id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-b2f7cbd2cfdfe5bc201c4cabe006b04efbd602b557863631ba96c60a3f2f2b67 | $4\sin ^4 3x-3\cos x+5=0$. | [
"$\\emptyset $"
] | [
"$4\\sin ^4 3x-3\\cos x+5=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5cd107a901eb1541b92c11582d176a90e9f684fc5692fc50ecd8774cb30d924a | $4\sin^2 x+\sin^2 2x=3$. | [
"$\\frac{\\pi }{4}+\\frac{\\pi }{2}k$"
] | [
"$4\\sin^2 x+\\sin^2 2x=3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bdf379b1fff8c70aa7c7e57ff8ca55952998bf68bed2d78632da78f4ce289689 | $4\tan \frac{x}{2}+\cot x=\frac{5}{\sin x}$. | [
"$\\emptyset $"
] | [
"$4\\tan \\frac{x}{2}+\\cot x=\\frac{5}{\\sin x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8efc7e8387cc101caa8c62458242b86a98c69d20b925f4b4e5e548e1cae77e3 | $4^x-4m2^x+2m+2=0$. | [
"$x=\\log _2 \\left(2m+\\sqrt {4m^2-2m-2}\\right)$ for $m<-1$; $x=1$ for $m=1$; $x_{1,2}=\\log _2 \\left(2m\\pm \\sqrt {4m^2-2m-2}\\right)$ for $m>1$; there are no solutions for the other values of $m$. Find the positive roots of the quadratic $y^2-4my+2m+2$. The discriminant $D=4m^2-2m-2$ is nonnegative for $m\\le... | [
"$4^x-4m2^x+2m+2=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f8247de712a114739dec899a39a20c515e138ac4c69f0db268130501ca4900c7 | $4^{\sin x}+m2^{\sin x}+m^2-1=0$, where $-1<m<1$. | [
"$x=(-1)^k \\arcsin \\log _2\\left[\\left(-m+\\sqrt {4-3m^3}\\right)/2\\right]+k\\pi $. Find the roots of the quadratic $f(y)=y^2+my+m^2-1$ such that $1/2\\le y\\le 2$. By virtue of $-1<m<1$ the discriminant $D=4-3m^2>0$ and the constant term is negative, so we can only be interested in the greater root. It does no... | [
"$4^{\\sin x}+m2^{\\sin x}+m^2-1=0$, where $-1<m<1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-287cea59dc3a136e6bf91886f9346f87180c75bc419b4dba3be6314e255019b9 | $4x^4+x^6=-\sin ^2 5x$. | [
"0"
] | [
"$4x^4+x^6=-\\sin ^2 5x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-50a2d65f934319a74978f92477c446b5f60ae42b3941edae24b48a30844c0e9a | $5\cos 2x=4\sin x$ | [
"$x=\\pi n+(-1)^n \\arcsin\\frac{-2\\pm \\sqrt {54}}{10}$"
] | [
"$5\\cos 2x=4\\sin x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da587bfe348b4ebf45dffd4c7ebe6401132eeb1be69a0a10e50c43f2a4cbc6b8 | $5\sin 2x-11(\sin x+\cos x)+7=0$. | [
"$\\frac{\\pi }{4}\\pm \\arccos \\frac{\\sqrt 2}{10}+2\\pi k$"
] | [
"$5\\sin 2x-11(\\sin x+\\cos x)+7=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-36ecabec7ade390df47a2ce5444f32699d1ae2143ebe0562788ee8b552aea39d | $5^{2023}$ is a 1415-digit number that starts with 1. How many smaller powers of 5 (from $5^0$ to $5^{2022}$ inclusive) also start with 1? | [
"608"
] | [
"$5^{2023}$ is a 1415-digit number that starts with 1. How many smaller powers of 5 (from $5^0$ to $5^{2022}$ inclusive) also start with 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-53e9f3de9563f7be1a9ea7d498d62b67d433d42e84b823b11325cd79a93e5cfb | $5x^2+5y^2+8xy+2x-2y+2=0$. | [
"$x=-1$, $y=1$"
] | [
"$5x^2+5y^2+8xy+2x-2y+2=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a47b58c73b72d4d24fee68d296ea7a0f23530479908a923c725bdf0c61b96d75 | $6$ straight lines are given in the plane. Let's look at all regular triangles whose sides lie on straight lines. What is the largest possible number of such triangles with no congruent triangles between them? | [
"8"
] | [
"$6$ straight lines are given in the plane. Let's look at all regular triangles whose sides lie on straight lines. What is the largest possible number of such triangles with no congruent triangles between them?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-07fd6311f55f1e445c357296b27ff5e13a5fd2a79785c66fa86f33d8d947cf0d | $6:\frac{1}{3}-0.8:\frac{1.5}{\frac{3}{2}\cdot 0.4\cdot \frac{50}{1:\frac{1}{2}}}+\frac{1}{4} +\frac{1+\frac{1}{2}\cdot \frac{1}{0.25}} {6-\frac{46}{1+2.2\cdot 10}}$ | [
"11"
] | [
"$6:\\frac{1}{3}-0.8:\\frac{1.5}{\\frac{3}{2}\\cdot 0.4\\cdot \\frac{50}{1:\\frac{1}{2}}}+\\frac{1}{4} +\\frac{1+\\frac{1}{2}\\cdot \\frac{1}{0.25}} {6-\\frac{46}{1+2.2\\cdot 10}}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-10634f59918a554e5cab9ef7ff7b8e7ffe0f57638efe60694c8b7dda32c8f572 | $6\sin^2 x+3\sin x\cos x-5\cos^2 x=2$ | [
"$x=\\pi n+\\frac{\\pi }{4};\\ x=\\pi n-\\arctan\\frac{7}{4}$"
] | [
"$6\\sin^2 x+3\\sin x\\cos x-5\\cos^2 x=2$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-464763e5a249d1fa6cdbe3a680a029867ea960f3cacc83b151e97da5a62cd886 | $6\tan x+5\cot 3x=\tan 2x$. | [
"$\\pm \\frac{1}{2}\\arccos \\frac{1}{3}+\\pi k;\\ \\pm \\frac{1}{2}\\arccos \\left(-\\frac{1}{4}\\right)+\\pi n$"
] | [
"$6\\tan x+5\\cot 3x=\\tan 2x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4f91766b000e445afdda1d1e23d5eddca0737eb3b49a4be4e4a77392681305e | $77$ stones weighing $1, 2, \dots, 77$ grams are divided into $k$ groups such that the total weights of each group are different from each other and each group contains fewer stones than groups with smaller total weights. For how many $k \in \{9, 10, 11, 12\}$ is such a division possible? | [
"None of above"
] | [
"$77$ stones weighing $1, 2, \\dots, 77$ grams are divided into $k$ groups such that the total weights of each group are different from each other and each group contains fewer stones than groups with smaller total weights. For how many $k \\in \\{9, 10, 11, 12\\}$ is such a division possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a9316ecd0a91a75cba0d274f83680d63a4f3bcb0e9b07363a7a8328653dbe54 | $8712$ is an integral multiple of its reversal, $2178$, as $8712 = 4 \times 2178$. Find another $4$-digit number which is a non-trivial integral multiple of its reversal. | [
"9801"
] | [
"$8712$ is an integral multiple of its reversal, $2178$, as $8712 = 4 \\times 2178$. Find another $4$-digit number which is a non-trivial integral multiple of its reversal."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e217397f82584048f73dbf41b1ebfebc3f256d7b0c8bb1f7592f01d3feed6ea6 | $8\cos^4 x-8\cos^2 x-\cos x+1=0$. | [
"$\\frac{2\\pi }{3}k;\\ \\pm \\arccos \\frac{-1\\pm \\sqrt 5}{4}+2\\pi n$"
] | [
"$8\\cos^4 x-8\\cos^2 x-\\cos x+1=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da9af6f87dcd0a8c3738acbc2ee374a3d911b757069baedd0d47f22e5b42f340 | $8\sin ^2 \frac{x}{2}-3\sin x-4=0$. | [
"$-\\arctan \\frac{4}{3}+\\pi k$"
] | [
"$8\\sin ^2 \\frac{x}{2}-3\\sin x-4=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f266c87631abf5f147f9ce29b6e6eaa8d21e751a619630442e041ecb91c8f952 | $8\sin^6 x+3\cos 2x+2\cos 4x+1=0$. | [
"$\\frac{\\pi }{4}+\\frac{\\pi }{2}k$"
] | [
"$8\\sin^6 x+3\\cos 2x+2\\cos 4x+1=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a27fc9eb1704a1a25f445fe7f62253ea1b5eeb4b7c037f32403784fdfaf9f85 | $8\tan^2 \frac{x}{2}=1+\sec x$ | [
"$x=2\\pi n\\pm \\arccos \\frac{1}{3}$"
] | [
"$8\\tan^2 \\frac{x}{2}=1+\\sec x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b9b4bab79d7099b3061291f218c58048cefef26367352234950aea2bc17eacfd | $99$ rays coming from one point $M$ are drawn on the plane. Among these rays, there are two forming an obtuse angle, and no ray is drawn inside this obtuse angle. What is the largest number of obtuse angles that these rays can form? | [
"2352"
] | [
