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-3d44f7fe91a6c4c6bd0c2b74002b9fb9f472d52c6c4ed0f09f7cd58805e53c41 | Two individuals A and B start at position 0 on the number line. Their positions at any time are given by \(a\) and \(b\) respectively. They take a step (at least one changes position) every second according to the equation \(a^2 + b^4 = t\), where \(t\) is the number of seconds elapsed since the start of the game. If t... | [
"3"
] | [
"Two individuals A and B start at position 0 on the number line. Their positions at any time are given by \\(a\\) and \\(b\\) respectively. They take a step (at least one changes position) every second according to the equation \\(a^2 + b^4 = t\\), where \\(t\\) is the number of seconds elapsed since the start of t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-a01fb3c88f7ad2ee6113706e3987b897500082e2d527a63702eb94d9b54d82f3 | Two individuals, $A$ and $B$, will play heads or tails with a fair coin. They agree to flip the coin 5 times, and the one who wins in 3 or more rolls wins the game. Each bets $2800. After the first two launches, in which both $A$ wins, they decide to end the game. From a probabilistic point of view, how should the $560... | [
"A gets $4900$, B gets $700$."
] | [
"Two individuals, $A$ and $B$, will play heads or tails with a fair coin. They agree to flip the coin 5 times, and the one who wins in 3 or more rolls wins the game. Each bets $2800. After the first two launches, in which both $A$ wins, they decide to end the game. From a probabilistic point of view, how should the... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 31
} | 39 | 0 | 0 |
omr-ws-7f6aa96fda75530c0eb9ad5cc249d9ddf9f45711ac1f0216fa53a0df93f66d0e | Two infinite geometric series \(a + ar + ar^2 + \ldots\) and \(b + br + br^2 + \ldots\) have the same sum. The second term of the second series is the first term of the first series. If \(r = \frac{1}{2}\) for the second series, find \(r\) for the first series. | [
"\\(\\frac{3}{4}\\)"
] | [
"Two infinite geometric series \\(a + ar + ar^2 + \\ldots\\) and \\(b + br + br^2 + \\ldots\\) have the same sum. The second term of the second series is the first term of the first series. If \\(r = \\frac{1}{2}\\) for the second series, find \\(r\\) for the first series."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-958958da4b6ca519439f9e6a3f98833e9687eac092971f40969933fc4b1f7274 | Two integers \( a \) and \( b \) are randomly chosen from the set \(\{1, 2, 13, 17, 19, 87, 115, 121\}\), with \( a > b \). What is the expected value of the number of factors of \( ab \)? | [
"\\(\\frac{171}{28}\\)"
] | [
"Two integers \\( a \\) and \\( b \\) are randomly chosen from the set \\(\\{1, 2, 13, 17, 19, 87, 115, 121\\}\\), with \\( a > b \\). What is the expected value of the number of factors of \\( ab \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-20258f497e4629c7054f9f5dd51fcabdf31618428ca2e2646889dbc8cb34caba | Two integers \( m \) and \( n \) are chosen at random with replacement from the natural numbers 1 through 9. What is the probability that \( m^2 - n^2 \) is divisible by 4? | [
"\\(\\frac{41}{81}\\)"
] | [
"Two integers \\( m \\) and \\( n \\) are chosen at random with replacement from the natural numbers 1 through 9. What is the probability that \\( m^2 - n^2 \\) is divisible by 4?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-49a4f71537a9b759de3a71d50e1306b1712086ecc865c49b210e2f580622aa78 | Two integers \( m \) and \( n \) are chosen at random with replacement from the set \(\{1, 2, \ldots, 9\}\). What is the probability that \( m^2 - n^2 \) is even? | [
"\\(\\frac{41}{81}\\)"
] | [
