n\\nAlternatively, express a\\xe2\\x81\\xb4 + b\\xe2\\x81\\xb4 in terms of S and P. Let's recall that:\\n\\na\\xe2\\x81\\xb4 + b\\xe2\\x81\\xb4 = (a\\xc2\\xb2 + b\\xc2\\xb2)^2 - 2a\\xc2\\xb2b\\xc2\\xb2 = (S\\xc2\\xb2 - 2P)^2 - 2P\\xc2\\xb2.\\n\\nTherefore, (S\\xc2\\xb2 - 2P)^2 - 2P\\xc2\\xb2 = 2c +2019 S.\\n\\nLet me compute left side:\\n\\n(S\\xc2\\xb2 - 2P)^2 -2P\\xc2\\xb2 = S^4 -4S\\xc2\\xb2 P +4P\\xc2\\xb2 -2P\\xc2\\xb2 = S^4 -4S\\xc2\\xb2 P +2P\\xc2\\xb2.\\n\\nTherefore, S^4 -4S\\xc2\\xb2 P +2P\\xc2\\xb2 = 2c +2019 S.\\n\\nSo, we have another equation: S^4 -4S\\xc2\\xb2 P +2P\\xc2\\xb2 =2c +2019 S.\\n\\nSo, now we have two equations:\\n\\n1) S^3 -2 S P =2019\\n\\n2) S^4 -4 S\\xc2\\xb2 P +2 P\\xc2\\xb2 =2c +2019 S\\n\\nOur goal is to express c in terms of S and P, then relate P to c.\\n\\nLet me try to solve equation 1) for S^3:\\n\\nFrom equation 1: S^3 =2019 +2 S P.\\n\\nThen, substitute S^3 into equation 2.\\n\\nEquation 2:\\n\\nS^4 -4 S\\xc2\\xb2 P +2 P\\xc2\\xb2 =2c +2019 S\\n\\nBut S^4 = S * S^3 = S*(2019 +2 S P) =2019 S +2 S\\xc2\\xb2 P.\\n\\nTherefore, substituting into equation 2:\\n\\n2019 S +2 S\\xc2\\xb2 P -4 S\\xc2\\xb2 P +2 P\\xc2\\xb2 =2c +2019 S\\n\\nSimplify:\\n\\n2019 S + (2 S\\xc2\\xb2 P -4 S\\xc2\\xb2 P) +2 P\\xc2\\xb2 =2c +2019 S\\n\\nWhich becomes:\\n\\n2019 S -2 S\\xc2\\xb2 P +2 P\\xc2\\xb2 =2c +2019 S\\n\\nSubtract 2019 S from both sides:\\n\\n-2 S\\xc2\\xb2 P +2 P\\xc2\\xb2 =2c\\n\\nDivide both sides by 2:\\n\\n- S\\xc2\\xb2 P + P\\xc2\\xb2 =c\\n\\nTherefore,\\n\\nc = P\\xc2\\xb2 - S\\xc2\\xb2 P = P (P - S\\xc2\\xb2)\\n\\nSo, c = P (P - S\\xc2\\xb2)\\n\\nBut from equation 1), S^3 =2019 +2 S P. Let's solve equation 1) for S P:\\n\\nFrom equation 1):\\n\\nS^3 -2 S P =2019 \\xe2\\x86\\x92 2 S P = S^3 -2019 \\xe2\\x86\\x92 S P = (S^3 -2019)/2\\n\\nTherefore, S P = (S^3 -2019)/2\\n\\nSo, we can write c = P (P - S\\xc2\\xb2) = P\\xc2\\xb2 - S\\xc2\\xb2 P\\n\\nBut S P = (S^3 -2019)/2, so S\\xc2\\xb2 P = S*(S P) = S*( (S^3 -2019)/2 ) = (S^4 -2019 S)/2\\n\\nTherefore, c = P\\xc2\\xb2 - (S^4 -2019 S)/2\\n\\nBut S^4 can be expressed from equation 2). Wait, maybe this is getting too convoluted. Let me see.\\n\\nAlternatively, from c = P (P - S\\xc2\\xb2). Let's substitute S\\xc2\\xb2 from equation 1.\\n\\nFrom equation 1: S^3 -2 S P =2019 \\xe2\\x86\\x92 S^3 =2019 +2 S P. Let's divide both sides by S (assuming S \\xe2\\x89\\xa00, which it is not, because a and b are of opposite signs, so S =a +b could be positive or negative. If a is negative and b is positive, S =a +b. Depending on the magnitudes, S could be positive or negative. Let me check.