{"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":"57bba79d-fdb2-58be-9fac-642b0cf59682","task_key":"precalculus--test--57bba79d-fdb2-58be-9fac-642b0cf59682","task_revision_id":"1","upstream_id":"","short_description":"Let $ABCD$ be a convex quadrilateral, and let $M$ and $N$ be the midpoints of…","config":"precalculus","split":"test","body":"{\"level\":\"Level 3\",\"problem\":\"Let $ABCD$ be a convex quadrilateral, and let $M$ and $N$ be the midpoints of $\\\\overline{AC}$ and $\\\\overline{BD},$ respectively.  Then there exists a constant $k$ so that\\n\\\\[AB^2 + BC^2 + CD^2 + DA^2 = AC^2 + BD^2 + k \\\\cdot MN^2.\\\\]Find $k.$\\n\\n[asy]\\nunitsize(0.8 cm);\\n\\npair A, B, C, D, M, N;\\n\\nA = (0,0);\\nB = (4,0.5);\\nC = (5,-3);\\nD = (-2,-2.5);\\nM = (A + C)/2;\\nN = (B + D)/2;\\n\\ndraw(A--B--C--D--cycle);\\ndraw(A--C);\\ndraw(B--D);\\ndraw(M--N);\\n\\nlabel(\\\"$A$\\\", A, NW);\\nlabel(\\\"$B$\\\", B, NE);\\nlabel(\\\"$C$\\\", C, SE);\\nlabel(\\\"$D$\\\", D, SW);\\ndot(\\\"$M$\\\", M, NE);\\ndot(\\\"$N$\\\", N, NW);\\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":[]}