"Two integers \\( m \\) and \\( n \\) are chosen at random with replacement from the set \\(\\{1, 2, \\ldots, 9\\}\\). What is the probability that \\( m^2 - n^2 \\) is even?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-42d2d1f7df5fd7f0f01e109b852be30efb29ea33349ab87a8e4e813823b83a92 | Two integers \(a\) and \(b\) are chosen such that both are less than 30. Professor A is given the sum of \(a\) and \(b\), and Professor B is given the product of \(a\) and \(b\). The following conversation occurs: - Professor A asks Professor B if they know \(a\) and \(b\). - Professor B responds that they do not know ... | [
"\\(a = 4, b = 13\\) or \\(a = 13, b = 4\\)"
] | [
"Two integers \\(a\\) and \\(b\\) are chosen such that both are less than 30. Professor A is given the sum of \\(a\\) and \\(b\\), and Professor B is given the product of \\(a\\) and \\(b\\). The following conversation occurs:\n- Professor A asks Professor B if they know \\(a\\) and \\(b\\).\n- Professor B responds... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 2,
"tir": 5
} | 12 | 0 | 0 |
omr-ws-5be0ca1ef04b52016a1891e09610ea5937fd1651b33e9d072d9c9d42b715f722 | Two integers are chosen at random between 0 and 10. What is the probability that they differ by no more than 5? | [
"\\(\\frac{91}{121}\\)"
] | [
"Two integers are chosen at random between 0 and 10. What is the probability that they differ by no more than 5?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-f182deeb4c0cfb4da8e24bf6e45b4573de6c06a91f5560d7dd84144f22bd3cc8 | Two integers are chosen independently at random from $\{1,2,\dots,n\}$. Then, independently at random, one of the divisors of each of these numbers is chosen. Let $p(n)$ be the probability that these two divisors are equal. Calculate the value of $\lim_{n\to\infty}np(n)$. | [
"1"
] | [
"Two integers are chosen independently at random from $\\{1,2,\\dots,n\\}$. Then, independently at random, one of the divisors of each of these numbers is chosen. Let $p(n)$ be the probability that these two divisors are equal. Calculate the value of $\\lim_{n\\to\\infty}np(n)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17c1a3f7bafd6844ec78a408002db87f11966ebfa2a254efe04d3a16babe8585 | Two integers are inserted into the list $3, 3, 8, 11, 28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers? | [
"60"
] | [
"Two integers are inserted into the list $3, 3, 8, 11, 28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 91,
"tir": 1
} | 92 | 0 | 0 |
omr-ws-9996c215912d9d2599c6af001bb78177e1183f67e8de48185fb9ac8bc5d54fb3 | Two integers are randomly selected from the set of integers greater than or equal to -5 and less than or equal to 5. The two numbers need not be different. What is the probability that the sum of the two integers is less than their product? | [
"\\(\\frac{49}{121}\\)"
] | [
"Two integers are randomly selected from the set of integers greater than or equal to -5 and less than or equal to 5. The two numbers need not be different. What is the probability that the sum of the two integers is less than their product?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4be70dc91f0c226ca5866bb8831bf3f06c24bae114472fbab85135970593df9a | Two integers are relatively prime if they have no common factors other than 1 or -1. What is the probability that a positive integer less than or equal to 30 is relatively prime to 30? Express your answer as a common fraction. | [
"\\frac{4}{15}"
] | [
"Two integers are relatively prime if they have no common factors other than 1 or -1. What is the probability that a positive integer less than or equal to 30 is relatively prime to 30? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61c3a196c7a645b809a920c2777becc6997cd65116f69aaaf3def79b3718d25f | Two integers are secretly drawn from the range ${2, 3, \ldots, 99}$. The sum of the two integers is given to Dr. S, and the product is given to Dr. P. Dr. P states, "I do not know the two numbers." Dr. S responds, "I knew that you didn't know the two numbers." Dr. P then says, "Now I know the two numbers." Dr. S conclu... | [
"13"
] | [
