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-03fb6b02b9abf1ecb7a387a2e35f94a8fa8a2693b0cf4df3d9ee894825634e9a | $A_0A_1A_2$ is a triangle, such that $A_0A_1=13,A_1A_2=14,A_2A_0=15$. $\{A_n\}_{n=0}^\infty$ is a sequence of points, such that for all integers $n\geq3$, $A_n$ is defined to be the orthocenter of triangle $A_{n-1}A_{n-2}A_{n-3}$. Find the area of triangle $A_{2018}A_{4037}A_{6056}$. | [
"0"
] | [
"$A_0A_1A_2$ is a triangle, such that $A_0A_1=13,A_1A_2=14,A_2A_0=15$. $\\{A_n\\}_{n=0}^\\infty$ is a sequence of points, such that for all integers $n\\geq3$, $A_n$ is defined to be the orthocenter of triangle $A_{n-1}A_{n-2}A_{n-3}$. Find the area of triangle $A_{2018}A_{4037}A_{6056}$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-14d29da5475411f7b57f06e6b1dc53d981af23fea56e01760104d0fd3429897b | $B$ and $C$ trisect $\overline{AD}$ and $M$ is the midpoint of $\overline{AD}$. $MC = 8$. How many units are in the length of $\overline{AD}$? | [
"48"
] | [
"$B$ and $C$ trisect $\\overline{AD}$ and $M$ is the midpoint of $\\overline{AD}$. $MC = 8$. How many units are in the length of $\\overline{AD}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ca7258b7bd9834a425c8e674e835d79f317da178ac86807d3bbe8fba5800412 | $B$ arrives at the hotel at 6:00 p.m. every day and is picked up by his wife, with the journey home taking one hour. One day, he arrives an hour early and starts walking home. He meets his wife on the way, and they reach home 10 minutes earlier than usual. Determine the time for which $B$ walked. | [
"55"
] | [
"$B$ arrives at the hotel at 6:00 p.m. every day and is picked up by his wife, with the journey home taking one hour. One day, he arrives an hour early and starts walking home. He meets his wife on the way, and they reach home 10 minutes earlier than usual. Determine the time for which $B$ walked."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3c765fc5587524a3645985657b3ac7e560d2caa72cc2e89699839e599d6fb745 | $C$ is on a semicircle with diameter $AB$ and center $O$. Circle radius $r_1$ is tangent to $OA, OC,$ and arc $AC$, and circle radius $r_2$ is tangent to $OB, OC,$ and arc $BC$. It is known that $\tan AOC=\frac{7}{24}$. The ratio $\frac{r_2}{r_1}$ can be expressed $\frac{m}{n}$, where $m, n$ are relatively prime positi... | [
"59"
] | [
"$C$ is on a semicircle with diameter $AB$ and center $O$. Circle radius $r_1$ is tangent to $OA, OC,$ and arc $AC$, and circle radius $r_2$ is tangent to $OB, OC,$ and arc $BC$. It is known that $\\tan AOC=\\frac{7}{24}$. The ratio $\\frac{r_2}{r_1}$ can be expressed $\\frac{m}{n}$, where $m, n$ are relatively pri... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bb3d23205f654c3c7674429e7dd145fed3fc91adaba974d387533f568f2b24bf | $C_0$ is a circle of radius $10$ meters tangent to the ground. $C_1$ is a circle of radius $1$ millimeter, external to $C_0$ and tangent to both $C_0$ and the ground. For $n > 1$, $C_n$ is a circle tangent to $C_0$, $C_{n-1}$, and the ground. How many circles can be placed in this way (before they get too large to be t... | [
"100"
] | [
"$C_0$ is a circle of radius $10$ meters tangent to the ground. $C_1$ is a circle of radius $1$ millimeter, external to $C_0$ and tangent to both $C_0$ and the ground. For $n > 1$, $C_n$ is a circle tangent to $C_0$, $C_{n-1}$, and the ground. How many circles can be placed in this way (before they get too large to... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 1,
"tir": 2
} | 11 | 0 | 0 |
omr-ws-76bb5125f70a191a3ecf60513629f0a8eddb2371a9d00b60c0892773d2f8261e | $C_{1}$ is a circle with radius $164$ and $C_{2}$ is a circle internally tangent to $C_{1}$ that passes through the center of $C_{1}$. $AB$ is a chord in $C_{1}$ of length $320$ tangent to $C_{2}$ at $D$ ($AD > BD$). Find $BD$. | [
"\\( 160 - 48\\sqrt{2} \\)"
] | [
"$C_{1}$ is a circle with radius $164$ and $C_{2}$ is a circle internally tangent to $C_{1}$ that passes through the center of $C_{1}$. $AB$ is a chord in $C_{1}$ of length $320$ tangent to $C_{2}$ at $D$ ($AD > BD$). Find $BD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 17
} | 37 | 0 | 0 |
omr-ws-97d7fba7c5452b645f5443cf9e5b72ffb9efba23fb02999bf93e46d2f9e96d32 | $E$ is a point inside a square $ABCD$ such that $AE=BE$. If $\angle AEB=150^\circ$, find $\angle CED$. | [
"\\(60^\\circ\\)"
] | [
"$E$ is a point inside a square $ABCD$ such that $AE=BE$. If $\\angle AEB=150^\\circ$, find $\\angle CED$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 19
} | 67 | 0 | 0 |
omr-ws-97f0ffe58636dc129c7261a75c3306114e89852ff716a4bf0cda6502d571c9af | $E$ is a point on side $AD$ of rectangle $ABCD$, so that $DE = 12$, while $DA = 16$, and $DC = 12$. If $CE$ extended meets the circumcircle of the rectangle at $F$, find the measure of chord $DF$. | [
"10\\sqrt{2}"
] | [
"$E$ is a point on side $AD$ of rectangle $ABCD$, so that $DE = 12$, while $DA = 16$, and $DC = 12$. If $CE$ extended meets the circumcircle of the rectangle at $F$, find the measure of chord $DF$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e5701513f49d01ccb34243a532413cd68acd2529f54c60d6b39ff8a0d1d389c0 | $E$ is the midpoint of side $BC$ of rectangle $ABCD$. A point $X$ is chosen on $BE$. $DX$ meets the extended $AB$ at $P$. If the length of $AB$ is $2a$ and $BC$ is $2b$, find the length $BX$ such that the sum of the areas of $\triangle BPX$ and $\triangle DXC$ is maximized. Express your answer in terms of $a$ and $b$. | [
"b"
] | [
"$E$ is the midpoint of side $BC$ of rectangle $ABCD$. A point $X$ is chosen on $BE$. $DX$ meets the extended $AB$ at $P$. If the length of $AB$ is $2a$ and $BC$ is $2b$, find the length $BX$ such that the sum of the areas of $\\triangle BPX$ and $\\triangle DXC$ is maximized. Express your answer in terms of $a$ an... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-e281b97f54be9ff2f671be5653640a4884875ebecb40e67ab837ca55d24c7b4b | $E$ is the midpoint of the side $\overline{BC}$ of a triangle $ABC$. $F$ is on $\overline{AC}$ such that $\overline{AC} = 3\overline{FC}$. Find the ratio of the area of quadrilateral $ABEF$ to the area of triangle $ABC$. | [
"\\(\\frac{5}{7}\\)"
] | [
"$E$ is the midpoint of the side $\\overline{BC}$ of a triangle $ABC$. $F$ is on $\\overline{AC}$ such that $\\overline{AC} = 3\\overline{FC}$. Find the ratio of the area of quadrilateral $ABEF$ to the area of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-580d02bf7c01b227af8129f8c56cbdd8dfb79fee9370c8a7a8eb5289649d2494 | $G$ is a regular convex $2022^2$-gon which has a circumcircle of radius $2022$. Let $R$ denote the set of all distinct non-degenerate rectangles with integer area whose four vertices coincide with vertices of $G$. What is the value of $\sqrt{|R|}$? | [
"1011"
] | [
"$G$ is a regular convex $2022^2$-gon which has a circumcircle of radius $2022$. Let $R$ denote the set of all distinct non-degenerate rectangles with integer area whose four vertices coincide with vertices of $G$. What is the value of $\\sqrt{|R|}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f6827d96c34099896258c3df89675eb956c6e774c0f09b1d95e31239666ceb07 | $H$ is the orthocenter of the acute-angled triangle $ABC$, the points $D$ and $E$ are the feet of the altitudes drawn from the vertices $B$ and $C$, respectively. A circle with diameter $DE$ intersects sides $AB$ and $AC$ again at points $F$ and $G$, respectively. Segments $FG$ and $AH$ intersect at point $K$. If $BC =... | [
