{"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":"4cfc045a-6bc6-5e7f-8754-56181da5421a","task_key":"precalculus--test--4cfc045a-6bc6-5e7f-8754-56181da5421a","task_revision_id":"1","upstream_id":"","short_description":"There exist two complex numbers $c$, say $c_1$ and $c_2$, so that $-5 + 3i$, $8…","config":"precalculus","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"There exist two complex numbers $c$, say $c_1$ and $c_2$, so that $-5 + 3i$, $8 - i$, and $c$ form the vertices of an equilateral triangle.  Find the product $c_1 c_2$.\\n\\n[asy]\\nunitsize(0.5 cm);\\n\\npair A, B;\\npair[] C;\\n\\nA = (2,2);\\nB = (5,1);\\nC[1] = rotate(60,A)*(B);\\nC[2] = rotate(60,B)*(A);\\n\\ndraw(A--C[1]--B--C[2]--cycle);\\ndraw(A--B);\\n\\ndot(\\\"$-5 + 3i$\\\", A, W);\\ndot(\\\"$8 - i$\\\", B, E);\\ndot(\\\"$c_1$\\\", C[1], N);\\ndot(\\\"$c_2$\\\", C[2], S);\\n[/asy]\",\"type\":\"Precalculus\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}