"Two integers are secretly drawn from the range ${2, 3, \\ldots, 99}$. The sum of the two integers is given to Dr. S, and the product is given to Dr. P. Dr. P states, \"I do not know the two numbers.\" Dr. S responds, \"I knew that you didn't know the two numbers.\" Dr. P then says, \"Now I know the two numbers.\" ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-346c1ae97e842430d2dac6d4d0133818727facbf5d16e0840411f76e0dc5d45d | Two integers between $-1024$ and $1023$ are chosen at random. What is the probability that the sum of the two integers is also between $-1024$ and $1023$? | [
"\\(\\frac{1021}{2043}\\)"
] | [
"Two integers between $-1024$ and $1023$ are chosen at random. What is the probability that the sum of the two integers is also between $-1024$ and $1023$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e80ba008bc46156c633cf1987bc28e907b5d6ab4a555dd83b784d7960b4c6d83 | Two integers greater than 1 have a sum less than 100. Serge knows their sum, and Polly knows their product. Polly initially cannot determine the numbers, but after Serge confirms he knew Polly wouldn't be able to determine them, Polly then figures out the numbers, and subsequently, so does Serge. Determine the two numb... | [
"4 and 13"
] | [
"Two integers greater than 1 have a sum less than 100. Serge knows their sum, and Polly knows their product. Polly initially cannot determine the numbers, but after Serge confirms he knew Polly wouldn't be able to determine them, Polly then figures out the numbers, and subsequently, so does Serge. Determine the two... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 19
} | 24 | 0 | 0 |
omr-ws-aa7148435ea67a8cb033b63ede926c82ae83e2b8d891515debe11639e6d21015 | Two intelligent players (A and B, with A playing first) take turns filling numbers $1, 2, \dots, 9$ in a $3 \times 3$ grid without repetition. A wins if the determinant of the grid is negative, B wins if the determinant is positive, and it's a tie if the determinant is zero. Determine who has a winning strategy. | [
"A"
] | [
"Two intelligent players (A and B, with A playing first) take turns filling numbers $1, 2, \\dots, 9$ in a $3 \\times 3$ grid without repetition. A wins if the determinant of the grid is negative, B wins if the determinant is positive, and it's a tie if the determinant is zero. Determine who has a winning strategy.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-bb828d6e761bb40e8ec9e221d8b6f4c2df4c8363e120522251d356fc84b984b2 | Two intelligent players (A and B, with A playing first) take turns filling numbers $1, 2, \dots, 9$ in a $3 \times 3$ grid without repetition. A wins if the determinant of the grid is positive, B wins if the determinant is negative, and it's a tie if the determinant is zero. Determine who has a winning strategy. | [
"A"
] | [
"Two intelligent players (A and B, with A playing first) take turns filling numbers $1, 2, \\dots, 9$ in a $3 \\times 3$ grid without repetition. A wins if the determinant of the grid is positive, B wins if the determinant is negative, and it's a tie if the determinant is zero. Determine who has a winning strategy.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-afede7c93bf04d647098e926033009db932b109d39d45b8d267a1552dd56cabd | Two interior angles $A$ and $B$ of pentagon $ABCDE$ are $60^{\circ}$ and $85^{\circ}$. Two of the remaining angles, $C$ and $D$, are equal and the fifth angle $E$ is $15^{\circ}$ more than twice $C$. Find the measure of the largest angle. | [
"205^\\circ"
] | [
"Two interior angles $A$ and $B$ of pentagon $ABCDE$ are $60^{\\circ}$ and $85^{\\circ}$. Two of the remaining angles, $C$ and $D$, are equal and the fifth angle $E$ is $15^{\\circ}$ more than twice $C$. Find the measure of the largest angle."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9857782908abc3c591dbca9bd1d9b5abafd60cdcd4bfff6410cf521788ec473 | Two internal combustion engines of the same power output were subjected to an efficiency test and it was found that one of them consumed 600 grams of petrol, while the other, which was in operation 2 hours less, consumed 384 grams. If the first engine had consumed as much petrol per hour as the second, and the second, ... | [
