{"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":"f57fcfb3-ecee-54f8-b5c5-65c35304be1e","task_key":"test--f57fcfb3-ecee-54f8-b5c5-65c35304be1e","task_revision_id":"1","upstream_id":"","short_description":"A binary tree is a tree in which each node has exactly two descendants. Suppose…","config":"","split":"test","body":"{\"problem\":\"A binary tree is a tree in which each node has exactly two descendants. Suppose that each node of the tree is coloured black with probability \\\\(p\\\\), and white otherwise, independently of all other nodes. For any path \\\\(\\\\pi\\\\) containing \\\\(n\\\\) nodes beginning at the root of the tree, let \\\\(B(\\\\pi)\\\\) be the number of black nodes in \\\\(\\\\pi\\\\), and let \\\\(X_{n}(k)\\\\) be the number of such paths \\\\(\\\\pi\\\\) for which \\\\(B(\\\\pi) \\\\geq k\\\\). (1) Show that there exists \\\\(\\\\beta_{c}\\\\) such that \\\\(\\\\lim _{n \\\\rightarrow \\\\infty} \\\\mathbb{E}\\\\left(X_{n}(\\\\beta n)\\\\right)= \\\\begin{cases}0, & \\\\text { if } \\\\beta>\\\\beta_{c} \\\\\\\\ \\\\infty, & \\\\text { if } \\\\beta<\\\\beta_{c}\\\\end{cases}\\\\) How to determine the value of \\\\(\\\\beta_{c}\\\\) ? (2) For \\\\(\\\\beta \\\\neq \\\\beta_{c}\\\\), find the limit \\\\(\\\\lim _{n \\\\rightarrow \\\\infty} \\\\mathbb{P}\\\\left(X_{n}(\\\\beta n) \\\\geq 1\\\\right)\\\\).\"}","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":[]}