"\\dfrac{216}{25}"
] | [
"$H$ is the orthocenter of the acute-angled triangle $ABC$, the points $D$ and $E$ are the feet of the altitudes drawn from the vertices $B$ and $C$, respectively. A circle with diameter $DE$ intersects sides $AB$ and $AC$ again at points $F$ and $G$, respectively. Segments $FG$ and $AH$ intersect at point $K$. If ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-aaf0af3f15b5ee56cbe99eb8f51476bca4dd03db2047c515d2788842d089ce2b | $I$ is the intersection point of the angle bisectors of the triangle $ABC$. It turned out that $CA + AI = BC$ and $\angle A = 70^\circ$. Find the rest angles of triangle $ABC$. | [
"75^\\circ"
] | [
"$I$ is the intersection point of the angle bisectors of the triangle $ABC$. It turned out that $CA + AI = BC$ and $\\angle A = 70^\\circ$. Find the rest angles of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d4160095273c6f2f54f2cd88ab68213f7da2cf872f9a55619496317a2e6e1d6a | $K$ students participated in an exam with 6 problems. Each problem was solved by exactly 1000 students, and no two students solved all of the problems together. Find the least value of $K$. | [
"2000"
] | [
"$K$ students participated in an exam with 6 problems. Each problem was solved by exactly 1000 students, and no two students solved all of the problems together. Find the least value of $K$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-b309131b7231246cbbf0d3fd1e1ec0a3c5d15e512a98174b84e6cfbf21abdafc | $M$ and $N$ are the midpoints of sides $GH$ and $FG$, respectively, of parallelogram $EFGH$. The area of triangle $ENM$ is $12$ $cm^{2}$. What is the area of the parallelogram $EFGH$? | [
"32"
] | [
"$M$ and $N$ are the midpoints of sides $GH$ and $FG$, respectively, of parallelogram $EFGH$. The area of triangle $ENM$ is $12$ $cm^{2}$. What is the area of the parallelogram $EFGH$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-9ac96ae31bba2f21fede8ee8c60be2422daa0ddbc77bcf4224224c5c411884b6 | $M(n)$ is defined as follows: $d_{1}$ is the greatest multiple of $1 \le n$. $d_{2}$ is the greatest multiple of $2 \le d_{1}$. $d_{3}$ is the greatest multiple of $3 \le d_{2}$. Continue this process until $d_{k} = k$. Compute the sum of all positive integers $n$ for which $M(n) = 10$. | [
"300"
] | [
"$M(n)$ is defined as follows: $d_{1}$ is the greatest multiple of $1 \\le n$. $d_{2}$ is the greatest multiple of $2 \\le d_{1}$. $d_{3}$ is the greatest multiple of $3 \\le d_{2}$. Continue this process until $d_{k} = k$. Compute the sum of all positive integers $n$ for which $M(n) = 10$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-67697029df18a1fc8f77bab102f6239c2eeb850a7536afe55ac944f1f6509b84 | $M, A, T, H$, and $L$ are all not necessarily distinct digits, with $M \ne 0$ and $L \ne 0$. Given that the sum $MATH + LMT$, where each letter represents a digit, equals $2012$, what is the average of all possible values of the three-digit integer $LMT$? | [
"514.5"
] | [
"$M, A, T, H$, and $L$ are all not necessarily distinct digits, with $M \\ne 0$ and $L \\ne 0$. Given that the sum $MATH + LMT$, where each letter represents a digit, equals $2012$, what is the average of all possible values of the three-digit integer $LMT$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-b2e889f4e88726b531e88396ed514b12a1ac937ceb4f14cc89763b00c0510f3a | $N$ baskets are in a line and numbered $1$ through $N$. $101$ eggs are distributed among the baskets such that each basket has at least $1$ egg and every $6$ consecutively numbered baskets has a total of $17$ eggs. Find all possible values of $N$ such that there is guaranteed to be at least $6$ baskets with exactly $1$... | [
"60"
] | [
"$N$ baskets are in a line and numbered $1$ through $N$. $101$ eggs are distributed among the baskets such that each basket has at least $1$ egg and every $6$ consecutively numbered baskets has a total of $17$ eggs. Find all possible values of $N$ such that there is guaranteed to be at least $6$ baskets with exactl... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0613afb69353fa4256b12405d01fb5fb5d2340ca7e159a907948f0b828add4d6 | $N$ coins are distributed among $M$ indistinguishable boxes. Billy buys boxes one at a time for one coin each and can examine their contents immediately after purchase. He knows the initial distribution of the coins. If Billy's strategy is to stop buying boxes once his net return (total coins collected minus the number... | [
"5"
] | [
"$N$ coins are distributed among $M$ indistinguishable boxes. Billy buys boxes one at a time for one coin each and can examine their contents immediately after purchase. He knows the initial distribution of the coins. If Billy's strategy is to stop buying boxes once his net return (total coins collected minus the n... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-165b2ced751b843520e64d7e9b532e2992d59f7ee4714a31de4d37d2231296d5 | $N$ cossacks are split into 3 groups. Cossack Taras moved from the first group to the second, cossack Andriy moved from the second to the third, and cossack Ostap moved from the third group to the first. It turned out that the average height of the cossacks in the first group decreased by 8 cm, while in the second and ... | [
"27"
] | [
"$N$ cossacks are split into 3 groups. Cossack Taras moved from the first group to the second, cossack Andriy moved from the second to the third, and cossack Ostap moved from the third group to the first. It turned out that the average height of the cossacks in the first group decreased by 8 cm, while in the second... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d1c0190030f9f4d93e42229e528f43291f5e789e16b5d1af6f2fae23d4bf786 | $N$ couples meet for a party and exchange gifts. Each of the $2N$ people writes their name on a piece of paper and puts it into a hat. They take turns drawing names and give their gift to the person whose name they drew. It is not allowed for anyone to draw either their own name or the name of their partner. How many d... | [
"\\sum_{k=0}^{N} (-1)^k \\binom{N}{k} \\cdot !(2N - 2k)"
] | [
"$N$ couples meet for a party and exchange gifts. Each of the $2N$ people writes their name on a piece of paper and puts it into a hat. They take turns drawing names and give their gift to the person whose name they drew. It is not allowed for anyone to draw either their own name or the name of their partner. How m... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-2a0ce38a5f5699ca33d613d82416bb9700d8ff6ce339ba55eef11a5f7596b9d8 | $N$ integers are placed around a circle. Each of these numbers is greater than the sum of the two numbers immediately after it in a clockwise direction. At least how many negative numbers can there be around the circle? | [
"N"
] | [
"$N$ integers are placed around a circle. Each of these numbers is greater than the sum of the two numbers immediately after it in a clockwise direction. At least how many negative numbers can there be around the circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-343dacc081da6fcf65c26d3aadaaa06aa3bab6836be788ddec2e5df0f1f817d0 | $N$ is a 50-digit number where all digits except the 25th from the left are 1. If $N$ is divisible by 13, find the unknown digit. | [