"The first engine consumes 60 g/h; the second, 48 g/h"
] | [
"Two internal combustion engines of the same power output were subjected to an efficiency test and it was found that one of them consumed 600 grams of petrol, while the other, which was in operation 2 hours less, consumed 384 grams. If the first engine had consumed as much petrol per hour as the second, and the sec... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f5d9209ebcd5cefadaabf242b20ee1f3f03ec18cac74a9be0ccbbcd1753f1972 | Two intersecting chords of a circle divide one chord into lengths $3$ and $6$, and the other chord into lengths $x$ and $y$. If $x$ and $y$ are both integers less than ten, and the two chords are not equal in length, what is $x + y$? | [
"11"
] | [
"Two intersecting chords of a circle divide one chord into lengths $3$ and $6$, and the other chord into lengths $x$ and $y$. If $x$ and $y$ are both integers less than ten, and the two chords are not equal in length, what is $x + y$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb610dc62ad471dfb53a3cca0035cbf3dc40cf38b1c091b34daf28375b06c061 | Two intersecting circles $L_1$ and $L_2$ have centers at $M$ and $N$ respectively. The radii of the circles $L_1$ and $L_2$ are $5$ units and $6$ units respectively. The circle $L_1$ passes through the point $N$ and intersects the circle $L_2$ at points $P$ and $Q$. The point $U$ lies on the circle $L_2$ such that the ... | [
"4"
] | [
"Two intersecting circles $L_1$ and $L_2$ have centers at $M$ and $N$ respectively. The radii of the circles $L_1$ and $L_2$ are $5$ units and $6$ units respectively. The circle $L_1$ passes through the point $N$ and intersects the circle $L_2$ at points $P$ and $Q$. The point $U$ lies on the circle $L_2$ such that... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dacf8b10a1f0486194c7a419d47f3642dfa779cf9f9266afc0ca08e1393a6825 | Two intersecting circles $\omega_1$ and $\omega_2$ are given. The lines $AB$ and $CD$ are common tangents to these circles (points $A$ and $C$ lie on $\omega_1$, and points $B$ and $D$ lie on $\omega_2$). Let $M$ be the midpoint of the segment $AB$. Tangents drawn from point $M$ to $\omega_1$ and $\omega_2$, other than... | [
"10"
] | [
"Two intersecting circles $\\omega_1$ and $\\omega_2$ are given. The lines $AB$ and $CD$ are common tangents to these circles (points $A$ and $C$ lie on $\\omega_1$, and points $B$ and $D$ lie on $\\omega_2$). Let $M$ be the midpoint of the segment $AB$. Tangents drawn from point $M$ to $\\omega_1$ and $\\omega_2$,... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-6daf57b7890291bd140e2145d54a74caefd31930a602fae541cc66ef80dc12c4 | Two intersecting circles \( C1 \) and \( C2 \) have a common tangent that touches \( C1 \) at \( P \) and \( C2 \) at \( Q \). The two circles intersect at \( M \) and \( N \), where \( N \) is nearer to \( PQ \) than \( M \). The line \( PN \) meets the circle \( C2 \) again at \( R \). If the measure of angle \( PMR ... | [
"\\theta"
] | [
"Two intersecting circles \\( C1 \\) and \\( C2 \\) have a common tangent that touches \\( C1 \\) at \\( P \\) and \\( C2 \\) at \\( Q \\). The two circles intersect at \\( M \\) and \\( N \\), where \\( N \\) is nearer to \\( PQ \\) than \\( M \\). The line \\( PN \\) meets the circle \\( C2 \\) again at \\( R \\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-e9a55ac3f018eef3dfd0105921c14a247423be0c87dbf0d3bca861ace885f913 | Two intersecting circles \( C1 \) and \( C2 \) have a common tangent which touches \( C1 \) at \( P \) and \( C2 \) at \( Q \). The two circles intersect at \( M \) and \( N \), where \( N \) is nearer to \( PQ \) than \( M \) is. If the length of \( MN \) is 10 units, calculate the sum of the lengths of \( MP \) and \... | [