\\n\\nIf a is negative and b is positive, then S = a +b. Since b is greater than 12.6, and a is negative, but how negative is a? Let's see. Let's consider the original equation for a: a\\xe2\\x81\\xb4 -2019a =c. If a is negative, then a\\xe2\\x81\\xb4 is positive, and -2019a is positive, so the left side is positive. So, c is positive. Similarly for b. So, the negative root a must satisfy a\\xe2\\x81\\xb4 -2019a =c. Let's take an example. Suppose c=1. Then, a\\xe2\\x81\\xb4 -2019a =1. For a negative, say a=-k, k>0. Then, k\\xe2\\x81\\xb4 +2019k =1. So, k\\xe2\\x81\\xb4 +2019k =1. Since k is positive, the left side is greater than 2019k. So, 2019k <1 \\xe2\\x86\\x92k <1/2019\\xe2\\x89\\x880.000495. So, k is very small. Then, a\\xe2\\x89\\x88-0.000495. Then, b would be the positive root of x\\xe2\\x81\\xb4 -2019x =1. Let's see, x\\xe2\\x81\\xb4 -2019x -1=0. For large x, x\\xe2\\x81\\xb4 dominates, so b is approximately (2019x +1)^(1/4). But actually, since x is large, 2019x is the dominant term. So, x\\xe2\\x89\\x88 (2019x)^(1/4) \\xe2\\x86\\x92 x^4 \\xe2\\x89\\x882019x \\xe2\\x86\\x92x^3\\xe2\\x89\\x882019 \\xe2\\x86\\x92x\\xe2\\x89\\x88(2019)^(1/3)\\xe2\\x89\\x8812.6. But since we have the equation x\\xe2\\x81\\xb4 -2019x =1, the positive root is slightly larger than 12.6. So, in this example, a is a very small negative number, and b is slightly larger than 12.6, so S =a +b \\xe2\\x89\\x88 -0.000495 +12.6\\xe2\\x89\\x8812.5995, which is positive.\\n\\nAlternatively, if c is very large, then a would be more negative. For example, suppose c=10,000. Then, a\\xe2\\x81\\xb4 -2019a =10,000. Let a=-k, k>0. Then, k\\xe2\\x81\\xb4 +2019k =10,000. Let's solve for k. Let's approximate:\\n\\nIf k=10: 10,000 +20190=30,190 >10,000\\n\\nk=5: 625 +10095=10720 >10,000\\n\\nk=4: 256 +8076=8332 <10,000\\n\\nSo, between 4 and5. Let's try k=4.5:\\n\\nk=4.5: (4.5)^4=4.5*4.5*4.5*4.5= (20.25)*(20.25)=410.0625. Then, 410.0625 +2019*4.5=410.0625 +9085.5\\xe2\\x89\\x889495.5625 <10,000\\n\\nk=4.75:\\n\\nk^4 \\xe2\\x89\\x88(4.75)^4. Let's compute:\\n\\n4.75^2=22.5625\\n\\n4.75^4=(22.5625)^2\\xe2\\x89\\x88508.0625 + 22.5625*2*22.5625 (Wait, actually, (a+b)^2= a\\xc2\\xb2 +2ab +b\\xc2\\xb2. Wait, 22.5625 squared is 22.5625*22.5625.