{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"omni-math","formal_name":"Omni-MATH","introduction":"Omni-MATH evaluates mathematical reasoning on Olympiad-level problems. Its official dataset contains 4,428 problems accompanied by domain and difficulty information.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://huggingface.co/datasets/KbsdJames/Omni-MATH","indexing_mode":"index","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[],"history":"","construction":"","limitations":"","saturation":"","scores":[]}},"task_id":"f8642477-f53a-57e6-bac0-f2000b7305e5","task_key":"test--f8642477-f53a-57e6-bac0-f2000b7305e5","task_revision_id":"1","upstream_id":"","short_description":"Let $n > 2$ be an integer and let $\\ell \\in \\{1, 2,\\dots, n\\}$. A collection…","config":"","split":"test","body":"{\"problem\":\"Let $n > 2$ be an integer and let $\\\\ell \\\\in \\\\{1, 2,\\\\dots, n\\\\}$. A collection $A_1,\\\\dots,A_k$ of (not necessarily distinct) subsets of $\\\\{1, 2,\\\\dots, n\\\\}$ is called $\\\\ell$-large if $|A_i| \\\\ge \\\\ell$ for all $1 \\\\le i \\\\le k$. Find, in terms of $n$ and $\\\\ell$, the largest real number $c$ such that the inequality\\n\\\\[  \\\\sum_{i=1}^k\\\\sum_{j=1}^k x_ix_j\\\\frac{|A_i\\\\cap A_j|^2}{|A_i|\\\\cdot|A_j|}\\\\ge c\\\\left(\\\\sum_{i=1}^k x_i\\\\right)^2 \\\\]\\nholds for all positive integer $k$, all nonnegative real numbers $x_1,x_2,\\\\dots,x_k$, and all $\\\\ell$-large collections $A_1,A_2,\\\\dots,A_k$ of subsets of $\\\\{1,2,\\\\dots,n\\\\}$.\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://huggingface.co/datasets/KbsdJames/Omni-MATH","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}