"10"
] | [
"Two intersecting circles \\( C1 \\) and \\( C2 \\) have a common tangent which touches \\( C1 \\) at \\( P \\) and \\( C2 \\) at \\( Q \\). The two circles intersect at \\( M \\) and \\( N \\), where \\( N \\) is nearer to \\( PQ \\) than \\( M \\) is. If the length of \\( MN \\) is 10 units, calculate the sum of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-d6cc018ed380f47070931d2903cb2b01aadfd95f217abbaa3ed07340da23e93f | Two intersecting circles \( C1 \) and \( C2 \) have a common tangent which touches \( C1 \) at \( P \) and \( C2 \) at \( Q \). The two circles intersect at \( M \) and \( N \), where \( N \) is nearer to \( PQ \) than \( M \) is. The line \( PN \) meets the circle \( C2 \) again at \( R \). If the measure of angle \( ... | [
"2\\theta"
] | [
"Two intersecting circles \\( C1 \\) and \\( C2 \\) have a common tangent which touches \\( C1 \\) at \\( P \\) and \\( C2 \\) at \\( Q \\). The two circles intersect at \\( M \\) and \\( N \\), where \\( N \\) is nearer to \\( PQ \\) than \\( M \\) is. The line \\( PN \\) meets the circle \\( C2 \\) again at \\( R... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-e79e91dd7f7d3d46f966850e4026b468274c49f046d61575bd18b67229ae8070 | Two intersecting circles \( C1 \) and \( C2 \) have a common tangent which touches \( C1 \) at \( P \) and \( C2 \) at \( Q \). The two circles intersect at \( M \) and \( N \), where \( N \) is nearer to \( PQ \) than \( M \). If the area of triangle \( MNP \) is \( A \), find the area of triangle \( MNQ \) in terms o... | [
"A"
] | [
"Two intersecting circles \\( C1 \\) and \\( C2 \\) have a common tangent which touches \\( C1 \\) at \\( P \\) and \\( C2 \\) at \\( Q \\). The two circles intersect at \\( M \\) and \\( N \\), where \\( N \\) is nearer to \\( PQ \\) than \\( M \\). If the area of triangle \\( MNP \\) is \\( A \\), find the area o... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-19eac2892b69fdec61ee812ac1bd8608140516330454d04deeeb157a1183054e | Two intersecting circles \( C_1 \) and \( C_2 \) have a common tangent intersecting \( C_1 \) at \( P \) and \( C_2 \) at \( Q \). The circles intersect at points \( M \) and \( N \), where \( N \) is closer to the line \( PQ \) than \( M \). Let \( A \) be the area of triangle \( MNP \) and \( B \) be the area of tria... | [
"0"
] | [
"Two intersecting circles \\( C_1 \\) and \\( C_2 \\) have a common tangent intersecting \\( C_1 \\) at \\( P \\) and \\( C_2 \\) at \\( Q \\). The circles intersect at points \\( M \\) and \\( N \\), where \\( N \\) is closer to the line \\( PQ \\) than \\( M \\). Let \\( A \\) be the area of triangle \\( MNP \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-6814f9d441759e483c93ad07037ad68b81241b0611ef01092bf688f22e810dcb | Two intersecting circles \( C_1 \) and \( C_2 \) have a common tangent which touches \( C_1 \) at \( P \) and \( C_2 \) at \( Q \). The two circles intersect at \( M \) and \( N \), where \( N \) is nearer to \( PQ \) than \( M \) is. Let \( A \) be the area of triangle \( MNP \) and \( B \) be the area of triangle \( ... | [
"0"
] | [
"Two intersecting circles \\( C_1 \\) and \\( C_2 \\) have a common tangent which touches \\( C_1 \\) at \\( P \\) and \\( C_2 \\) at \\( Q \\). The two circles intersect at \\( M \\) and \\( N \\), where \\( N \\) is nearer to \\( PQ \\) than \\( M \\) is. Let \\( A \\) be the area of triangle \\( MNP \\) and \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4dc343f7c021a391199d81150c57b22eff1f817678483f9341e424c20863e23d | Two intersecting circles \( C_1 \) and \( C_2 \) have a common tangent which touches \( C_1 \) at \( P \) and \( C_2 \) at \( Q \). The two circles intersect at \( M \) and \( N \), where \( N \) is nearer to \( PQ \) than \( M \) is. The line \( PN \) meets the circle \( C_2 \) again at \( R \). If the measure of angl... | [