"9"
] | [
"$N$ is a 50-digit number where all digits except the 25th from the left are 1. If $N$ is divisible by 13, find the unknown digit."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-c0788574935709dae0973f0eebba8f826a218a0ae95344cec5a1bb17c228e45b | $N$ is a 50-digit number where all digits except the 26th digit (from the left) are 1. If $N$ is divisible by 13, find the 26th digit. | [
"3"
] | [
"$N$ is a 50-digit number where all digits except the 26th digit (from the left) are 1. If $N$ is divisible by 13, find the 26th digit."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-a810f1ca6b36693dacd1c212d8a57d1d707d93c35ce308f37686f9ac30106859 | $N$ is a positive integer whose units digit is $4$. If the $4$ is moved to the front of the number to form a new integer $M$, which is four times $N$, determine the number of digits in the smallest $N$ that satisfies the condition. | [
"6"
] | [
"$N$ is a positive integer whose units digit is $4$. If the $4$ is moved to the front of the number to form a new integer $M$, which is four times $N$, determine the number of digits in the smallest $N$ that satisfies the condition."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5dac5c7bac4dfb37f7023ea86a05e55688542ea3a6b37dfc5398e3519cb98f39 | $N$ is a six-digit palindrome which is divisible by $6$. The number obtained after deleting the leftmost and the rightmost digit of $N$ is divisible by $4$. How many values does $N$ have? | [
"33"
] | [
"$N$ is a six-digit palindrome which is divisible by $6$. The number obtained after deleting the leftmost and the rightmost digit of $N$ is divisible by $4$. How many values does $N$ have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d82699a792c088e9d5adffba938fb857e4c9273f88b828490c42f2f1d371d2d9 | $N$ people are standing in a circle, and Person 1 has the die. If someone rolls an $N$-sided die and gets $K$, then player $K$ receives the die. The first player to roll their own number twice in a row wins. If the probability that Person 1 wins is $\frac{7}{127}$, find the value of $N$. | [
"19"
] | [
"$N$ people are standing in a circle, and Person 1 has the die. If someone rolls an $N$-sided die and gets $K$, then player $K$ receives the die. The first player to roll their own number twice in a row wins. If the probability that Person 1 wins is $\\frac{7}{127}$, find the value of $N$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-243e7408f92ad10e9794599782977c5d487f453a7862ec2b83e8d46267ccd8e7 | $N$ people sit in a circle. Each round, each person randomly looks to their left or right. If two adjacent people are facing each other, both of them exit the circle. Find a formula, in terms of $N$, for the expected number of rounds until no one is left in the circle. | [
"N-1"
] | [
"$N$ people sit in a circle. Each round, each person randomly looks to their left or right. If two adjacent people are facing each other, both of them exit the circle. Find a formula, in terms of $N$, for the expected number of rounds until no one is left in the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-51f99a3d112d1cd2271913b3e12f9d667be7ba111d702c011e5397b40a4cf2b0 | $N$ points are marked on a circle. Two players take turns connecting two points with a chord, ensuring no chords intersect. The player who cannot make such a move loses. Who wins when played correctly? | [
"\\text{First Player}"
] | [
"$N$ points are marked on a circle. Two players take turns connecting two points with a chord, ensuring no chords intersect. The player who cannot make such a move loses. Who wins when played correctly?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-363b2b4df574bab7bc1337457f8d536e6cb28d59b300e09ec95b80a1505a0060 | $N$ prisoners are arranged in a circle. In succession, every second prisoner is killed; the last prisoner remaining is set free. We start with prisoner $1$, so that prisoner $2$ is the first to die. Which prisoner will be set free? | [
"\\((N - 2^{\\lfloor \\log_{2}N \\rfloor}) \\cdot 2 + 1\\)"
] | [
"$N$ prisoners are arranged in a circle. In succession, every second prisoner is killed; the last prisoner remaining is set free. We start with prisoner $1$, so that prisoner $2$ is the first to die. Which prisoner will be set free?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-b94c601c9937f898ef8074e7a834721f0451885f536b8c1e5bf4d065d9270343 | $N$ sociologists are sitting around a circular table. Some always tell the truth and the rest always lie. During the first round, each sociologist asks their left neighbor whether $2+2=4$. In each subsequent round, each sociologist asks their left neighbor if they received a "yes" answer in the preceding round. Find th... | [
"\\(\\frac{N(N+1)}{2}\\)"
] | [
"$N$ sociologists are sitting around a circular table. Some always tell the truth and the rest always lie. During the first round, each sociologist asks their left neighbor whether $2+2=4$. In each subsequent round, each sociologist asks their left neighbor if they received a \"yes\" answer in the preceding round. ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 2
} | 19 | 0 | 0 |
omr-ws-3fe9c1681f3c3211625592bee84998f0ea9f3e7325ebabb97b4913fa23bc3d46 | $N$ sociologists are sitting at a round table. Some always tell the truth and others always lie. During the first round, each sociologist asks their left neighbor whether $2+2=4$. In each subsequent round, each sociologist asks their left neighbor if they received a "yes" answer in the preceding round. Find the maximal... | [
"\\( N^2 \\)"
] | [
"$N$ sociologists are sitting at a round table. Some always tell the truth and others always lie. During the first round, each sociologist asks their left neighbor whether $2+2=4$. In each subsequent round, each sociologist asks their left neighbor if they received a \"yes\" answer in the preceding round. Find the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68fd15f6a0b9069e53e0f50ed3786ec860c4304362aba92d1e1b1a88c429e98c | $N$ visitors of the LycoReco cafe each give an integer rating from 1 to 10 on Takina Inoue's chocolate parfait. Given that the sum of all ratings given is 83 and the mean of the ratings equals the median of the ratings, what is the sum of all possible values of $N$? | [
"83"
] | [
"$N$ visitors of the LycoReco cafe each give an integer rating from 1 to 10 on Takina Inoue's chocolate parfait. Given that the sum of all ratings given is 83 and the mean of the ratings equals the median of the ratings, what is the sum of all possible values of $N$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-aed1cad15a7c7b9be5d63e3664964783200d3c15383605a54623677171c39704 | $OA$ and $OB$ are two roads enclosing an angle of $120^\circ$. $X$ and $Y$ start from $O$ at the same time. $X$ travels along $OA$ with a speed of $4$ km/hour and $Y$ travels along $OB$ with a speed of $3$ km/hour. Find the rate at which the shortest distance between $X$ and $Y$ is increasing after one hour. | [
"$\\sqrt{37} \\frac{km}{h}$"
] | [
