{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"math","formal_name":"MATH","introduction":"MATH organises 12,500 high-school competition problems into seven subjects and five difficulty levels, each with a full worked solution. It underlies much of the later mathematics evaluation work; this catalogue imports the full test split: 5,000 problems across all seven subjects.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://github.com/hendrycks/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":"27469141-959f-590b-ac7d-56a6fa50885f","task_key":"intermediate~5falgebra--test--27469141-959f-590b-ac7d-56a6fa50885f","task_revision_id":"1","upstream_id":"","short_description":"For $1 \\leq i \\leq 215$ let $a_i = \\dfrac{1}{2^{i}}$ and $a_{216} =…","config":"intermediate_algebra","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"For $1 \\\\leq i \\\\leq 215$ let $a_i = \\\\dfrac{1}{2^{i}}$ and $a_{216} = \\\\dfrac{1}{2^{215}}$. Let $x_1, x_2, \\\\dots, x_{216}$ be positive real numbers such that $\\\\sum_{i=1}^{216} x_i=1$ and\\n\\\\[\\\\sum_{1 \\\\leq i < j \\\\leq 216} x_ix_j = \\\\dfrac{107}{215} + \\\\sum_{i=1}^{216} \\\\dfrac{a_i x_i^{2}}{2(1-a_i)}.\\\\]Find the maximum possible value of $x_2.$\",\"type\":\"Intermediate Algebra\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}