{"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":"3930aa0a-d90e-533e-b1c7-4c947073039a","task_key":"geometry--test--3930aa0a-d90e-533e-b1c7-4c947073039a","task_revision_id":"1","upstream_id":"","short_description":"A hexagon is drawn with its vertices at $$(0,0),(1,0),(2,1),(2,2),(1,2), \\text{…","config":"geometry","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"A hexagon is drawn with its vertices at $$(0,0),(1,0),(2,1),(2,2),(1,2), \\\\text{ and } (0,1),$$ and all of its diagonals are also drawn, as shown below. The diagonals cut the hexagon into $24$ regions of various shapes and sizes. These $24$ regions are shown in pink and yellow below. If the smallest region (by area) has area $a$, and the largest has area $b$, then what is the ratio $a:b$? Give your answer in lowest terms. [asy]\\npair a=(0,0); pair b=(1,0); pair c=(2,1); pair d=(2,2); pair e=(1,2); pair f=(0,1);\\npair g=(1,1);\\npair h=(a+g)/2; pair i=(2*h+b)/3; pair j=(b+g)/2; pair k=(2*j+c)/3; pair l=(c+g)/2; pair m=(2*l+d)/3;\\npair n=2*g-h; pair o=2*g-i; pair p=2*g-j; pair q=2*g-k; pair r=2*g-l; pair s=2*g-m;\\nfill(a--h--i--cycle,pink);\\nfill(a--i--b--cycle,yellow);\\nfill(i--b--j--cycle,pink);\\nfill(j--b--k--cycle,yellow);\\nfill(b--k--c--cycle,pink);\\nfill(k--c--l--cycle,yellow);\\nfill(l--c--m--cycle,pink);\\nfill(m--c--d--cycle,yellow);\\nfill(m--d--n--cycle,pink);\\nfill(n--d--o--cycle,yellow);\\nfill(o--d--e--cycle,pink);\\nfill(o--e--p--cycle,yellow);\\nfill(p--e--q--cycle,pink);\\nfill(q--e--f--cycle,yellow);\\nfill(f--r--q--cycle,pink);\\nfill(f--r--s--cycle,yellow);\\nfill(f--s--a--cycle,pink);\\nfill(a--s--h--cycle,yellow);\\nfill(g--r--q--p--cycle,yellow);\\nfill(g--p--o--n--cycle,pink);\\nfill(g--n--m--l--cycle,yellow);\\nfill(g--l--k--j--cycle,pink);\\nfill(g--j--i--h--cycle,yellow);\\nfill(g--h--s--r--cycle,pink);\\ndraw(a--b--c--d--e--f--a,black+2);\\ndraw(a--c--e--a);\\ndraw(b--d--f--b);\\ndraw(a--d);\\ndraw(b--e);\\ndraw(c--f);\\n[/asy]\",\"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":[]}