"$OA$ and $OB$ are two roads enclosing an angle of $120^\\circ$. $X$ and $Y$ start from $O$ at the same time. $X$ travels along $OA$ with a speed of $4$ km/hour and $Y$ travels along $OB$ with a speed of $3$ km/hour. Find the rate at which the shortest distance between $X$ and $Y$ is increasing after one hour."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 30
} | 40 | 0 | 0 |
omr-ws-1c82404cf5e1681a3368b0b03fc8989c80990b1dfcc80cfb07c47efb635a12b2 | $P$ is a point inside $\triangle ABC$ such that $m\angle PBC=30^\circ$, $m\angle PBA=8^\circ$, and $m\angle PAB=m\angle PAC=22^\circ$. Find $m\angle APC$ in degrees. | [
"142"
] | [
"$P$ is a point inside $\\triangle ABC$ such that $m\\angle PBC=30^\\circ$, $m\\angle PBA=8^\\circ$, and $m\\angle PAB=m\\angle PAC=22^\\circ$. Find $m\\angle APC$ in degrees."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-812865e0e078f9af8207d8357a02c1e5e8cd51803a2cdf0f204484b734e1e24e | $P$ is a point inside an equilateral triangle $\triangle ABC$ such that $AP = 5$, $BP = 4$, and $CP = 3$. What is the length of the side of $\triangle ABC$? | [
"\\(\\sqrt{25 + 12\\sqrt{3}}\\)"
] | [
"$P$ is a point inside an equilateral triangle $\\triangle ABC$ such that $AP = 5$, $BP = 4$, and $CP = 3$. What is the length of the side of $\\triangle ABC$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 30
} | 77 | 0 | 0 |
omr-ws-1ee04e2dba784b1eb572b69e5a0a145fc83fc3f2f7b9276bdddc1ee3dc37a988 | $P$ is a point inside equilateral triangle $ABC$ such that $\angle BPC=120^\circ$. If the area of $\triangle ABP$ is $18$ and the area of $\triangle ACP$ is $8$, find the area of $\triangle ABC$. | [
"38"
] | [
"$P$ is a point inside equilateral triangle $ABC$ such that $\\angle BPC=120^\\circ$. If the area of $\\triangle ABP$ is $18$ and the area of $\\triangle ACP$ is $8$, find the area of $\\triangle ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-04acec097f84ae88488190eb75e6eea2228f8e7d981daafeba848e5d801133cf | $P$ is a point inside rectangle $ABCD$. Given $PA=x-3$, $PB=2x$, $PC=x-4$, and $PD=x+5$, compute the value of $x$. | [
"-8"
] | [
"$P$ is a point inside rectangle $ABCD$. Given $PA=x-3$, $PB=2x$, $PC=x-4$, and $PD=x+5$, compute the value of $x$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-54b73a21be24cd984d6286c874596e69002b0bef4a436de38bf03dd57727e94a | $P$ is a point inside the triangle $\triangle ABC$. Lines are drawn through $P$ parallel to the sides of the triangle. The areas of the three resulting triangles $\triangle PMN$, $\triangle PLK$, and $\triangle PRS$ are $9$, $25$, and $81$, respectively. What is the area of $\triangle ABC$? | [
"289"
] | [
"$P$ is a point inside the triangle $\\triangle ABC$. Lines are drawn through $P$ parallel to the sides of the triangle. The areas of the three resulting triangles $\\triangle PMN$, $\\triangle PLK$, and $\\triangle PRS$ are $9$, $25$, and $81$, respectively. What is the area of $\\triangle ABC$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-69633605a918887329a3a52f2ee19fc38612bd659298534fe7e6e95a54ebac8a | $P$ is a point inside the triangle $\triangle{ABC}$. Lines are drawn through $P$ parallel to the sides of the triangle. The areas of the three resulting triangles with a vertex at $P$ have areas $4$, $9$, and $49$. What is the area of $\triangle{ABC}$? | [
"144"
] | [
"$P$ is a point inside the triangle $\\triangle{ABC}$. Lines are drawn through $P$ parallel to the sides of the triangle. The areas of the three resulting triangles with a vertex at $P$ have areas $4$, $9$, and $49$. What is the area of $\\triangle{ABC}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-8603d707a640bdefa06e9ba2a1ac00093400094fb73896028a6e9e1c8beca0f4 | $P$ is a point inside triangle $ABC$ such that $\angle PBC = 30^\circ$, $\angle PBA = 8^\circ$, and $\angle PAB = \angle PAC = 22^\circ$. Find $\angle APC$, in degrees. | [
"142"
] | [
"$P$ is a point inside triangle $ABC$ such that $\\angle PBC = 30^\\circ$, $\\angle PBA = 8^\\circ$, and $\\angle PAB = \\angle PAC = 22^\\circ$. Find $\\angle APC$, in degrees."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 20
} | 68 | 0 | 0 |
omr-ws-10395d04be6b02450ccc0f682bdfce90739554d2d3b4251e8595be3e8d6c2f9c | $P$ is a point inside triangle $ABC$ such that $\angle PBC=30^\circ$, $\angle PBA=8^\circ$, and $\angle PAB=\angle PAC=22^\circ$. Find $\angle APC$. | [
"142^\\circ"
] | [
"$P$ is a point inside triangle $ABC$ such that $\\angle PBC=30^\\circ$, $\\angle PBA=8^\\circ$, and $\\angle PAB=\\angle PAC=22^\\circ$. Find $\\angle APC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-e0bf6e1301f32986af5849beb88661ecc8c3f4d9dbcf93c2fb91b78a9c7be199 | $P$ is a point inside triangle $ABC$. The distance from any vertex of the triangle to its opposite side is $m$ times the distance from $P$ to that side. What is the maximum value of $m$? | [
"3"
] | [
"$P$ is a point inside triangle $ABC$. The distance from any vertex of the triangle to its opposite side is $m$ times the distance from $P$ to that side. What is the maximum value of $m$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-10a3c12654e672948445fc3cab48cd4714b349886363e1444ab05ed4f0110028 | $P$ is an arbitrary point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, and $A(-c,0)$ and $B(c,0)$ are the foci of the ellipse. Draw a line segment from $B$ to $P$ and extend it to $Q$ such that the distance from $P$ to $Q$ is twice the distance from $B$ to $P$. Find the locus of $Q$. | [
"\\dfrac{(x + 2c)^2}{9a^2} + \\dfrac{y^2}{9b^2} = 1"
] | [
"$P$ is an arbitrary point on the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$, and $A(-c,0)$ and $B(c,0)$ are the foci of the ellipse. Draw a line segment from $B$ to $P$ and extend it to $Q$ such that the distance from $P$ to $Q$ is twice the distance from $B$ to $P$. Find the locus of $Q$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4e3a47f27446fbb32339d4655d1267c908cb489e96fbc05616b06b5bbeb3461d | $P$, $Q$, $R$ and $S$ are respectively the midpoints of the sides $AB$, $BC$, $CD$, and $DA$ of a parallelogram $ABCD$. Find the area of the figure bounded by the straight lines $AQ$, $BR$, $CS$ and $DP$ knowing that the area of the parallelogram is equal to $a^2$. | [
"\\(\\frac{1}{5}a^2\\)"
] | [
"$P$, $Q$, $R$ and $S$ are respectively the midpoints of the sides $AB$, $BC$, $CD$, and $DA$ of a parallelogram $ABCD$. Find the area of the figure bounded by the straight lines $AQ$, $BR$, $CS$ and $DP$ knowing that the area of the parallelogram is equal to $a^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-855a952a542c8ec34cf886f334d1b57617bd21ea7373cf619e2dfe3523572ca2 | $P(x)$ and $Q(x)$ are two polynomials such that \[ P(P(x)) = P(x)^{16} + x^{48} + Q(x). \] Find the smallest possible degree of $Q$. | [
"35"
] | [
"$P(x)$ and $Q(x)$ are two polynomials such that \n\\[\nP(P(x)) = P(x)^{16} + x^{48} + Q(x).\n\\]\nFind the smallest possible degree of $Q$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-2b5e20bb96696e373dba856fd1c4ef987f403ba20b2806627666a42cd484015b | $P(x)$ is a polynomial of degree $3n$ such that \begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{eqnarray*} Determine $n$. | [
"4"
] | [
"$P(x)$ is a polynomial of degree $3n$ such that \n\\begin{eqnarray*} P(0) = P(3) = \\cdots &=& P(3n) = 2, \\\\ P(1) = P(4) = \\cdots &=& P(3n-2) = 1, \\\\ P(2) = P(5) = \\cdots &=& P(3n-1) = 0, \\quad\\text{ and }\\\\ && P(3n+1) = 730.\\end{eqnarray*} \nDetermine $n$."
