{"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":"61fea923-c617-5e6f-86d1-328d1202b7cf","task_key":"test--61fea923-c617-5e6f-86d1-328d1202b7cf","task_revision_id":"1","upstream_id":"","short_description":"We say that a finite set $\\mathcal{S}$ of points in the plane is [i]balanced[/i]…","config":"","split":"test","body":"{\"problem\":\"We say that a finite set $\\\\mathcal{S}$ of points in the plane is [i]balanced[/i] if, for any two different points $A$ and $B$ in $\\\\mathcal{S}$, there is a point $C$ in $\\\\mathcal{S}$ such that $AC=BC$. We say that $\\\\mathcal{S}$ is [i]centre-free[/i] if for any three different points $A$, $B$ and $C$ in $\\\\mathcal{S}$, there is no points $P$ in $\\\\mathcal{S}$ such that $PA=PB=PC$.\\n\\n(a) Show that for all integers $n\\\\ge 3$, there exists a balanced set consisting of $n$ points.\\n\\n(b) Determine all integers $n\\\\ge 3$ for which there exists a balanced centre-free set consisting of $n$ points.\"}","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":[]}