"$99$ rays coming from one point $M$ are drawn on the plane. Among these rays, there are two forming an obtuse angle, and no ray is drawn inside this obtuse angle. What is the largest number of obtuse angles that these rays can form?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44436eef029f8b8099cffe7d798b9f3cd7bde4be57ec1047db9dad97200d125f | $9ax + 7y = 12$ and $4x + 2by = 9$ represent the same line. $a$ and $b$ can both be expressed as fractions in lowest terms. What is the numerator of $a$ plus the denominator of $b$ equal to? | [
"24"
] | [
"$9ax + 7y = 12$ and $4x + 2by = 9$ represent the same line. $a$ and $b$ can both be expressed as fractions in lowest terms. What is the numerator of $a$ plus the denominator of $b$ equal to?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-84374b76cf964f5ffe574d4f513e5d3a44d010a2ff12c2ca8707c7741b700d55 | $A BCD$ is a parallelogram, $O$ is its center, $Q$ is an arbitrary point of space. Express the vector $\overrightarrow{QO}$ in terms of the vectors $\overrightarrow{QA}={\bf a}$, $\overrightarrow{CD}={\bf b}$ and $\overrightarrow{AD}={\bf c}$. | [
"$\\overrightarrow{QO}={\\bf a}-\\frac{1}{2}{\\bf b}+\\frac{1}{2}{\\bf c}$"
] | [
"$A BCD$ is a parallelogram, $O$ is its center, $Q$ is an arbitrary point of space. Express the vector $\\overrightarrow{QO}$ in terms of the vectors $\\overrightarrow{QA}={\\bf a}$, $\\overrightarrow{CD}={\\bf b}$ and $\\overrightarrow{AD}={\\bf c}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4be11517b4cd92e2a5b1657feba6458cad0f850113f0b63c81502ebfca51630 | $A$ and $B$ are intersection points of circles $S_1$ and $S_2$ such that the center of the circle $S_1$ lies on the circle $S_2$ and the center of the circle $S_2$ lies on the circle $S_1$. Point $C$, different from point $B$, is chosen on the circle $S_1$ such that $AC=AB$. Find the angles of triangle $ABC$. | [
"60^\\circ"
] | [
"$A$ and $B$ are intersection points of circles $S_1$ and $S_2$ such that the center of the circle $S_1$ lies on the circle $S_2$ and the center of the circle $S_2$ lies on the circle $S_1$. Point $C$, different from point $B$, is chosen on the circle $S_1$ such that $AC=AB$. Find the angles of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d4b2ce497b84fd4dcfdb51568a7e705b39ea72eecdce8fb5ff49bf285b017f12 | $A$ and $B$ are playing a game. $A$ chooses a collection of $n$ points in the plane. $B$ colors each of these points red or blue. $A$ then tries to draw $l$ lines dividing the plane into regions, such that no region contains two points of different colors. If $A$ can do so, $A$ wins. What is the largest $n$ for which p... | [
"3"
] | [
"$A$ and $B$ are playing a game. $A$ chooses a collection of $n$ points in the plane. $B$ colors each of these points red or blue. $A$ then tries to draw $l$ lines dividing the plane into regions, such that no region contains two points of different colors. If $A$ can do so, $A$ wins. What is the largest $n$ for wh... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-4c19020fb390957b3f2fe39fd3a2d8e7a6635e5d540836b0270c15d3335bb679 | $A$ and $B$ have a duel with three equally likely scenarios: $A$ shoots first and $B$ second, $B$ shoots first and $A$ second, or they shoot simultaneously. The probability that $A$ strikes is $0.7$, and the probability that $B$ strikes is $0.8$. What is the probability that $A$ will be killed? | [
"\\( 0.6133 \\)"
] | [
"$A$ and $B$ have a duel with three equally likely scenarios: $A$ shoots first and $B$ second, $B$ shoots first and $A$ second, or they shoot simultaneously. The probability that $A$ strikes is $0.7$, and the probability that $B$ strikes is $0.8$. What is the probability that $A$ will be killed?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 30
} | 33 | 0 | 0 |
omr-ws-bdbb9b810b8b884417cd9cebe973d581c71d630540d084f0a810f7259e812a83 | $A$ and $B$ play a game where $A$ chooses a positive integer $n$ from the set $S = \{1, 2, \dots, 1001\}$. $B$ must guess the value of $n$ by selecting subsets of $S$ and asking how many of these subsets contain $n$. $B$ can do this three times, selecting $k_1$, $k_2$, and $k_3$ subsets each time. What is the least val... | [
"28"
] | [
"$A$ and $B$ play a game where $A$ chooses a positive integer $n$ from the set $S = \\{1, 2, \\dots, 1001\\}$. $B$ must guess the value of $n$ by selecting subsets of $S$ and asking how many of these subsets contain $n$. $B$ can do this three times, selecting $k_1$, $k_2$, and $k_3$ subsets each time. What is the l... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 5
} | 53 | 0 | 0 |
omr-ws-82201a70688ebdd388a5ccd4cf2a102cf6dc894f801102ce9b404eef890fe04c | $A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has traveled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, then what is the circumference of the track in ... | [
"480"
] | [
"$A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has traveled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, then what is the circumference of the trac... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 80,
"tir": 4
} | 84 | 0 | 0 |
omr-ws-0fc8b815479fd497fc5d8b877ec37ff680af08ac6d708710f7fabc02558de268 | $A$ can finish a job $t$ days later than $B$ and $T$ days later than $C$; $A$ and $B$, working together, can do the job in as many days as $C$ working alone. How long does it take each of them to complete the job if they work separately? For what ratio of $t$ and $T$ does the problem have a solution? | [
"$T+\\sqrt {T(T-t)},\\ T-t+\\sqrt {T(T-t)},\\ \\sqrt {T(T-t)},\\ (T>t)$"
] | [
"$A$ can finish a job $t$ days later than $B$ and $T$ days later than $C$; $A$ and $B$, working together, can do the job in as many days as $C$ working alone. How long does it take each of them to complete the job if they work separately? For what ratio of $t$ and $T$ does the problem have a solution?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3515297bba857d2d4f888b3490275a427c751aaf668af5f57a2bd05fa41370ae | $A$ chooses a natural number $x$ such that $1 \le x \le 100$. At every step, $B$ chooses two natural numbers $m$ and $n$ (where $1 \le m, n \le 100$), and $A$ declares $\gcd(x+m, n)$. What is the minimum number of steps $T_{100}$ that $B$ needs to determine $x$? | [
"3"
] | [
"$A$ chooses a natural number $x$ such that $1 \\le x \\le 100$. At every step, $B$ chooses two natural numbers $m$ and $n$ (where $1 \\le m, n \\le 100$), and $A$ declares $\\gcd(x+m, n)$. What is the minimum number of steps $T_{100}$ that $B$ needs to determine $x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-487928d09ad586e1019e55a1ed00ef45f15641c3bc3c48db03a11d11a227ed99 | $A$ gives $B$ as many cents as $B$ has and $C$ as many cents as $C$ has. Similarly, $B$ then gives $A$ and $C$ as many cents as each then has. $C$, similarly, then gives $A$ and $B$ as many cents as each then has. If each finally has $16$ cents, with how many cents does $A$ start? | [
"26"
] | [