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-7ca5f580e0ca6ed9f5ae579a7527b5adf29210b55ec42fc4f71106cbd5d09cf3 | $P(x)$ is a polynomial of degree at most 6 such that $P(1)$, $P(2)$, $P(3)$, $P(4)$, $P(5)$, $P(6)$, and $P(7)$ are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ in some order. What is the maximum possible value of $P(8)$? | [
"312"
] | [
"$P(x)$ is a polynomial of degree at most 6 such that $P(1)$, $P(2)$, $P(3)$, $P(4)$, $P(5)$, $P(6)$, and $P(7)$ are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ in some order. What is the maximum possible value of $P(8)$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-7782e60ffb1cff5435157c2d9c62989c06ed79b7e6f801f614174e614ba60c9e | $P(x)$ is a polynomial with real coefficients such that $P(a_1) = 0, P(a_{i+1}) = a_i$ ($i = 1, 2, \ldots$) where $\{a_i\}_{i=1,2,\ldots}$ is an infinite sequence of distinct natural numbers. Determine the possible values of the degree of $P(x)$. | [
"1"
] | [
"$P(x)$ is a polynomial with real coefficients such that $P(a_1) = 0, P(a_{i+1}) = a_i$ ($i = 1, 2, \\ldots$) where $\\{a_i\\}_{i=1,2,\\ldots}$ is an infinite sequence of distinct natural numbers. Determine the possible values of the degree of $P(x)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bf85e117a96d820d7e6cd199a9f3961a47a8803c8a587866ebfa5df6a10c5dc7 | $P(x)$ is an odd function, and when $P(x)$ is divided by $x - 3$, it leaves a remainder of $6$. What is the remainder when $P(x)$ is divided by $x^2 - 9$? | [
"2x"
] | [
"$P(x)$ is an odd function, and when $P(x)$ is divided by $x - 3$, it leaves a remainder of $6$. What is the remainder when $P(x)$ is divided by $x^2 - 9$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eafa5e3dafa3b125759dc0026c2f00991ebde24f0a56455163078193ea9eed9f | $P-ABCD$ is a right pyramid with a square base $ABCD$ where each edge of the base has length 6, and $PA=PB=PC=PD=6\sqrt{2}$. The probability that a randomly selected point inside the pyramid is at least $\frac{\sqrt{6}}{3}$ units away from each face can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relat... | [
"9"
] | [
"$P-ABCD$ is a right pyramid with a square base $ABCD$ where each edge of the base has length 6, and $PA=PB=PC=PD=6\\sqrt{2}$. The probability that a randomly selected point inside the pyramid is at least $\\frac{\\sqrt{6}}{3}$ units away from each face can be expressed in the form $\\frac{m}{n}$ where $m$ and $n$ ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-70675b4f1176cf73ca9fcd22ae5d869b1f6477b688a66a68f2b9e927d3f176b4 | $P-ABCD$ is a right pyramid with square base $ABCD$ edge length 6, and $PA=PB=PC=PD=6\sqrt{2}$. The probability that a randomly selected point inside the pyramid is at least $\frac{\sqrt{6}}{3}$ units away from each face can be expressed in the form $\frac{m}{n}$, where $m, n$ are relatively prime positive integers. Fi... | [
"5041"
] | [
"$P-ABCD$ is a right pyramid with square base $ABCD$ edge length 6, and $PA=PB=PC=PD=6\\sqrt{2}$. The probability that a randomly selected point inside the pyramid is at least $\\frac{\\sqrt{6}}{3}$ units away from each face can be expressed in the form $\\frac{m}{n}$, where $m, n$ are relatively prime positive int... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e666da13c9fdd4092538670bdd875b33884932677d2baf0aee31e07079e777be | $PQ$ and $RS$ are two perpendicular chords of the hyperbola $xy = c^2$, with the center at $C$. Find the maximum value of the product of the slopes of $CP$, $CQ$, $CR$, and $CS$. | [
"\\(-1\\)"
] | [
"$PQ$ and $RS$ are two perpendicular chords of the hyperbola $xy = c^2$, with the center at $C$. Find the maximum value of the product of the slopes of $CP$, $CQ$, $CR$, and $CS$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19fec9248dd3fadd86fbcf0e32ee1037657ed3932ca008df068c894d43787c23 | $PQRS$ is a variable parallelogram inscribed in quadrilateral $ABCD$. $O$ is the point of intersection of the diagonals of $PQRS$. Find the locus of $O$. | [
"\\text{the segment connecting the midpoints of } AC \\text{ and } BD"
] | [
"$PQRS$ is a variable parallelogram inscribed in quadrilateral $ABCD$. $O$ is the point of intersection of the diagonals of $PQRS$. Find the locus of $O$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9d257fdcd5923e8cdbf292e437bbffa92f7922d5252a59c91c8f9eab46e4b8ba | $P_1$ is a parabola with a line of symmetry parallel to the $x$-axis, vertex at $(0, 1)$, and passing through $(2, 2)$. $P_2$ is a parabola with a line of symmetry parallel to the $y$-axis, vertex at $(1, 0)$, and passing through $(2, 2)$. Find all four points of intersection between $P_1$ and $P_2$. | [
"\\left(\\frac{3 - \\sqrt{5}}{4}, \\frac{3 + \\sqrt{5}}{4}\\right)"
] | [
"$P_1$ is a parabola with a line of symmetry parallel to the $x$-axis, vertex at $(0, 1)$, and passing through $(2, 2)$. $P_2$ is a parabola with a line of symmetry parallel to the $y$-axis, vertex at $(1, 0)$, and passing through $(2, 2)$. Find all four points of intersection between $P_1$ and $P_2$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-f5dfa50e8df393550cfe417b36175860e33112f58e9bbf4ed65a261bc574de4f | $P_{1}P_{2}P_{3}...P_{2007}$ is a regular $2007$-gon with $P_{1}$ at $(1, 0)$, and the farthest point on the circumference of the polygon from $P_{1}$ is at $(3, 0)$. Let $x_{n}$ and $y_{n}$ be the $x$ and $y$ coordinates of $P_{n}$, respectively. Let $z_{n}= x_{n}+iy_{n}$. Compute the remainder when $|z_{1}z_{2}z_{3}\... | [