"0"
] | [
"Two intersecting circles \\( C_1 \\) and \\( C_2 \\) have a common tangent which touches \\( C_1 \\) at \\( P \\) and \\( C_2 \\) at \\( Q \\). The two circles intersect at \\( M \\) and \\( N \\), where \\( N \\) is nearer to \\( PQ \\) than \\( M \\) is. The line \\( PN \\) meets the circle \\( C_2 \\) again at ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-43636e9c446151d2c99077924a40091d0deab03c31ed94effd8e75c58bb076aa | Two intersecting circles \( C_{1} \) and \( C_{2} \) have a common tangent which touches \( C_{1} \) at \( P \) and \( C_{2} \) at \( Q \). The two circles intersect at \( M \) and \( N \), where \( N \) is nearer to \( PQ \) than \( M \) is. If the area of triangle \( MNP \) is \( A \), find the area of triangle \( MN... | [
"A"
] | [
"Two intersecting circles \\( C_{1} \\) and \\( C_{2} \\) have a common tangent which touches \\( C_{1} \\) at \\( P \\) and \\( C_{2} \\) at \\( Q \\). The two circles intersect at \\( M \\) and \\( N \\), where \\( N \\) is nearer to \\( PQ \\) than \\( M \\) is. If the area of triangle \\( MNP \\) is \\( A \\), ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d794c37ed9b1504408cecc1343ac12350e600774fbffaf85018b0c7c10bfd184 | Two intersecting circles \(C_1\) and \(C_2\) have a common tangent which touches \(C_1\) at \(P\) and \(C_2\) at \(Q\). The two circles intersect at \(M\) and \(N\), where \(N\) is nearer to \(PQ\) than \(M\) is. If the length of \(MN\) is 10 units, calculate the sum of the areas of triangles \(MNP\) and \(MNQ\). | [
"50"
] | [
"Two intersecting circles \\(C_1\\) and \\(C_2\\) have a common tangent which touches \\(C_1\\) at \\(P\\) and \\(C_2\\) at \\(Q\\). The two circles intersect at \\(M\\) and \\(N\\), where \\(N\\) is nearer to \\(PQ\\) than \\(M\\) is. If the length of \\(MN\\) is 10 units, calculate the sum of the areas of triangl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-e728c3bf3297a6b14870bfbbfb3bcbb7cf2c71b65098785dad06404485f85437 | Two intersecting circles \(C_1\) and \(C_2\) have a common tangent which touches \(C_1\) at \(P\) and \(C_2\) at \(Q\). The two circles intersect at \(M\) and \(N\), where \(N\) is nearer to \(PQ\) than \(M\) is. The line \(PN\) meets the circle \(C_2\) again at \(R\). If the measure of angle \(PMQ\) is \(2\theta\) and... | [
"\\dfrac{5}{12}"
] | [
"Two intersecting circles \\(C_1\\) and \\(C_2\\) have a common tangent which touches \\(C_1\\) at \\(P\\) and \\(C_2\\) at \\(Q\\). The two circles intersect at \\(M\\) and \\(N\\), where \\(N\\) is nearer to \\(PQ\\) than \\(M\\) is. The line \\(PN\\) meets the circle \\(C_2\\) again at \\(R\\). If the measure of... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-7cae8a28ed7a76991f006935a2dd10dfa57579e9bb02795f83054ec1707c374c | Two intersecting circles \(\omega_1\) and \(\omega_2\) are given. Lines \(AB\) and \(CD\) are common tangents of \(\omega_1\) and \(\omega_2\) with \(A, C \in \omega_1\) and \(B, D \in \omega_2\). Let \(M\) be the midpoint of \(AB\). Tangents through \(M\) to \(\omega_1\) and \(\omega_2\) (other than \(AB\)) intersect ... | [
"1"
] | [
"Two intersecting circles \\(\\omega_1\\) and \\(\\omega_2\\) are given. Lines \\(AB\\) and \\(CD\\) are common tangents of \\(\\omega_1\\) and \\(\\omega_2\\) with \\(A, C \\in \\omega_1\\) and \\(B, D \\in \\omega_2\\). Let \\(M\\) be the midpoint of \\(AB\\). Tangents through \\(M\\) to \\(\\omega_1\\) and \\(\\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-703d58f3426f168b55acb058867df5adbafa505258e70eb2822c55f34552ce42 | Two intersecting circles are given in the plane. Let \( A \) be one of the points where the circles intersect. In each circle, a diameter is drawn that is parallel to the tangent to the other circle at point \( A \), and these diameters do not intersect. Let \( B, C, D, \) and \( E \) be the four endpoints of these dia... | [