"$A$ gives $B$ as many cents as $B$ has and $C$ as many cents as $C$ has. Similarly, $B$ then gives $A$ and $C$ as many cents as each then has. $C$, similarly, then gives $A$ and $B$ as many cents as each then has. If each finally has $16$ cents, with how many cents does $A$ start?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-9f6f195045bf756fd8be6c69485d9528c8c9bf90921005cdde12c7bc32bac0f9 | $A$ is 78 km distant from $B$. A cyclist leaves $A$ in the direction of $B$. One hour later, another cyclist leaves $B$ in the direction of $A$ and cycles 4 km/h faster than the first one. They meet 36 km from $B$. How long is each one en route prior to the encounter and what are their speeds? | [
"The speed of the first cyclist is equal to 14 km/h and that of the second, to 18 km/h; the first is 3 hours en route prior to the encounter, the second, 2 hours"
] | [
"$A$ is 78 km distant from $B$. A cyclist leaves $A$ in the direction of $B$. One hour later, another cyclist leaves $B$ in the direction of $A$ and cycles 4 km/h faster than the first one. They meet 36 km from $B$. How long is each one en route prior to the encounter and what are their speeds?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-968fee7abc701dc0f7708345319c673150deff8cc903c1ad6012b9563345ecff | $A$ is a point inside circle $\Gamma$. The diameter containing $A$ intersects $\Gamma$ at points $B$ and $C$, where $AB = 3$ and $AC = 9$. Point $D$ is on circle $\Gamma$ such that $\overline{AD} \perp \overline{BC}$. Tangents from points $B$ and $D$ to circle $\Gamma$ intersect at $E$. A line from $E$ parallel to $BC$... | [
"17"
] | [
"$A$ is a point inside circle $\\Gamma$. The diameter containing $A$ intersects $\\Gamma$ at points $B$ and $C$, where $AB = 3$ and $AC = 9$. Point $D$ is on circle $\\Gamma$ such that $\\overline{AD} \\perp \\overline{BC}$. Tangents from points $B$ and $D$ to circle $\\Gamma$ intersect at $E$. A line from $E$ para... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5f2e693918387ca83aa743a62ec32e55d7931dda163ead015aeef5482f13f521 | $A$ normally arrives at the hotel at 6:00 p.m. and is picked up by his wife. One day, he arrives an hour earlier and starts walking home. He meets his wife on the way, and they return home together in the car, arriving 10 minutes earlier than usual. Find the ratio of $A$'s walking speed to his wife's driving speed. | [
"\\dfrac{1}{11}"
] | [
"$A$ normally arrives at the hotel at 6:00 p.m. and is picked up by his wife. One day, he arrives an hour earlier and starts walking home. He meets his wife on the way, and they return home together in the car, arriving 10 minutes earlier than usual. Find the ratio of $A$'s walking speed to his wife's driving speed... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-01850ee66e0a1f055425ef1f5636e0fae69ff0e11a85830985a7b42a46caa1ef | $A$ tosses 2 fair coins and $B$ tosses 3 fair coins. The game is won by the person who throws the greater number of heads. In case of a tie, the game is continued under identical rules until someone wins. Find the probability of $A$ winning the game. | [
"\\(\\frac{3}{7}\\)"
] | [
"$A$ tosses 2 fair coins and $B$ tosses 3 fair coins. The game is won by the person who throws the greater number of heads. In case of a tie, the game is continued under identical rules until someone wins. Find the probability of $A$ winning the game."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fbb4d64c6243cd739b6a2b0b8e0ebf04a1e09dba3d2d889572c11dd3ceb89a15 | $A$ writes, at his choice, $8$ ones and $8$ twos on a $4 \times 4$ board. Then $B$ covers the board with $8$ dominoes and for each domino, she finds the smaller of the two numbers that the domino covers. Finally, $A$ adds these $8$ numbers and the result is her score. What is the highest score $A$ can secure, no matter... | [
"8"
] | [
"$A$ writes, at his choice, $8$ ones and $8$ twos on a $4 \\times 4$ board. Then $B$ covers the board with $8$ dominoes and for each domino, she finds the smaller of the two numbers that the domino covers. Finally, $A$ adds these $8$ numbers and the result is her score. What is the highest score $A$ can secure, no ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae998adc4d2bed0a19ad3a90bd824a3a4725af8dbefb0817b74557b5c18785dc | $A$, $B$, $C$, $D$ are successive vertices of a parallelogram. Points $E$, $F$, $P$, $H$ lie on the sides $AB$, $BC$, $CD$, $AD$, respectively. The segment $A E$ is equal to 1/3 of the side $AB$, the segment $BF$ is equal to 1/3 of the side $BC$, and the points $P$ and $H$ bisect the sides on which they lie. Find the r... | [
"$\\frac{37}{72}$"
] | [
"$A$, $B$, $C$, $D$ are successive vertices of a parallelogram. Points $E$, $F$, $P$, $H$ lie on the sides $AB$, $BC$, $CD$, $AD$, respectively. The segment $A E$ is equal to 1/3 of the side $AB$, the segment $BF$ is equal to 1/3 of the side $BC$, and the points $P$ and $H$ bisect the sides on which they lie. Find ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f3bc4e51ee418912ab0acff9d286aff539a1b4ca048ebfb6aca98a25c428ae5c | $A$, $B$, $C$, and $D$ are four distinct coplanar points in Euclidean 3-space such that the plane containing them is not perpendicular to the xy plane. Project $A$, $B$, $C$, and $D$ onto the xy plane to points $W$, $X$, $Y$, and $Z$ respectively. Given that there exists a point $P$ not in the xy plane such that $AP=AW... | [
"circle"
] | [
"$A$, $B$, $C$, and $D$ are four distinct coplanar points in Euclidean 3-space such that the plane containing them is not perpendicular to the xy plane. Project $A$, $B$, $C$, and $D$ onto the xy plane to points $W$, $X$, $Y$, and $Z$ respectively. Given that there exists a point $P$ not in the xy plane such that $... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8f22b4a94f97d65dbb84f63bbdc69410e2aacb484301565d889172d6b996b09b | $A, B, C$ and $D$ are distinct positive integers such that the product $AB = 60$, the product $CD = 60$ and $A - B = C + D$ . What is the value of $A$? | [
"20"
] | [
"$A, B, C$ and $D$ are distinct positive integers such that the product $AB = 60$, the product $CD = 60$ and $A - B = C + D$ . What is the value of $A$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f1bcd3e5f7646a798edb88bf436b49566d114da39f24b2dc030982de85585385 | $A, B, C$ are collinear with $B$ between $A$ and $C$. $K_{1}$ is the circle with diameter $AB$, and $K_{2}$ is the circle with diameter $BC$. Another circle touches $AC$ at $B$ and meets $K_{1}$ again at $P$ and $K_{2}$ again at $Q$. The line $PQ$ meets $K_{1}$ again at $R$ and $K_{2}$ again at $S$. Let $T$ be the inte... | [
"\\dfrac{24}{7}"
] | [
"$A, B, C$ are collinear with $B$ between $A$ and $C$. $K_{1}$ is the circle with diameter $AB$, and $K_{2}$ is the circle with diameter $BC$. Another circle touches $AC$ at $B$ and meets $K_{1}$ again at $P$ and $K_{2}$ again at $Q$. The line $PQ$ meets $K_{1}$ again at $R$ and $K_{2}$ again at $S$. Let $T$ be the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-9e29a51dd25349c2eaaacc1732ba1faeebf0f203ba92a2616592aff6c5fe8091 | $A, B, C, D$ all lie on a circle with $\overline{AB} = \overline{BC} = \overline{CD}$. If the distance between any two of these points is a positive integer, what is the smallest possible perimeter of quadrilateral $ABCD$? | [
"17"
] | [