\\n\\nLet me compute 22*22=484, 22*0.5625=12.375, 0.5625*22=12.375, 0.5625*0.5625\\xe2\\x89\\x880.3164. So total:\\n\\n484 +12.375 +12.375 +0.3164\\xe2\\x89\\x88508.0664. So, k^4\\xe2\\x89\\x88508.0664. Then, 2019k=2019*4.75\\xe2\\x89\\x882019*4 +2019*0.75=8076 +1514.25\\xe2\\x89\\x889590.25. So total k^4 +2019k\\xe2\\x89\\x88508.0664 +9590.25\\xe2\\x89\\x8810098.3164. Which is very close to 10,000. So, k\\xe2\\x89\\x884.75 gives k^4 +2019k\\xe2\\x89\\x8810,098.32. So, to get 10,000, k\\xe2\\x89\\x884.75 - a small amount. So, approximately, k\\xe2\\x89\\x884.75, so a\\xe2\\x89\\x88-4.75. Then, the positive root b for c=10,000 would be x\\xe2\\x81\\xb4 -2019x=10,000. Let's see, when x=20: 160,000 -40380=119,620>10,000. When x=15: 50625 -30285=20340>10,000. When x=14: 38416 -28266=10150\\xe2\\x89\\x8810,150. So, x\\xe2\\x89\\x8814. So, b\\xe2\\x89\\x8814. So, S=a +b\\xe2\\x89\\x88-4.75 +14=9.25>0. So, in this case, S is positive. So, depending on c, S can be positive or not? Wait, when c is very small, a is a very small negative number, so S\\xe2\\x89\\x88b>0. When c is large, a is more negative, but b is also larger. Wait, in the previous example, when c=10,000, a\\xe2\\x89\\x88-4.75, b\\xe2\\x89\\x8814, so S\\xe2\\x89\\x889.25. So, S is positive. Wait, is there a scenario where S could be negative?\\n\\nSuppose a is a very large negative number, then b would be a very large positive number. But given that a and b satisfy a\\xe2\\x81\\xb4 -2019a =c and b\\xe2\\x81\\xb4 -2019b =c, how do their magnitudes compare?\\n\\nFor large |x|, x\\xe2\\x81\\xb4 dominates. So, if a is a large negative number, then a\\xe2\\x81\\xb4 \\xe2\\x89\\x88c, so |a|\\xe2\\x89\\x88c^{1/4}. Similarly, b\\xe2\\x81\\xb4\\xe2\\x89\\x88c, so b\\xe2\\x89\\x88c^{1/4}. But for large c, both a and b would be approximately c^{1/4}, but a is negative and b is positive. So, their sum S =a +b would be approximately -c^{1/4} +c^{1/4}=0. But in reality, a is more negative because -2019a adds to a\\xe2\\x81\\xb4. Wait, for a negative, a\\xe2\\x81\\xb4 -2019a =c. So, a\\xe2\\x81\\xb4 + |2019a| =c. Therefore, if a is large in magnitude, the term a\\xe2\\x81\\xb4 dominates, so a\\xe2\\x89\\x88-c^{1/4}. Similarly, for b positive, b\\xe2\\x81\\xb4 -2019b =c. For large b, b\\xe2\\x81\\xb4\\xe2\\x89\\x88c, so b\\xe2\\x89\\x88c^{1/4}. So, S =a +b\\xe2\\x89\\x88-c^{1/4} +c^{1/4}=0. Therefore, for large c, S approaches zero. But in our previous example with c=10,000, S\\xe2\\x89\\x889.25. Hmm, that seems contradictory. Wait, but when c=10,000, c^{1/4}=10. So, a\\xe2\\x89\\x88-10 and b\\xe2\\x89\\x8810. But in reality, with c=10,000, a\\xe2\\x89\\x88-4.75 and b\\xe2\\x89\\x8814. So, S\\xe2\\x89\\x889.25. So, maybe the approximation a\\xe2\\x89\\x88-c^{1/4} is not accurate here because of the linear term. Hmm. So, perhaps S remains positive for all c>0.