{"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":"800f6e35-3b97-5749-94cc-f9c495055885","task_key":"test--800f6e35-3b97-5749-94cc-f9c495055885","task_revision_id":"1","upstream_id":"","short_description":"Let \\(\\left\\{X_{n}\\right\\}_{n \\geq 1}\\) be i.i.d. random variables such that…","config":"","split":"test","body":"{\"problem\":\"Let \\\\(\\\\left\\\\{X_{n}\\\\right\\\\}_{n \\\\geq 1}\\\\) be i.i.d. random variables such that \\\\(\\\\mathbb{P}\\\\left(X_{1}=1\\\\right)=1-\\\\mathbb{P}\\\\left(X_{1}=-1\\\\right)=p>\\\\frac{1}{2}\\\\). Let \\\\(S_{0}=0, S_{n}=\\\\sum_{i=1}^{n} X_{i}\\\\). Define the range of \\\\(\\\\left\\\\{S_{n}\\\\right\\\\}_{n \\\\geq 0}\\\\) by \\\\(R_{n}=\\\\#\\\\left\\\\{S_{0}, S_{1}, S_{2}, \\\\cdots, S_{n}\\\\right\\\\}\\\\), which is the number of distinct points visited by the random walk \\\\(\\\\left\\\\{S_{n}\\\\right\\\\}_{n \\\\geq 0}\\\\) up to time \\\\(n\\\\). (1) Prove \\\\(\\\\mathbb{E}\\\\left(R_{n}\\\\right)=\\\\mathbb{E}\\\\left(R_{n-1}\\\\right)+P\\\\left(S_{1} S_{2} \\\\cdots S_{n} \\\\neq 0\\\\right), \\\\quad n=1,2, \\\\cdots\\\\). (2) Find \\\\(\\\\lim _{n \\\\rightarrow \\\\infty} \\\\frac{1}{n} \\\\mathbb{E}\\\\left(R_{n}\\\\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":[]}