"$A, B, C, D$ all lie on a circle with $\\overline{AB} = \\overline{BC} = \\overline{CD}$. If the distance between any two of these points is a positive integer, what is the smallest possible perimeter of quadrilateral $ABCD$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 8
} | 26 | 0 | 0 |
omr-ws-12ff0f49085b3f5fcefa420c4948d81cd3f7ae4630311400efc3d3dc5fb4a9ed | $A, B, C, D$ are points along a circle, in that order. $AC$ intersects $BD$ at $X$. If $BC = 6$, $BX = 4$, $XD = 5$, and $AC = 11$, find $AB$. | [
"6"
] | [
"$A, B, C, D$ are points along a circle, in that order. $AC$ intersects $BD$ at $X$. If $BC = 6$, $BX = 4$, $XD = 5$, and $AC = 11$, find $AB$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-222fb7e6c4cd188c398893601ff4d0927ece873b64b7b0b9dcabd0d2a1c53586 | $A, B, C, D$ are successive vertices of a rectangle. A circle passes through $A$ and $B$ and touches the side $CD$ at its midpoint. A straight line drawn through $D$ touches the same circle at a point $E$ and then cuts the extension of the side $AB$ at a point $K$. Find the area of the trapezoid $BCDK$ if it is known t... | [
"210"
] | [
"$A, B, C, D$ are successive vertices of a rectangle. A circle passes through $A$ and $B$ and touches the side $CD$ at its midpoint. A straight line drawn through $D$ touches the same circle at a point $E$ and then cuts the extension of the side $AB$ at a point $K$. Find the area of the trapezoid $BCDK$ if it is kn... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-96ccca878e3158a21d0079c65b8948f9c78f90614707d168d36821827f7b159f | $A, B, C, D,$ and $E$ are collinear in that order such that $AB = BC = 1, CD = 2,$ and $DE = 9$. If $P$ can be any point in space, what is the smallest possible value of $AP^2 + BP^2 + CP^2 + DP^2 + EP^2$? | [
"110"
] | [
"$A, B, C, D,$ and $E$ are collinear in that order such that $AB = BC = 1, CD = 2,$ and $DE = 9$. If $P$ can be any point in space, what is the smallest possible value of $AP^2 + BP^2 + CP^2 + DP^2 + EP^2$?\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c30d1a21b373e2d19db4367c196bc8afcb582f575e4570a671aa807c005f7d6a | $AB$ and $CD$ are diameters of the same circle. The tangent that touches the circle at $D$ intersects $AB$ at point $E$ (where $E$ is on the different side of $B$ from $A$). $BC$ and $DE$ intersect at $F$. Given that $\frac{EB}{AB} = \frac{5}{2}$ and $DF = 4$, find the length of $EF$. | [
"10"
] | [
"$AB$ and $CD$ are diameters of the same circle. The tangent that touches the circle at $D$ intersects $AB$ at point $E$ (where $E$ is on the different side of $B$ from $A$). $BC$ and $DE$ intersect at $F$. Given that $\\frac{EB}{AB} = \\frac{5}{2}$ and $DF = 4$, find the length of $EF$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-eb96d428f884ca6c337a673e4c682309205daf9c2b8d9f2f444472ab3cd8b1bc | $AB$ is a diameter of a circle, and $AD$ and $BC$ are tangents to the circle at points $A$ and $B$, respectively. The lines $AC$ and $BD$ intersect at a point on the circle. Given that $AD = a$ and $BC = b$, with $a \neq b$, find the length of $AB$. | [
"\\(\\sqrt{ab}\\)"
] | [
"$AB$ is a diameter of a circle, and $AD$ and $BC$ are tangents to the circle at points $A$ and $B$, respectively. The lines $AC$ and $BD$ intersect at a point on the circle. Given that $AD = a$ and $BC = b$, with $a \\neq b$, find the length of $AB$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-6a6560a688254471c91fb1dbd594f494b6bc8b2f19a679859023671f48ffd25a | $AB$ is a diameter of circle $O$ with $AB = 6$. $AB$ is extended to $C$ such that $BC = 2$. Point $D$ is on circle $O$ such that $CD$ is tangent to the circle. Find the length of $BD$. | [
"\\(\\frac{6\\sqrt{5}}{5}\\)"
] | [
"$AB$ is a diameter of circle $O$ with $AB = 6$. $AB$ is extended to $C$ such that $BC = 2$. Point $D$ is on circle $O$ such that $CD$ is tangent to the circle. Find the length of $BD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-2630c9224ef89936aa832b0f9062e4dd8cd20ebcc090e68e6cb62d6f21ddac9e | $AB$ is the diameter of circle $O$. A random point $P$ is selected on $O$ such that $AP = 4$ and $BP = 3$. Points $C$ and $D$ are drawn on circle $O$ so that $OC$ bisects $AP$ and $OD$ bisects $BP$. What is the degree measure of $\angle COD$? | [
"90"
] | [
"$AB$ is the diameter of circle $O$. A random point $P$ is selected on $O$ such that $AP = 4$ and $BP = 3$. Points $C$ and $D$ are drawn on circle $O$ so that $OC$ bisects $AP$ and $OD$ bisects $BP$. What is the degree measure of $\\angle COD$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3942181eb52919e23e7e44e661beba1057ce81ce4b8b7bd640f7270728bd1bae | $ABC$ and $A'B'C'$ are equilateral triangles with parallel sides and the same center. The distance between side $BC$ and side $B'C'$ is $\frac{1}{6}$ the altitude of $\triangle ABC$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$? | [
"\\(\\frac{1}{4}\\)"
] | [
"$ABC$ and $A'B'C'$ are equilateral triangles with parallel sides and the same center. The distance between side $BC$ and side $B'C'$ is $\\frac{1}{6}$ the altitude of $\\triangle ABC$. What is the ratio of the area of $\\triangle A'B'C'$ to the area of $\\triangle ABC$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 18
} | 65 | 0 | 0 |
omr-ws-a290cdae77407a57a3abd0ac141a1c12fa706d68daae141a9fb951950b5a7e11 | $ABC$ and $CDE$ are two isosceles right triangles with hypotenuse lengths $BC = 7$ and $CE = 14$. Point $C$ lies on the segment $BE$, and points $A$ and $D$ are on the same side of line $BE$. $O$ is the intersection point of segments $AE$ and $BD$. Find the area of triangle $ODE$. | [
"42"
] | [
"$ABC$ and $CDE$ are two isosceles right triangles with hypotenuse lengths $BC = 7$ and $CE = 14$. Point $C$ lies on the segment $BE$, and points $A$ and $D$ are on the same side of line $BE$. $O$ is the intersection point of segments $AE$ and $BD$. Find the area of triangle $ODE$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-fdc41512ccf1a394bf56b86241cfd908f144dacd61f0c11198b71748df3fe524 | $ABC$ and $PQR$ are both equilateral triangles with an area of 1. The center $M$ of $PQR$ lies on the perimeter of $ABC$. Determine the minimal area of the intersection of the two triangles. | [
"\\dfrac{1}{4}"
] | [
"$ABC$ and $PQR$ are both equilateral triangles with an area of 1. The center $M$ of $PQR$ lies on the perimeter of $ABC$. Determine the minimal area of the intersection of the two triangles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-79f3344777f880523750760e0176b2f8946c8901c021d3551804eb1d503c5856 | $ABC$ is a right-angled triangle with $\angle A = 90^\circ$ and $D$ is the midpoint of $BC$. A point $F$ is chosen on $AB$. $CA$ and $DF$ meet at $G$ and $GB \parallel AD$. $CF$ and $AD$ meet at $O$ and $AF = FO$. $GO$ meets $BC$ at $R$. Find the sides of $ABC$ if the area of $GDR$ is $\frac{2}{\sqrt{15}}$. | [
"\\dfrac{4\\sqrt{10}}{5}"
] | [
"$ABC$ is a right-angled triangle with $\\angle A = 90^\\circ$ and $D$ is the midpoint of $BC$. A point $F$ is chosen on $AB$. $CA$ and $DF$ meet at $G$ and $GB \\parallel AD$. $CF$ and $AD$ meet at $O$ and $AF = FO$. $GO$ meets $BC$ at $R$. Find the sides of $ABC$ if the area of $GDR$ is $\\frac{2}{\\sqrt{15}}$."
... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-16db1cae5c0974fca2f275f6fc136b9f286379a8b28374a3e673604cf9321a33 | $ABC$ is a scalene triangle. The circle with diameter $\overline{AB}$ intersects $\overline{BC}$ at $D$, and $E$ is the foot of the altitude from $C$. $P$ is the intersection of $\overline{AD}$ and $\overline{CE}$. Given that $AP = 136$, $BP = 80$, and $CP = 26$, determine the circumradius of $ABC$. | [
"85"
] | [
"$ABC$ is a scalene triangle. The circle with diameter $\\overline{AB}$ intersects $\\overline{BC}$ at $D$, and $E$ is the foot of the altitude from $C$. $P$ is the intersection of $\\overline{AD}$ and $\\overline{CE}$. Given that $AP = 136$, $BP = 80$, and $CP = 26$, determine the circumradius of $ABC$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-ce5f96e227307c1d38afa4a26434eb8b0d7792f09af6d68f18ee7ab8c9e6d1a2 | $ABC$ is an equilateral triangle and $l$ is a line such that the distances from $A, B,$ and $C$ to $l$ are $39, 35,$ and $13$, respectively. Find the largest possible value of $AB$. | [
"76"
] | [
"$ABC$ is an equilateral triangle and $l$ is a line such that the distances from $A, B,$ and $C$ to $l$ are $39, 35,$ and $13$, respectively. Find the largest possible value of $AB$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 7,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-e1a0ba9243d9a1d187c5ba90f7669d284559b9fc5627f54a23e193bcb3fc276e | $ABC$ is an equilateral triangle with side length $12$. Let $O_A$ be the point inside $ABC$ that is equidistant from $B$ and $C$ and is $\sqrt{3}$ units from $A$. Define $O_B$ and $O_C$ symmetrically. Find the area of the intersection of triangles $O_ABC$, $AO_BC$, and $ABO_C$. | [
"\\dfrac{324\\sqrt{3}}{49}"
] | [
"$ABC$ is an equilateral triangle with side length $12$. Let $O_A$ be the point inside $ABC$ that is equidistant from $B$ and $C$ and is $\\sqrt{3}$ units from $A$. Define $O_B$ and $O_C$ symmetrically. Find the area of the intersection of triangles $O_ABC$, $AO_BC$, and $ABO_C$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-6e0f54c90282909560494f17078b9b84b50b685e6be309f473f94d08047ea091 | $ABC$ is an equilateral triangle. $H$ lies in the interior of $ABC$, and points $X$, $Y$, $Z$ lie on sides $AB, BC, CA$, respectively, such that $HX \perp AB$, $HY \perp BC$, $HZ \perp CA$. Furthermore, $HX = 2$, $HY = 3$, $HZ = 4$. Find the area of triangle $ABC$. | [
"\\(\\frac{27\\sqrt{3}}{4}\\)"
] | [
"$ABC$ is an equilateral triangle. $H$ lies in the interior of $ABC$, and points $X$, $Y$, $Z$ lie on sides $AB, BC, CA$, respectively, such that $HX \\perp AB$, $HY \\perp BC$, $HZ \\perp CA$. Furthermore, $HX = 2$, $HY = 3$, $HZ = 4$. Find the area of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3995a304f24d62fd6d722cb42eee258f350722b286af0157973b936311e05fa7 | $ABC$ is an equilateral triangle. On one side of the plane $ABC$, perpendicular segments $AA' = AB$ and $BB' = \frac{AB}{2}$ are laid. Find the angle between the planes $ABC$ and $A'B'C$. | [
"\\(90^\\circ\\)"
] | [
"$ABC$ is an equilateral triangle. On one side of the plane $ABC$, perpendicular segments $AA' = AB$ and $BB' = \\frac{AB}{2}$ are laid. Find the angle between the planes $ABC$ and $A'B'C$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e41eabd356cc0fd5e44bd505cc61c21ea702a774d12b25891f4d118a00fc1fa6 | $ABCD$ is a cyclic quadrilateral with $AC=56$, $BD=65$, $BC>DA$, and $\frac{AB}{BC} = \frac{CD}{DA}$. Find the ratio of the area of triangle $ABC$ to the area of triangle $ADC$. | [
"\\dfrac{49}{16}"
] | [
"$ABCD$ is a cyclic quadrilateral with $AC=56$, $BD=65$, $BC>DA$, and $\\frac{AB}{BC} = \\frac{CD}{DA}$. Find the ratio of the area of triangle $ABC$ to the area of triangle $ADC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7c1b00d367b88f04b26c3df5add08e946939f666c732b8f9b399e17bd483f5fd | $ABCD$ is a cyclic quadrilateral with perpendicular diagonals of lengths $10\sqrt{3}$ and $12$. The radius of the circumscribed circle is $10$. Find the length of the largest side of the quadrilateral. | [
"4\\sqrt{15} + 2\\sqrt{5}"
] | [
"$ABCD$ is a cyclic quadrilateral with perpendicular diagonals of lengths $10\\sqrt{3}$ and $12$. The radius of the circumscribed circle is $10$. Find the length of the largest side of the quadrilateral."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-55fec7abde05930b3ceeffbf5ee8e2f9c985e51690b9c8d86a61ee24328b1602 | $ABCD$ is a parallelogram and $P$ is a point inside triangle $BAD$. If the area of triangle $PAB$ is $2$ and the area of triangle $PCB$ is $5$, what is the area of triangle $PBD$? | [
"3"
] | [
"$ABCD$ is a parallelogram and $P$ is a point inside triangle $BAD$. If the area of triangle $PAB$ is $2$ and the area of triangle $PCB$ is $5$, what is the area of triangle $PBD$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7282b9739062fd433e3d18d84353b056d1e2d63295ae8d6499d3d7e983535b01 | $ABCD$ is a parallelogram with $AB = 7$, $BC = 2$, and $\angle DAB = 120^\circ$. Parallelogram $ECFA$ is contained within $ABCD$ and is similar to it. Find the ratio of the area of $ECFA$ to the area of $ABCD$. | [
"$\\frac{39}{67}$"
] | [
"$ABCD$ is a parallelogram with $AB = 7$, $BC = 2$, and $\\angle DAB = 120^\\circ$. Parallelogram $ECFA$ is contained within $ABCD$ and is similar to it. Find the ratio of the area of $ECFA$ to the area of $ABCD$."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-502921b913e3e44918c06db93d2679a725ee028ac56c96624828e83e48a903a7 | $ABCD$ is a quadrilateral with $AB = 183$, $BC = 300$, $CD = 55$, $DA = 244$, and $BD = 305$. Find $AC$. | [
"273"
] | [
"$ABCD$ is a quadrilateral with $AB = 183$, $BC = 300$, $CD = 55$, $DA = 244$, and $BD = 305$. Find $AC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 4
} | 21 | 0 | 0 |
omr-ws-2c375a1ba0b68f2ef2b19e4de0be79881e677aa383cace559a22395a0e250162 | $ABCD$ is a quadrilateral with $AB = BC = 25$, $CD = DA = 60$, and $\angle BAD$ and $\angle BCD$ are both right angles. $P$ is the point in the plane of $ABCD$ which minimizes the sum $PA + PB + PC + PD$. Find this minimum sum. | [
"130"
] | [
"$ABCD$ is a quadrilateral with $AB = BC = 25$, $CD = DA = 60$, and $\\angle BAD$ and $\\angle BCD$ are both right angles. $P$ is the point in the plane of $ABCD$ which minimizes the sum $PA + PB + PC + PD$. Find this minimum sum."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a41ef2728d214e64788f6b5eeb5016b26fcf88102a24459de8058d1946771ea0 | $ABCD$ is a quadrilateral with right angles at $A$ and $C$. Points $E$ and $F$ are on $AC$, and $DE$ and $BF$ are perpendicular to $AC$. If $AE=3$, $DE=5$, and $CE=7$, then $BF=$ | [
"4.2"
] | [
"$ABCD$ is a quadrilateral with right angles at $A$ and $C$. Points $E$ and $F$ are on $AC$, and $DE$ and $BF$ are perpendicular to $AC$. If $AE=3$, $DE=5$, and $CE=7$, then $BF=$"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 14
} | 62 | 0 | 0 |