\\n\\nTherefore, S =a +b is positive. So, S>0.\\n\\nSo, S>0, and we have S^3 -2 S P =2019.\\n\\nAlso, c =P\\xc2\\xb2 - S\\xc2\\xb2 P.\\n\\nOur variables are S and P, with S>0 and P=ab<0.\\n\\nWe need to show that -\\xe2\\x88\\x9ac -\\xe2\\x88\\x9ac.\\n\\nGiven that c=P\\xc2\\xb2 -S\\xc2\\xb2 P, and we need to relate P and \\xe2\\x88\\x9ac.\\n\\nLet me try to manipulate c =P\\xc2\\xb2 -S\\xc2\\xb2 P. Let's write c =P(P - S\\xc2\\xb2). Since P is negative (because ab<0), and S\\xc2\\xb2 is positive, then (P - S\\xc2\\xb2) is negative. So, c = negative * negative = positive, which matches that c is positive.\\n\\nSo, c = P\\xc2\\xb2 - S\\xc2\\xb2 P. Let's rearrange:\\n\\nc = P\\xc2\\xb2 - S\\xc2\\xb2 P\\n\\nLet me think. If we can express this in terms of P and then relate it to \\xe2\\x88\\x9ac.\\n\\nAlternatively, let's consider the equation c = P\\xc2\\xb2 - S\\xc2\\xb2 P and the equation S^3 -2 S P =2019.\\n\\nWe need to find a relation between P and c. Let's try to eliminate S.\\n\\nFrom S^3 -2 S P =2019, solve for S^3 =2019 +2 S P.\\n\\nExpress S in terms of P? Not straightforward.\\n\\nAlternatively, note that c = P\\xc2\\xb2 - S\\xc2\\xb2 P. Let's solve for S\\xc2\\xb2:\\n\\nS\\xc2\\xb2 = (P\\xc2\\xb2 -c)/P\\n\\nBut S\\xc2\\xb2 must be positive. Since P is negative (ab<0), (P\\xc2\\xb2 -c)/P. P\\xc2\\xb2 is positive, c is positive. So, (positive - positive)/negative. Depending on the values, but since S\\xc2\\xb2 must be positive, (P\\xc2\\xb2 -c)/P must be positive. Since P is negative, denominator is negative. Therefore, (P\\xc2\\xb2 -c) must be negative. Therefore, P\\xc2\\xb2 -c <0 \\xe2\\x86\\x92c >P\\xc2\\xb2. Therefore, c >P\\xc2\\xb2. So, since c is positive, and P is negative, then c >P\\xc2\\xb2. Which gives that |P| <\\xe2\\x88\\x9ac. But P=ab is negative, so |P|=-P. Therefore, -P <\\xe2\\x88\\x9ac \\xe2\\x86\\x92 P >-\\xe2\\x88\\x9ac. Which is exactly the lower bound we need to prove.\\n\\nWait, that seems straightforward. Let me check that again.\\n\\nFrom c =P\\xc2\\xb2 - S\\xc2\\xb2 P\\n\\nBut since S\\xc2\\xb2 is positive, and P is negative, then -S\\xc2\\xb2 P is positive. Therefore, c =P\\xc2\\xb2 + positive term. So, c >P\\xc2\\xb2.\\n\\nBut if c >P\\xc2\\xb2, then \\xe2\\x88\\x9ac >|P|. Since P is negative, |P|=-P. Therefore, \\xe2\\x88\\x9ac >-P \\xe2\\x86\\x92 P >-\\xe2\\x88\\x9ac.\\n\\nWhich is the required lower bound: ab > -\\xe2\\x88\\x9ac.\\n\\nTherefore, combining c >P\\xc2\\xb2 and P <0, we get P > -\\xe2\\x88\\x9ac.