{"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":"96288173-e0b2-567b-86f6-3b28ce6b8a8b","task_key":"geometry--test--96288173-e0b2-567b-86f6-3b28ce6b8a8b","task_revision_id":"1","upstream_id":"","short_description":"In the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are…","config":"geometry","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"In the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sides of $\\\\triangle ABC$, as shown. [asy]\\nsize(250);\\npair A, B, C, P, Q, R, S;\\nR=(0,0);\\nQ=(-2,0);\\nS=(2,0);\\nP=(1,1.732);\\nB=(-5.73,-1);\\nC=(3.732,-1);\\nA=(1.366,3.098);\\ndraw(A--B--C--A);\\ndraw(circle(P, 1));\\ndraw(circle(Q, 1));\\ndraw(circle(R, 1));\\ndraw(circle(S, 1));\\nlabel(\\\"A\\\", A, N);\\nlabel(\\\"B\\\", B, SW);\\nlabel(\\\"C\\\", C, SE);\\ndot(P);\\ndot(Q);\\ndot(R);\\ndot(S);\\nlabel(\\\"P\\\", P, N);\\nlabel(\\\"Q\\\", Q, SW);\\nlabel(\\\"R\\\", R, SW);\\nlabel(\\\"S\\\", S, SE);\\n[/asy]\\n\\nThe radius of the circle with center $R$ is decreased so that\\n\\n$\\\\bullet$ the circle with center $R$ remains tangent to $BC$,\\n\\n$\\\\bullet$ the circle with center $R$ remains tangent to the other three circles, and\\n\\n$\\\\bullet$ the circle with center $P$ becomes tangent to the other three circles.\\n\\nThe radii and tangencies of the other three circles stay the same. This changes the size and shape of $\\\\triangle ABC$. $r$ is the new radius of the circle with center $R$. $r$ is of the form $\\\\frac{a+\\\\sqrt{b}}{c}$. Find $a+b+c$.\",\"type\":\"Geometry\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}