omr-ws-0ecee5ca5d7c25a6ea96f833148f38d3234eeb1a3f37775619e73ce0ad657d22 | $ABCD$ is a rectangle with $AB = 8$ and $BC = 5$. Extend $AD$ to $P$ and $CB$ to $Q$ such that $P$ and $Q$ are on opposite sides of $AB$, $PD = 3$, and $BQ = 6$. The segment $PQ$ intersects $AB$ at $E$ and $CD$ at $F$. The length of $EF$ can be written as $\frac{m}{n}\sqrt{k}$, where $m$ and $n$ are coprime, and $k$ is... | [
"77"
] | [
"$ABCD$ is a rectangle with $AB = 8$ and $BC = 5$. Extend $AD$ to $P$ and $CB$ to $Q$ such that $P$ and $Q$ are on opposite sides of $AB$, $PD = 3$, and $BQ = 6$. The segment $PQ$ intersects $AB$ at $E$ and $CD$ at $F$. The length of $EF$ can be written as $\\frac{m}{n}\\sqrt{k}$, where $m$ and $n$ are coprime, and... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d6b09857dad478742b0c2516e3c45741bd54e41d37494728749c29a0e9138f46 | $ABCD$ is a rectangle with $AB = CD = 2$. A circle centered at $O$ is tangent to $BC$, $CD$, and $AD$ (and hence has radius $1$). Another circle, centered at $P$, is tangent to circle $O$ at point $T$ and is also tangent to $AB$ and $BC$. If line $AT$ is tangent to both circles at $T$, find the radius of circle $P$. | [
"\\(\\frac{3-\\sqrt{5}}{2}\\)"
] | [
"$ABCD$ is a rectangle with $AB = CD = 2$. A circle centered at $O$ is tangent to $BC$, $CD$, and $AD$ (and hence has radius $1$). Another circle, centered at $P$, is tangent to circle $O$ at point $T$ and is also tangent to $AB$ and $BC$. If line $AT$ is tangent to both circles at $T$, find the radius of circle $P... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 28,
"genselect": 7,
"tir": 14
} | 49 | 0 | 0 |
omr-ws-c06d969e304c390fcf1416e006785bc16d759561c9a715f82793442f3b704173 | $ABCD$ is a rectangle with $E$ as the midpoint of $AB$. Points $F$ and $G$ are on sides $BC$ and $CD$ respectively such that $CG = CF = 2$. Point $H$ is on $AG$ such that $EFGH$ is also a rectangle. Find the area of rectangle $ABCD$. | [
"48"
] | [
"$ABCD$ is a rectangle with $E$ as the midpoint of $AB$. Points $F$ and $G$ are on sides $BC$ and $CD$ respectively such that $CG = CF = 2$. Point $H$ is on $AG$ such that $EFGH$ is also a rectangle. Find the area of rectangle $ABCD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-4a01830044848467a46b5c73475ce94d7bdab765848ee184ee7c101cfc5ff580 | $ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD.$ The crease is $EF,$ where $E$ is on $AB$ and $F$ is on $CD.$ The dimensions $AE=8, BE=17,$ and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n,$ where $m$ and $n$ are relatively prime posi... | [
"293"
] | [
"$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD.$ The crease is $EF,$ where $E$ is on $AB$ and $F$ is on $CD.$ The dimensions $AE=8, BE=17,$ and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n,$ where $m$ and $n$ are relatively prime... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-4feff063e7cb9e750c94415a62f6f03a82820c64f85971cba64d2169c6be1a9d | $ABCD$ is a rectangular sheet of paper. $E$ and $F$ are points on $AB$ and $CD$ respectively such that $BE < CF$. If $BCFE$ is folded over $EF$, $C$ maps to $C'$ on $AD$ and $B$ maps to $B'$ such that $\angle{AB'C'} \cong \angle{B'EA}$. If $AB' = 5$ and $BE = 23$, then the area of $ABCD$ can be expressed as $a + b\sqrt... | [
"338"
] | [
"$ABCD$ is a rectangular sheet of paper. $E$ and $F$ are points on $AB$ and $CD$ respectively such that $BE < CF$. If $BCFE$ is folded over $EF$, $C$ maps to $C'$ on $AD$ and $B$ maps to $B'$ such that $\\angle{AB'C'} \\cong \\angle{B'EA}$. If $AB' = 5$ and $BE = 23$, then the area of $ABCD$ can be expressed as $a ... | [
"MATH_training_set"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-05786b5816e7b09299b29498aa9b6c83822b4f2347086410e5b0a23b8f405603 | $ABCD$ is a regular tetrahedron (right triangular pyramid). If $M$ is the midpoint of $\overline{CD}$, then what is $\cos \angle AMB$? | [
"\\frac{1}{3}"
] | [
"$ABCD$ is a regular tetrahedron (right triangular pyramid). If $M$ is the midpoint of $\\overline{CD}$, then what is $\\cos \\angle AMB$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6af8c207ed3a7ddac3f03e5a8e9f748fecf83ec7b346008c429d000694953d6c | $ABCD$ is a regular tetrahedron with side length $1$. Spheres $O_1,$ $O_2$, $O_3$, and $O_4$ have equal radii and are positioned inside $ABCD$ such that they are internally tangent to three of the faces at a time, and all four spheres intersect at a single point. If the radius of $O_1$ can be expressed as $\frac{a \sqr... | [
"23"
] | [
"$ABCD$ is a regular tetrahedron with side length $1$. Spheres $O_1,$ $O_2$, $O_3$, and $O_4$ have equal radii and are positioned inside $ABCD$ such that they are internally tangent to three of the faces at a time, and all four spheres intersect at a single point. If the radius of $O_1$ can be expressed as $\\frac{... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-d2d086c2489fd05bf111a2d3b890d96408ed435081774e34252f7076a1d5a7d9 | $ABCD$ is a regular tetrahedron with side length $4$. Four congruent spheres are inside $ABCD$ such that each sphere is tangent to exactly three of the faces, the spheres have distinct centers, and the four spheres are concurrent at one point. Let $v$ be the volume of one of the spheres. If $v^2$ can be written as $\tf... | [
"371"
] | [
"$ABCD$ is a regular tetrahedron with side length $4$. Four congruent spheres are inside $ABCD$ such that each sphere is tangent to exactly three of the faces, the spheres have distinct centers, and the four spheres are concurrent at one point. Let $v$ be the volume of one of the spheres. If $v^2$ can be written as... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5ad57ec77d86df02c67d625fb743635603591ff1b30101fd98cd5925b8d9b173 | $ABCD$ is a regular tetrahedron with side length 1. Points $X,$ $Y,$ and $Z,$ distinct from $A,$ $B,$ and $C,$ respectively, are drawn such that $BCDX,$ $ACDY,$ and $ABDZ$ are also regular tetrahedra. If the volume of the polyhedron with faces $ABC,$ $XYZ,$ $BXC,$ $XCY,$ $CYA,$ $YAZ,$ $AZB,$ and $ZBX$ can be written as... | [
"115"
] | [
"$ABCD$ is a regular tetrahedron with side length 1. Points $X,$ $Y,$ and $Z,$ distinct from $A,$ $B,$ and $C,$ respectively, are drawn such that $BCDX,$ $ACDY,$ and $ABDZ$ are also regular tetrahedra. If the volume of the polyhedron with faces $ABC,$ $XYZ,$ $BXC,$ $XCY,$ $CYA,$ $YAZ,$ $AZB,$ and $ZBX$ can be writt... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-e9986a2f11ed59505e81e1fb54d493886307c58b8236b3b231564ab90cd7af90 | $ABCD$ is a regular tetrahedron with side length 1. Spheres $O_1, O_2, O_3$, and $O_4$ have equal radii and are positioned inside $ABCD$ such that they are internally tangent to three of the faces at a time, and all four spheres intersect at a single point. If the radius of $O_1$ can be expressed as $\frac{a \sqrt{b}}{... | [
"23"
] | [