\\n\\nTherefore, -\\xe2\\x88\\x9ac P\\xc2\\xb2. Taking square roots on both sides (since c and P\\xc2\\xb2 are positive), we get \\xe2\\x88\\x9ac >|P|. Since P is negative, |P|=-P, so \\xe2\\x88\\x9ac >-P \\xe2\\x86\\x92 P >-\\xe2\\x88\\x9ac. Hence, -\\xe2\\x88\\x9ac \\n\\nGiven two distinct real numbers \\\\(a\\\\) and \\\\(b\\\\) and a positive real number \\\\(c\\\\) such that \\\\(a^4 - 2019a = b^4 - 2019b = c\\\\), we need to prove that \\\\(-\\\\sqrt{c} < ab < 0\\\\).\\n\\n1. **Key Equations and Relationships**:\\n - Both \\\\(a\\\\) and \\\\(b\\\\) satisfy the equation \\\\(x^4 - 2019x = c\\\\).\\n - Subtracting the equations for \\\\(a\\\\) and \\\\(b\\\\), we get:\\n \\\\[\\n a^4 - b^4 = 2019(a - b)\\n \\\\]\\n - Factoring the left-hand side:\\n \\\\[\\n (a^2 - b^2)(a^2 + b^2) = 2019(a - b)\\n \\\\]\\n - Since \\\\(a \\\\neq b\\\\), we can divide both sides by \\\\(a - b\\\\):\\n \\\\[\\n (a + b)(a^2 + b^2) = 2019\\n \\\\]\\n\\n2. **Introducing Variables**:\\n - Let \\\\(S = a + b\\\\) and \\\\(P = ab\\\\).\\n - Express \\\\(a^2 + b^2\\\\) in terms of \\\\(S\\\\) and \\\\(P\\\\):\\n \\\\[\\n a^2 + b^2 = S^2 - 2P\\n \\\\]\\n - Substituting into the key equation:\\n \\\\[\\n S(S^2 - 2P) = 2019 \\\\implies S^3 - 2SP = 2019\\n \\\\]\\n\\n3. **Expressing \\\\(c\\\\) in Terms of \\\\(S\\\\) and \\\\(P\\\\)**:\\n - Using the original equations \\\\(a^4 - 2019a = c\\\\) and \\\\(b^4 - 2019b = c\\\\), we derive:\\n \\\\[\\n a^4 + b^4 = 2c + 2019(a + b)\\n \\\\]\\n - Express \\\\(a^4 + b^4\\\\) in terms of \\\\(S\\\\) and \\\\(P\\\\):\\n \\\\[\\n a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2 = (S^2 - 2P)^2 - 2P^2\\n \\\\]\\n - Simplifying and substituting:\\n \\\\[\\n S^4 - 4S^2P + 2P^2 = 2c + 2019S\\n \\\\]\\n - Using \\\\(S^3 = 2019 + 2SP\\\\), we substitute \\\\(S^4\\\\) and simplify to find:\\n \\\\[\\n c = P^2 - S^2P\\n \\\\]\\n\\n4. **Bounding \\\\(ab\\\\)**:\\n - Since \\\\(c = P^2 - S^2P\\\\) and \\\\(S^2\\\\) is positive, \\\\(c > P^2\\\\) because \\\\(-S^2P\\\\) adds a positive term to \\\\(P^2\\\\).\\n - Taking square roots, \\\\(\\\\sqrt{c} > |P|\\\\). Since \\\\(P = ab\\\\) is negative, this implies:\\n \\\\[\\n |P| < \\\\sqrt{c} \\\\implies -\\\\sqrt{c} < P < 0\\n \\\\]\\n\\nThus, we conclude that \\\\(-\\\\sqrt{c} < ab < 0\\\\).\\n\\n\\\\[\\n\\\\boxed{-\\\\sqrt{c} < ab < 0}\\n\\\\]\"]", "default/generations_count": 2, "default/correctness": {"default/correctness/llama_verification": "[True, True, True]", "default/correctness/math_verify_answer": "[True, True, True]", "default/correctness/math_verify_reparsed_answer": "[True, True, True]"}, "default/reparsed_answers": "None"}