"127"
] | [
"$P_{1}P_{2}P_{3}...P_{2007}$ is a regular $2007$-gon with $P_{1}$ at $(1, 0)$, and the farthest point on the circumference of the polygon from $P_{1}$ is at $(3, 0)$. Let $x_{n}$ and $y_{n}$ be the $x$ and $y$ coordinates of $P_{n}$, respectively. Let $z_{n}= x_{n}+iy_{n}$. Compute the remainder when $|z_{1}z_{2}z... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-862cde02f10d3086d561d14d52278ee1eadc37f628c397c7002f15b1a314180d | $R_1, R_2,$ and $R_3$ are three rectangles of equal area. The length of $R_1$ is 12 inches more than its width, the length of $R_2$ is 32 inches more than its width, and the length of $R_3$ is 44 inches more than its width. If all dimensions are integers, find the dimensions of each rectangle. | [
"\\( R_1: (48, 60) \\), \\( R_2: (40, 72) \\), \\( R_3: (36, 80) \\)"
] | [
"$R_1, R_2,$ and $R_3$ are three rectangles of equal area. The length of $R_1$ is 12 inches more than its width, the length of $R_2$ is 32 inches more than its width, and the length of $R_3$ is 44 inches more than its width. If all dimensions are integers, find the dimensions of each rectangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 29
} | 36 | 0 | 0 |
omr-ws-2ce7dd74c35d6ecff20280f189bf84d7260f225c63136c16eb2e47090198be5b | $S$ is a board containing all unit squares in the $xy$ plane whose vertices have integer coordinates and which lie entirely inside the circle $x^2 + y^2 = 1998^2$. In each square of $S$, the number $+1$ is written. An allowed move is to change the sign of every square in a given row, column, or diagonal. Let $N$ be the... | [
"no_valid_answer_found"
] | [
"$S$ is a board containing all unit squares in the $xy$ plane whose vertices have integer coordinates and which lie entirely inside the circle $x^2 + y^2 = 1998^2$. In each square of $S$, the number $+1$ is written. An allowed move is to change the sign of every square in a given row, column, or diagonal. Let $N$ b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-886706a6e3ed8309d12289b52e9024c52a519ab6c776ad9840f0e124fd632677 | $S$ is a set of $9$ consecutive positive integers such that the sum of the squares of the $5$ smallest integers in the set is the sum of the squares of the remaining $4$. What is the sum of all $9$ integers? | [
"360"
] | [
"$S$ is a set of $9$ consecutive positive integers such that the sum of the squares of the $5$ smallest integers in the set is the sum of the squares of the remaining $4$. What is the sum of all $9$ integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14a5150191250c082838a9804437891783b98952c012cde44efc8b7bdd688b94 | $S$ is a set of positive integers with the following properties: (a) There are exactly 3 positive integers missing from $S$. (b) If $a$ and $b$ are elements of $S$, then $a + b$ is an element of $S$. (We allow $a$ and $b$ to be the same.) How many possibilities are there for the set $S$? | [
"4"
] | [
"$S$ is a set of positive integers with the following properties:\n(a) There are exactly 3 positive integers missing from $S$.\n(b) If $a$ and $b$ are elements of $S$, then $a + b$ is an element of $S$. (We allow $a$ and $b$ to be the same.)\nHow many possibilities are there for the set $S$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-cdb8a7734155a812d8d7511f6d4b0e9141fe71def208b0efc889bf3f46ddecff | $T$ is an equilateral triangle with a perimeter of 120 mm. $V$ is a solid such that the distance from any point in $V$ to the perimeter of $T$ is at most 1.8 mm. What is the volume of $V$ in cubic millimeters? | [
"392.688\\pi"
] | [
"$T$ is an equilateral triangle with a perimeter of 120 mm. $V$ is a solid such that the distance from any point in $V$ to the perimeter of $T$ is at most 1.8 mm. What is the volume of $V$ in cubic millimeters?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-53755eec1f26673f98e8936528a4bf8b32ef5a33d9c7fb1ffb87ea9476583e74 | $WXYZ$ is a rectangular sheet of paper with $WX = 10$ and $XY = 12$. The paper is folded so that $W$ coincides with the midpoint of $\overline{XY}$. What is the length of the fold? | [
"\\(\\frac{34\\sqrt{34}}{15}\\)"
] | [
"$WXYZ$ is a rectangular sheet of paper with $WX = 10$ and $XY = 12$. The paper is folded so that $W$ coincides with the midpoint of $\\overline{XY}$. What is the length of the fold?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-eb730abf3eb5099758761be70bd73d6cd0a9a826011778d1c34c4733eda77db1 | $X_n$ denotes the number which is arranged by the digit $X$ written (concatenated) $n$ times. As an example, $2_{(3)} = 222$ and $5_{(2)} = 55$. For $A, B, C \in \{1, 2, \ldots, 9\}$ and $1 \leq n \leq 2019$, determine the number of ordered quadruples $(A, B, C, n)$ satisfying: \[ A_{(2n)} = 2 \left ( B_{(n)} \right ) ... | [
"6063"
] | [
"$X_n$ denotes the number which is arranged by the digit $X$ written (concatenated) $n$ times. As an example, $2_{(3)} = 222$ and $5_{(2)} = 55$. For $A, B, C \\in \\{1, 2, \\ldots, 9\\}$ and $1 \\leq n \\leq 2019$, determine the number of ordered quadruples $(A, B, C, n)$ satisfying:\n\\[ A_{(2n)} = 2 \\left ( B_{... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 17
} | 34 | 0 | 0 |
omr-ws-c72994f23d368062043357e717c90326f0162c7f6acdc5abe6a371514cfe2809 | $[-\pi ,0]$ which satisfy the equation $$\sin \alpha +\int _\alpha ^{2\alpha }\cos 2xdx=0.$$ | [