"$ABCD$ is a regular tetrahedron with side length 1. Spheres $O_1, O_2, O_3$, and $O_4$ have equal radii and are positioned inside $ABCD$ such that they are internally tangent to three of the faces at a time, and all four spheres intersect at a single point. If the radius of $O_1$ can be expressed as $\\frac{a \\sq... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-d45e977a2e70afc702c69e8535f4c6478f2d0010a461089bfa3900114f7252e6 | $ABCD$ is a rhombus with side length equal to 26. Circle R has center A and passes through C. Circle Q has center B and passes through D. Given that R and Q are tangential to each other, what is the area of the rhombus? | [
"507"
] | [
"$ABCD$ is a rhombus with side length equal to 26. Circle R has center A and passes through C. Circle Q has center B and passes through D. Given that R and Q are tangential to each other, what is the area of the rhombus?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-34f05c46a0fd481a6cc324c39d21fd6df7189e2f23e5c85d8310cc601bd3f018 | $ABCD$ is a square such that $AB$ lies on the line $y=x+4$ and points $C$ and $D$ lie on the graph of the parabola $y^2 = x$. Compute the sum of all possible areas of $ABCD$. | [
"68"
] | [
"$ABCD$ is a square such that $AB$ lies on the line $y=x+4$ and points $C$ and $D$ lie on the graph of the parabola $y^2 = x$. Compute the sum of all possible areas of $ABCD$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4fb2ce8173ee94c64ba0e90d3243111ffdc947e249764ff8023528f7b5b66f95 | $ABCD$ is a square with $AB = 2$. The circle with center $C$ and radius $CB$ intersects the circle with diameter $AB$ at $E \ne B$. Determine the length of $AE$. | [
"\\(\\frac{2\\sqrt{5}}{5}\\)"
] | [
"$ABCD$ is a square with $AB = 2$. The circle with center $C$ and radius $CB$ intersects the circle with diameter $AB$ at $E \\ne B$. Determine the length of $AE$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-147d9836d2aff7fa7e59f3bc2695dc1316396a6a06979c7426891af1f0307d06 | $ABCD$ is a square with $AB=1$. Four circles are drawn with centers at $A$, $B$, $C$, and $D$, each with a radius of $1$. Find the area of the region that is common to all four circles. | [
"1 + \\dfrac{\\pi}{3} - \\sqrt{3}"
] | [
"$ABCD$ is a square with $AB=1$. Four circles are drawn with centers at $A$, $B$, $C$, and $D$, each with a radius of $1$. Find the area of the region that is common to all four circles."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-b7a7424e988ed9ea955e114530354c720a602c21a389b67fe34878474e9f36b5 | $ABCD$ is a square with side length $1$. With its vertices as centers, four circular arcs of $90^\circ$, each with radius $1$, are drawn. Find the area of the region common to all four arcs. | [
"\\( 1 - \\sqrt{3} + \\frac{\\pi}{3} \\)"
] | [
"$ABCD$ is a square with side length $1$. With its vertices as centers, four circular arcs of $90^\\circ$, each with radius $1$, are drawn. Find the area of the region common to all four arcs."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 16
} | 37 | 0 | 0 |
omr-ws-771caa0c9f89fd82bcba003173513f7da1f4c52d1538dafd76ea9e68a36044ab | $ABCD$ is a square with side length $8$ cm. Circles are drawn with diameters $AB$ and $BC$. What is the area inside the square not covered by either of the circles? | [
"\\( 48 - 8\\pi \\)"
] | [
"$ABCD$ is a square with side length $8$ cm. Circles are drawn with diameters $AB$ and $BC$. What is the area inside the square not covered by either of the circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-ec5257e6a0af19ebf39903082a23f14e400d484a9c386313cb9395adeaf270a2 | $ABCD$ is a square with side length $8$. Points $E$ and $F$ are the midpoints of sides $BC$ and $CD$, respectively. A circle is inscribed in quadrilateral $AECF$. Find the radius of this circle. | [
"\\( 2\\sqrt{5} - 2 \\)"
] | [
"$ABCD$ is a square with side length $8$. Points $E$ and $F$ are the midpoints of sides $BC$ and $CD$, respectively. A circle is inscribed in quadrilateral $AECF$. Find the radius of this circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-4b5189e664344ad93641c9ae975bc5141df6cbadd0aba4ce74145970bbac85a6 | $ABCD$ is a square with side length 1 unit. A circle passes through vertices $A$ and $B$ of the square, with the remaining two vertices $C$ and $D$ lying outside the circle. The length of the tangent drawn from vertex $D$ to the circle is 2 units. What is the radius of the circle? | [
"$\\frac{\\sqrt{10}}{2}$"
] | [
"$ABCD$ is a square with side length 1 unit. A circle passes through vertices $A$ and $B$ of the square, with the remaining two vertices $C$ and $D$ lying outside the circle. The length of the tangent drawn from vertex $D$ to the circle is 2 units. What is the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-7023aa935b4dfbb4e32320c4c86b60925dc9f4f2e1451b9418722e7faf22a659 | $ABCD$ is a square with side length 1. Points $E$ and $F$ are on sides $AD$ and $AB$ respectively such that $AE:ED = 3:1$ and $BF:FA = 3:1$. Find the area of quadrilateral $EOCD$. | [
"\\(\\frac{77}{200}\\)"
] | [
"$ABCD$ is a square with side length 1. Points $E$ and $F$ are on sides $AD$ and $AB$ respectively such that $AE:ED = 3:1$ and $BF:FA = 3:1$. Find the area of quadrilateral $EOCD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 6
} | 24 | 0 | 0 |
omr-ws-add2836b7abf16aeeaf3e09fb56e5aeec39cc08e617f312839145345a4fc36cb | $ABCD$ is a square with side length 1. Points $P$ and $Q$ are on sides $AB$ and $BC$, respectively, such that $\widehat{PDQ} = 45^{\circ}$. Find the perimeter of $\Delta PBQ$. | [
"2"
] | [
"$ABCD$ is a square with side length 1. Points $P$ and $Q$ are on sides $AB$ and $BC$, respectively, such that $\\widehat{PDQ} = 45^{\\circ}$. Find the perimeter of $\\Delta PBQ$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d9b5a0e7478f67840de8d17637bc002cb776ae0442d02cfba2535818c5ba9478 | $ABCD$ is a square, and $AF$ and $CH$ are tangents to the circle with center $E$. If the circle passes through point $D$, find the ratio $\frac{ED}{AD}$. | [
"\\(\\frac{\\sqrt{2}}{2}\\)"
] | [
"$ABCD$ is a square, and $AF$ and $CH$ are tangents to the circle with center $E$. If the circle passes through point $D$, find the ratio $\\frac{ED}{AD}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 3,
"tir": 2
} | 10 | 0 | 0 |
omr-ws-4fafee7ae1a810b82039fe47171b2b937c39b8fc640ed298e7076b4dcbc3e017 | $ABCD$ is a trapezoid with $AD \parallel BC$ and $\angle ADC = 90^\circ$. $M$ is the midpoint of $AB$ and $CM = 6.5$. Given that $BC + CD + DA = 17$, find the area of $ABCD$. | [
"30"
] | [
"$ABCD$ is a trapezoid with $AD \\parallel BC$ and $\\angle ADC = 90^\\circ$. $M$ is the midpoint of $AB$ and $CM = 6.5$. Given that $BC + CD + DA = 17$, find the area of $ABCD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 18
} | 66 | 0 | 0 |
omr-ws-e9a30cafe2ec94a6a99e45572f98eb56d52b04e18690cff7c2c58bcb75dff4e9 | $ABCD$ is a trapezoid with $AD \perp BC$ and $\angle ADC = 90^\circ$. $M$ is the midpoint of $AB$ and $CM = 6.5$. Given that $BC + CD + DA = 17$, find the area of trapezoid $ABCD$. | [
"30"
] | [
