{"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":"42d871f3-dd46-5b00-b02b-d081f87d2328","task_key":"intermediate~5falgebra--test--42d871f3-dd46-5b00-b02b-d081f87d2328","task_revision_id":"1","upstream_id":"","short_description":"The foci of the ellipse $\\frac{x^2}{49} + \\frac{y^2}{33} = 1$ are $F_1$ and…","config":"intermediate_algebra","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"The foci of the ellipse $\\\\frac{x^2}{49} + \\\\frac{y^2}{33} = 1$ are $F_1$ and $F_2,$ as shown below.  Let $P$ be a point on the circle $x^2 + (y - 3)^2 = 4.$  Line $F_2 P$ intersects the ellipse again at $Q,$ where the $y$-coordinate of $Q$ is positive.  Find the maximum value of $PQ + F_1 Q.$\\n\\n[asy]\\nunitsize(0.4 cm);\\n\\npair P, Q;\\npair[] F;\\n\\npath ell = yscale(sqrt(33))*xscale(7)*Circle((0,0),1);\\nF[1] = (4,0);\\nF[2] = (-4,0);\\nP = (0,3) + 2*dir(240);\\nQ = intersectionpoint(P--interp(F[2],P,5),ell);\\n\\ndraw(ell);\\ndraw((-8,0)--(8,0));\\ndraw((0,-7)--(0,7));\\ndraw(Circle((0,3),2));\\ndraw(F[1]--Q--F[2]);\\n\\ndot(\\\"$F_1$\\\", F[1], S);\\ndot(\\\"$F_2$\\\", F[2], S);\\ndot(\\\"$P$\\\", P, S);\\nlabel(\\\"$Q$\\\", Q, NE);\\n[/asy]\",\"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":[]}