"$-\\pi ,\\ -\\frac{\\pi }{3},\\ 0$"
] | [
"$[-\\pi ,0]$ which satisfy the equation $$\\sin \\alpha +\\int _\\alpha ^{2\\alpha }\\cos 2xdx=0.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-112249caf35846b3093709c21953ee1d53a6e53caf6c5283ef5eadc8ee5246e3 | $\ ABCDEF$ is a non-regular hexagon where all six sides touch a circle and are of equal length. If $\angle A = 140^\circ$, find $\angle D$. | [
"100"
] | [
"$\\ ABCDEF$ is a non-regular hexagon where all six sides touch a circle and are of equal length. If $\\angle A = 140^\\circ$, find $\\angle D$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-6e8eb722ad77c78aa68f05adaa827f1f56689af405da27c17c873e5ae149c766 | $\angle APB$ is in the plane, such that a circle is tangent to $PA$ and $PB$ at $A$ and $B$ and $PA = PB = 21$. The angle bisector of $\angle APB$ hits the circle for the first time at $C$. If $PC = 9$, what is the area of the circle? | [
"400\\pi"
] | [
"$\\angle APB$ is in the plane, such that a circle is tangent to $PA$ and $PB$ at $A$ and $B$ and $PA = PB = 21$. The angle bisector of $\\angle APB$ hits the circle for the first time at $C$. If $PC = 9$, what is the area of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d457d154995d9a3e17df599ce4cfc3bf21f85152d0e628f838e7b439e139c98c | $\arccos x=\arctan x$. | [
"$\\sqrt {\\frac{\\sqrt 5-1}{2}}$"
] | [
"$\\arccos x=\\arctan x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cff624e94d6e1f98ebcd9d6310c702e2d7cb69ef9556e9c3fef36d9626943f78 | $\arcsin\frac{x}{2}+\arccos \left(x+\frac{\sqrt 3}{2}\right)=\frac{\pi }{6}$. | [
"0"
] | [
"$\\arcsin\\frac{x}{2}+\\arccos \\left(x+\\frac{\\sqrt 3}{2}\\right)=\\frac{\\pi }{6}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c7863dbb2b015a049c61a8b4010ffef4148abad67d49e68e8968dfa194c50483 | $\cal A$ and $\cal B$ take turns throwing a die, with $\cal A$ starting first. The first player to reach a sum of $10$ wins. Find the probability that $\cal B$ wins. | [
"\\dfrac{5}{11}"
] | [
"$\\cal A$ and $\\cal B$ take turns throwing a die, with $\\cal A$ starting first. The first player to reach a sum of $10$ wins. Find the probability that $\\cal B$ wins."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-933d6cc80bc99b973e3ef54a2d2b28936cf8d8ce36ac76c5de204e559f495771 | $\cos (x-y)-2\sin x+2\sin y=3$. | [
"$\\left(-\\frac{\\pi }{2}+2\\pi k;\\frac{\\pi }{2}+2\\pi n\\right)$"
] | [
"$\\cos (x-y)-2\\sin x+2\\sin y=3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-993e2d978c8d20531e162f79005d537efdce15a88185909e662252b6f11d90ba | $\cos 2x-\cos 8x+\cos 6x = 1$ | [
"$x=\\frac{\\pi }{8}(2n+1);\\ x=\\frac{\\pi n}{3}$"
] | [
"$\\cos 2x-\\cos 8x+\\cos 6x = 1$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e1835c641bd3ef1407b6638c71f91ae04f2d403e3155d56b616cc16d4349424 | $\cos 2x\left(1-\frac{3}{4}\sin ^2 2x\right)=1$. | [
"$\\pi k$"
] | [
"$\\cos 2x\\left(1-\\frac{3}{4}\\sin ^2 2x\\right)=1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-083f456db22f53f4ba2f8928b89bcd0dd14ae7c24b4d93f993b12cc69093900d | $\cos 4x+2\sin 4x=1$. | [
"$\\frac{\\pi }{2}k;\\ \\frac{1}{2}\\arctan 2+\\pi n$"
] | [
"$\\cos 4x+2\\sin 4x=1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4277f63416c5dcb939cb112cd18427116955fbd702e96218b213e075b46f499 | $\cos 4x=-2\cos^2 x$ | [
"$x=180^\\circ n\\pm 45^\\circ;\\ x=180^\\circ n\\pm 60^\\circ$"
] | [
"$\\cos 4x=-2\\cos^2 x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-84075cf4e85c9790397e356221d92222252080bf885f4e0e14deeb3ea7acf34b | $\cos 5x=\sin 15x$. | [
"$\\frac{\\pi }{40}+\\frac{\\pi }{10}k;\\ \\frac{\\pi }{20}+\\frac{\\pi }{5}n$"
] | [
"$\\cos 5x=\\sin 15x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cb8c631d9f8af1f689af835ec9d08f3ba17d16931f5ca7b9372a0a5cae53080e | $\cos \alpha +\sin 2\alpha -\cos 3\alpha $ | [
"$4\\sin 2\\alpha \\sin\\left(\\frac{\\alpha }{2}+15^\\circ\\right) \\cos\\left(\\frac{\\alpha }{2}-15^\\circ\\right)$"
] | [
"$\\cos \\alpha +\\sin 2\\alpha -\\cos 3\\alpha $"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f6bb8470fe450eb801732da8eb8fb791a8951935a918432fa7e4f116287464f9 | $\cos \frac{4x}{3}=\cos ^2 x$. | [
"$3\\pi k;\\ \\pm \\frac{\\pi }{4}+\\frac{3\\pi }{2}n$"
] | [
"$\\cos \\frac{4x}{3}=\\cos ^2 x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af699ad3d7b774598fe528b2d4493ae44a32da6cd57b2ee33daeb4fcb20be158 | $\cos ^6 2x=1+\sin ^4 x$. | [
"$\\pi k$"
] | [
"$\\cos ^6 2x=1+\\sin ^4 x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-97477088d9a77df0c2b88c4397853486bf73ad34cb620dcc62999ca116ddb010 | $\cos ^8 2x+\cos ^8 2x=\frac{41}{128}$. | [
"$\\pm \\frac{\\pi }{12}+\\frac{\\pi }{4}k$"
] | [
"$\\cos ^8 2x+\\cos ^8 2x=\\frac{41}{128}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-476581f942c51cca8ef4e6bbe221f5fe42feea347a8f26c4caf97919ca83d27e | $\cos x+\sqrt {\sin ^2 x-2\sin 2x+4\cos ^2 x}=0$. | [
"$\\frac{5\\pi }{4}+2\\pi k;\\ \\arctan 3+\\pi (2n+1)$"
] | [
"$\\cos x+\\sqrt {\\sin ^2 x-2\\sin 2x+4\\cos ^2 x}=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-073bc3acceb9e26b670c4376e7ddf45fd1456f4e0e91b765c0ab1f9a2735328d | $\cos x-\cos 2x=\sin 3x$ | [
"$x=\\displaystyle\\frac{2\\pi n}{3};\\ x=\\displaystyle\\frac{\\pi }{2}(4n-1);\\ x=\\displaystyle\\frac{\\pi }{4}(4n+1)$"
] | [
"$\\cos x-\\cos 2x=\\sin 3x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6eb296cdff453600371a05fd5e944eeb5e56ed56fbfbaea22f596758edd8e347 | $\cos x-y^2-\sqrt {y-x^2-1}\ge 0$. | [