"$ABCD$ is a trapezoid with $AD \\perp BC$ and $\\angle ADC = 90^\\circ$. $M$ is the midpoint of $AB$ and $CM = 6.5$. Given that $BC + CD + DA = 17$, find the area of trapezoid $ABCD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-2813fc865fdb4e88e68c7469f3eaea7689314a4fe66db49b15ad99c86e3f7d98 | $ABCD$, a rectangle with $AB = 12$ and $BC = 16$, is the base of pyramid $P$, which has a height of $24$. A plane parallel to $ABCD$ is passed through $P$, dividing $P$ into a frustum $F$ and a smaller pyramid $P'$. Let $X$ denote the center of the circumsphere of $F$, and let $T$ denote the apex of $P$. If the volume ... | [
"177"
] | [
"$ABCD$, a rectangle with $AB = 12$ and $BC = 16$, is the base of pyramid $P$, which has a height of $24$. A plane parallel to $ABCD$ is passed through $P$, dividing $P$ into a frustum $F$ and a smaller pyramid $P'$. Let $X$ denote the center of the circumsphere of $F$, and let $T$ denote the apex of $P$. If the vo... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a371d10677857829e59ca4280e63c9eeee828a7bb72d03592f8006b2dd7fa04a | $ABCDE$ is a convex pentagon with $AB = CD = EA = 1$, $\angle ABC = \angle DEA = 90^\circ$, and $BC + DE = 1$. What is the area of the pentagon? | [
"1"
] | [
"$ABCDE$ is a convex pentagon with $AB = CD = EA = 1$, $\\angle ABC = \\angle DEA = 90^\\circ$, and $BC + DE = 1$. What is the area of the pentagon?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 2
} | 2 | 0 | 0 |
omr-ws-7c43d1ac6cf7b63922cd53512147a77345699523a47fdf00c682e0adac02a58e | $ABCDE$ is a cyclic pentagon with $AC=2$, $AD=3$, $BD=5$, $BE=1$, and the ratio $CD/DE = 10/3$. If the ratio $BC/CE = a/b$ where $a$ and $b$ are coprime, find the value of $a-b$. | [
"-1"
] | [
"$ABCDE$ is a cyclic pentagon with $AC=2$, $AD=3$, $BD=5$, $BE=1$, and the ratio $CD/DE = 10/3$. If the ratio $BC/CE = a/b$ where $a$ and $b$ are coprime, find the value of $a-b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-47b6f5cc6c5f3d1950360171757867acbcda6b7e8fd8e41628b3e745b9a139aa | $ABCDE$ is a five-digit number where $A < B < C < D < E$. If all such numbers are arranged in ascending order, what is the 100th number? | [
"24789"
] | [
"$ABCDE$ is a five-digit number where $A < B < C < D < E$. If all such numbers are arranged in ascending order, what is the 100th number?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-b5e4baddf12b21dc65e874193c96a18a4f5b66bda55abe079e7206dd42b1f2e3 | $ABCDE$ is inscribed in a circle with $AB = BC = CD = DE = 4$ and $AE = 1.$ Compute $(1 - \cos \angle B)(1 - \cos \angle ACE).$ | [
"\\frac{1}{64}"
] | [
"$ABCDE$ is inscribed in a circle with $AB = BC = CD = DE = 4$ and $AE = 1.$ Compute $(1 - \\cos \\angle B)(1 - \\cos \\angle ACE).$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-9db539b5120d91d82ab2d71cfcdc40b699bc603169ac0623cef5314f78db76f8 | $ABCDEF$ is a regular hexagon whose center is the origin of the coordinate plane. If the measure of a side of the hexagon is $6$ units, $\overline{AB}$ and $\overline{DE}$ are parallel to the $x$-axis and $B$ is in Quadrant I, what are the coordinates of $B$? Leave your answer in simplest radical form. | [
"(3, 3\\sqrt{3})"
] | [
"$ABCDEF$ is a regular hexagon whose center is the origin of the coordinate plane. If the measure of a side of the hexagon is $6$ units, $\\overline{AB}$ and $\\overline{DE}$ are parallel to the $x$-axis and $B$ is in Quadrant I, what are the coordinates of $B$? Leave your answer in simplest radical form."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-149bf711d213f06f904cfd7ab79b2f92c0f592290aa0c1264f6002de890bc86c | $ABCDEF$ is an equiangular hexagon with side lengths $AB=2015$, $BC=2016$, $CD=2017$, and $DE=2018$. Find the perimeter of $ABCDEF$. | [
"12099"
] | [
"$ABCDEF$ is an equiangular hexagon with side lengths $AB=2015$, $BC=2016$, $CD=2017$, and $DE=2018$. Find the perimeter of $ABCDEF$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 3
} | 26 | 0 | 0 |
omr-ws-922874f0499399eb0c2d427073f6d52424ae9f27fdbfead726fc17bd07847e28 | $ABCDEF$ is an inscribed hexagon with $AB=CD=EF=6$ and $BC=DE=FA=10$. If the area of the hexagon is $m\sqrt{3}$, find $m$. | [
"72"
] | [
"$ABCDEF$ is an inscribed hexagon with $AB=CD=EF=6$ and $BC=DE=FA=10$. If the area of the hexagon is $m\\sqrt{3}$, find $m$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40c43c808a5662936b25271c0bbc641070f52403e1b4523461c05f4655fdceb7 | $AC$ and $BD$ are perpendicular diameters of circle $\Omega$. A line passing through point $A$ intersects diameter $BD$ at point $X$ and arc $BC$ at point $Y$. Find the radius of the circle if $AX = 8$ and $XY = 1$. | [
"\\( 6\\sqrt{2} \\)"
] | [
"$AC$ and $BD$ are perpendicular diameters of circle $\\Omega$. A line passing through point $A$ intersects diameter $BD$ at point $X$ and arc $BC$ at point $Y$. Find the radius of the circle if $AX = 8$ and $XY = 1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b630607c406526cc78c9c3302ab8b265cc1052077dcbd3f80d887216f987b6d | $AC$, $CB$, and $BA$ are the diameters of the circles centered at $E$, $F$, and $D$, respectively. $|AC| = 1$ and $|BC| = 1+\sqrt{2}$. The area of the green region can be expressed as $\frac{m \pi}{n} (p\sqrt{q} + r) $, where $\gcd (m,n) = 1, \gcd(p, r) = 1$ and $q$ is square free. Find the value of $m^2 + n^2 - p + q ... | [
"7"
] | [
"$AC$, $CB$, and $BA$ are the diameters of the circles centered at $E$, $F$, and $D$, respectively. $|AC| = 1$ and $|BC| = 1+\\sqrt{2}$. The area of the green region can be expressed as $\\frac{m \\pi}{n} (p\\sqrt{q} + r) $, where $\\gcd (m,n) = 1, \\gcd(p, r) = 1$ and $q$ is square free. Find the value of $m^2 + n... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-6978976c042aaefabd48719ef02c357cb82c9d15ed931fd4dbbbf158974ebaef | $ACEG$ is a quadrilateral circumscribed about a circle and tangent to the circle at points $B$, $D$, $F$, and $H$. If $AB=3$, $CD=4$, $EF=5$, and $GH=6$, find the radius of the circle. | [
"\\(\\sqrt{19}\\)"
] | [
"$ACEG$ is a quadrilateral circumscribed about a circle and tangent to the circle at points $B$, $D$, $F$, and $H$. If $AB=3$, $CD=4$, $EF=5$, and $GH=6$, find the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 13
} | 23 | 0 | 0 |
omr-ws-e8d6a989b334fe81a4f332f63fedd4f613dd1969b63c8b798ef8ae659a950b03 | $AD$ and $BE$ are angle bisectors of triangle $ABC$. It is known that $\angle ADE = 24^\circ$ and $\angle BED = 18^\circ$. Find the angles of triangle $ABC$. | [
"\\( A = 60^\\circ, B = 24^\\circ, C = 96^\\circ \\)"
] | [
"$AD$ and $BE$ are angle bisectors of triangle $ABC$. It is known that $\\angle ADE = 24^\\circ$ and $\\angle BED = 18^\\circ$. Find the angles of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0de711df3d09c23a47edccbaed1ce0d16c2166d31fcb3d6ec03ab6b4c0a85b87 | $AM$ is the median of triangle $ABC$. Find the angle between the angle bisectors of triangles $AMB$ and $AMC$ drawn from the vertex $M$. | [
"90^\\circ"
] | [
"$AM$ is the median of triangle $ABC$. Find the angle between the angle bisectors of triangles $AMB$ and $AMC$ drawn from the vertex $M$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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