{"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":"7a21b010-aa45-5dba-b8e3-fd23b7498f15","task_key":"algebra--test--7a21b010-aa45-5dba-b8e3-fd23b7498f15","task_revision_id":"1","upstream_id":"","short_description":"The complete graph of $y=f(x),$ which consists of five line segments, is shown…","config":"algebra","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"The complete graph of $y=f(x),$ which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1.$)\\n\\nLet $a$ and $b$ be the largest negative integer and the smallest positive integer, respectively, such that the functions $g(x)=f(x)+ax$ and $h(x)=f(x)+bx$ are invertible. What is $a^2+b^2?$\\n\\n[asy]\\nsize(150);\\nreal ticklen=3;\\nreal tickspace=2;\\n\\nreal ticklength=0.1cm;\\nreal axisarrowsize=0.14cm;\\npen axispen=black+1.3bp;\\nreal vectorarrowsize=0.2cm;\\nreal tickdown=-0.5;\\nreal tickdownlength=-0.15inch;\\nreal tickdownbase=0.3;\\nreal wholetickdown=tickdown;\\nvoid rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {\\n\\nimport graph;\\n\\nreal i;\\n\\nif(complexplane) {\\n\\nlabel(\\\"$\\\\textnormal{Re}$\\\",(xright,0),SE);\\n\\nlabel(\\\"$\\\\textnormal{Im}$\\\",(0,ytop),NW);\\n\\n} else {\\n\\nlabel(\\\"$x$\\\",(xright+0.4,-0.5));\\n\\nlabel(\\\"$y$\\\",(-0.5,ytop+0.2));\\n\\n}\\n\\nylimits(ybottom,ytop);\\n\\nxlimits( xleft, xright);\\n\\nreal[] TicksArrx,TicksArry;\\n\\nfor(i=xleft+xstep; i<xright; i+=xstep) {\\n\\nif(abs(i) >0.1) {\\n\\nTicksArrx.push(i);\\n\\n}\\n\\n}\\n\\nfor(i=ybottom+ystep; i<ytop; i+=ystep) {\\n\\nif(abs(i) >0.1) {\\n\\nTicksArry.push(i);\\n\\n}\\n\\n}\\n\\nif(usegrid) {\\n\\nxaxis(BottomTop(extend=false), Ticks(\\\"%\\\", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);\\n\\nyaxis(LeftRight(extend=false),Ticks(\\\"%\\\", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);\\n\\n}\\n\\nif(useticks) {\\n\\nxequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks(\\\"%\\\",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));\\n\\nyequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks(\\\"%\\\",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));\\n\\n} else {\\n\\nxequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));\\n\\nyequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));\\n\\n}\\n};\\nrr_cartesian_axes(-5,5,-5,5);\\ndraw((-5,-4)--(-2,5)--(-1,3)--(1,-5)--(3,2)--(5,2),red+1);\\ndot((-5,-4),red);\\ndot((-2,5),red);\\ndot((-1,3),red);\\ndot((1,-5),red);\\ndot((3,2),red);\\ndot((5,2),red);\\n[/asy]\",\"type\":\"Algebra\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}