"$x=0$, $y=1$. If the inequality is true, then $y\\ge x^2+1\\ge 1$. On the other hand, $\\cos x\\ge y^2+\\sqrt {y-x^2-1}\\ge y^2$, that is, $\\cos x\\ge 1$. Consequently, all the inequalities written down become equalities: $y=x^2+1=1$, whence $x=0$, $y=1$. A check shows that this number pair does indeed satisfy th... | [
"$\\cos x-y^2-\\sqrt {y-x^2-1}\\ge 0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f354512581869b1147ced1206d5ae5acabb5b65d289f31f52e61d8acd5784704 | $\cos x=-1/3$ $\tan x=2\sqrt 2$. | [
"$x=\\arccos \\frac{1}{3}+(2k+1)\\pi $"
] | [
"$\\cos x=-1/3$ $\\tan x=2\\sqrt 2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-789ab0f157ec60489caea2112090cc3b689c6bc3d74f1bf5895a49a22d0f0401 | $\cos x\cos 2x\cos 4x\cos 8x=\frac{1}{16}$. | [
"$\\frac{2\\pi k}{15}\\ \\left(k\\ne \\frac{15l}{2}\\right);\\ \\frac{\\pi }{17}+\\frac{2\\pi }{17}n\\ (n\\ne 17m+8)$"
] | [
"$\\cos x\\cos 2x\\cos 4x\\cos 8x=\\frac{1}{16}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be1acc9ffc57344b3663c028eaca3380913bdfb2a767d1c24dcaafc812df667f | $\cos x\sin 7x=\cos 3x\sin 5x$ | [
"$x=\\frac{\\pi n}{2};\\ x=\\frac{\\pi }{8}(2n+1)$"
] | [
"$\\cos x\\sin 7x=\\cos 3x\\sin 5x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2602e75d3b64cf7ba4443b6d8df6f2060e30ce3c118effd20f082509356b317a | $\cos x\tan 3x=0$. | [
"$\\frac{\\pi }{3}k$"
] | [
"$\\cos x\\tan 3x=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3bd51327f39a4d500e3acbc72a47e1c92c6df59b3895775076c1fa7d9d34c228 | $\cos^2 x+3\sin^2 x+2\sqrt 3\sin x\cos x=1$ | [
"$x=\\pi n;\\ x=\\frac{\\pi }{3}(3n-1)$"
] | [
"$\\cos^2 x+3\\sin^2 x+2\\sqrt 3\\sin x\\cos x=1$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2567f981328b88cf2401654de881c3100153c25fb6ba27fd64234f141689448e | $\cot \alpha +\cot 2\alpha +\csc 2\alpha $ | [
"$2\\cot \\alpha $"
] | [
"$\\cot \\alpha +\\cot 2\\alpha +\\csc 2\\alpha $"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3ee2122bc14cb5bc874a83cf94346c86121a9e44cc9818a2a3884f57790e163b | $\cot x+\frac{\sin x}{1+\cos x}=2$ | [
"$x=\\pi n+(-1)^n \\frac{\\pi }{6}$"
] | [
"$\\cot x+\\frac{\\sin x}{1+\\cos x}=2$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95be3b80f071a5554a729f80b927ce32579601021e2f2cf10e84da8046cac04f | $\cot x+\frac{\sin x}{1+\cos x}=2$. | [
"$(-1)^k \\frac{\\pi }{6}+\\pi k$"
] | [
"$\\cot x+\\frac{\\sin x}{1+\\cos x}=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-834073281a30ad9f31692e2a68f710ebb310a09ffc76dccaab30613044781e09 | $\frac{(2.1-1.965):(1.2\cdot 0.045)}{0.00325:0.013}-\frac{1:0.25}{1.6\cdot 0.625}$ | [
"6"
] | [
"$\\frac{(2.1-1.965):(1.2\\cdot 0.045)}{0.00325:0.013}-\\frac{1:0.25}{1.6\\cdot 0.625}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-38052175ef55676e4fdeaca7f74f9403db5e7891f83d7749ab38602a6dc53678 | $\frac{1+\cos \alpha +\cos 2\alpha +\cos 3\alpha }{\cos \alpha +2\cos ^2 \alpha -1}$ | [
"$2\\cos \\alpha $"
] | [
"$\\frac{1+\\cos \\alpha +\\cos 2\\alpha +\\cos 3\\alpha }{\\cos \\alpha +2\\cos ^2 \\alpha -1}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f5fcce3fa3ba18f13476b97231412daff9c2faec71a5082020d075b30aa7e711 | $\frac{1+\sin x}{1+\cos x}=\frac{1}{2}$. | [
"$2\\pi k;\\ 2\\arctan (-2)+2\\pi n$"
] | [
"$\\frac{1+\\sin x}{1+\\cos x}=\\frac{1}{2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42951f8e6763ff81554064eb30c60c904e649924b0b6d19e4e891371fc492763 | $\frac{1+\sin\alpha -\cos \alpha }{\sin\frac{\alpha }{2}}$ | [
"$2\\sqrt 2\\cos\\left(45^\\circ-\\frac{\\alpha }{2}\\right)$"
] | [
"$\\frac{1+\\sin\\alpha -\\cos \\alpha }{\\sin\\frac{\\alpha }{2}}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-660c984ddcd50acd9a0251f8f99fd73007fbb5569fd643e3002e493f62071946 | $\frac{1+x^2+xy}{x+y}=2-y$ $\log _{2^{1-y}}2^{x^2}=1+y$. | [
"$x_1=1,\\ y=0$"
] | [
"$\\frac{1+x^2+xy}{x+y}=2-y$ $\\log _{2^{1-y}}2^{x^2}=1+y$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-267eb9b5f78f063804d97e46875ba449629ebf012a2e164a5f35c6762fe07fef | $\frac{172\frac{5}{6}-170\frac{1}{3}+3\frac{5}{12}}{0.8\cdot 0.25}$ | [
"$29\\frac{7}{12}$"
] | [
"$\\frac{172\\frac{5}{6}-170\\frac{1}{3}+3\\frac{5}{12}}{0.8\\cdot 0.25}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-963d6123d4968ff6af3b99f5a54f21515f0c8490e52cafd2d8b4b814acafd027 | $\frac{1}{1+\sqrt 2+\sqrt 3}$. | [
"$\\frac{\\sqrt 2\\left(1+\\sqrt 2-\\sqrt 3\\right)}{4}$"
] | [
"$\\frac{1}{1+\\sqrt 2+\\sqrt 3}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aa7389cc006ee1211e6a6e5a397aef02fcb745f4c4c1129ed5c7ce7efecf95f7 | $\frac{1}{2}\sin 3x+\frac{\sqrt 3}{2}\cos 3x=\sin 5x$. | [
"$\\frac{\\pi }{6}+\\pi k;\\ \\frac{\\pi }{12}+\\frac{\\pi }{4}n$"
] | [
"$\\frac{1}{2}\\sin 3x+\\frac{\\sqrt 3}{2}\\cos 3x=\\sin 5x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fce0034feee119adc7b90395189e1ff1afcaa2ba9dc060c29e4871b0387886d8 | $\frac{1}{3}:\frac{2}{3}+0.228:\left[\left(1.5291 -\frac{14.53662}{3-0.095}\cdot 0.305\right):0.12\right]$ | [
"10"
] | [
"$\\frac{1}{3}:\\frac{2}{3}+0.228:\\left[\\left(1.5291 -\\frac{14.53662}{3-0.095}\\cdot 0.305\\right):0.12\\right]$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9c769dc4904f4bdd2f89f0e029e403faaa75b3b8aba80b72e34e465e36765330 | $\frac{1}{\sqrt [3]{4}+\sqrt [3]{6}+\sqrt [3]{9}}$. | [
"$\\sqrt [3]{3}-\\sqrt [3]{2}$"
] | [
"$\\frac{1}{\\sqrt [3]{4}+\\sqrt [3]{6}+\\sqrt [3]{9}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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