"2"
] | [
"Two intersecting circles are given in the plane. Let \\( A \\) be one of the points where the circles intersect. In each circle, a diameter is drawn that is parallel to the tangent to the other circle at point \\( A \\), and these diameters do not intersect. Let \\( B, C, D, \\) and \\( E \\) be the four endpoints... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-d7550609406d98186e52120e00922291b05a5784004a2dc0f2c4f8c1eca00253 | Two intersecting circles each have radius $6$, and the distance between their centers is $6\sqrt{3}$. Find the area of the region that lies inside both circles. | [
"\\(12\\pi - 18\\sqrt{3}\\)"
] | [
"Two intersecting circles each have radius $6$, and the distance between their centers is $6\\sqrt{3}$. Find the area of the region that lies inside both circles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 2,
"tir": 30
} | 43 | 0 | 0 |
omr-ws-2ef8ec54a91eec43a235a7d9b91ae528291b4fd08274bb236c91a01d30226b36 | Two intersecting circles have radii $\sqrt{2}$ cm and $\sqrt{17}$ cm, with the distance between their centers being $5$ cm. A straight line intersects these circles at points $A$, $B$, $C$, and $D$ such that the lengths of segments $AB$, $BC$, and $CD$ are equal. Find the length of these segments. | [
"2"
] | [
"Two intersecting circles have radii $\\sqrt{2}$ cm and $\\sqrt{17}$ cm, with the distance between their centers being $5$ cm. A straight line intersects these circles at points $A$, $B$, $C$, and $D$ such that the lengths of segments $AB$, $BC$, and $CD$ are equal. Find the length of these segments."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-50135fa1b32eba2f5497023f100f8119f85b6dfa24ecc5431da9a2ae3aa3fc91 | Two intersecting lines, $k$ and $l$, form an angle with vertex $A$. Two points, $B$ and $C$, are given inside this angle. We choose every pair of points, $K$ and $L$, lying on $k$ and $l$ respectively, such that the quadrilateral $AKPL$, where $P$ is the point of intersection of the lines $BL$ and $CK$, is cyclic. If $... | [
"2 (B_y - C_y) x + 2 (C_x - B_x) y = B_x C_y - B_y C_x"
] | [
"Two intersecting lines, $k$ and $l$, form an angle with vertex $A$. Two points, $B$ and $C$, are given inside this angle. We choose every pair of points, $K$ and $L$, lying on $k$ and $l$ respectively, such that the quadrilateral $AKPL$, where $P$ is the point of intersection of the lines $BL$ and $CK$, is cyclic.... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4b241e3d3b05443cae684a0db0598b3425cb38102158719f1b608d3d0fb0d2c | Two investors, A and B, each have $10,000. They split their money among three accounts with annual yields of 8%, 6%, and 5%. Investor A only puts half as much in the 8% and 6% accounts as investor B. If investor A earns $580 and investor B earns $660, how much did investor B invest at 8%? | [
"4000"
] | [
"Two investors, A and B, each have $10,000. They split their money among three accounts with annual yields of 8%, 6%, and 5%. Investor A only puts half as much in the 8% and 6% accounts as investor B. If investor A earns $580 and investor B earns $660, how much did investor B invest at 8%?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-aa73baa123328067deaae08554ce97391ed8b44681086e39405aed6697c1c565 | Two is 10% of \( x \) and 20% of \( y \). What is \( x - y \)? | [
"10"
] | [
"Two is 10% of \\( x \\) and 20% of \\( y \\). What is \\( x - y \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-043b0cead9174b8d548dc4989cb3e8576c417b3fb302f5e88efd00c55786664c | Two isosceles triangles with the same angles at the base equal to $\alpha$ have a common side equal to $a$ (the triangles are connected externally). Find the area of the resulting quadrangle. | [
"\\( a^2 \\tan(\\alpha) \\)"
] | [
"Two isosceles triangles with the same angles at the base equal to $\\alpha$ have a common side equal to $a$ (the triangles are connected externally). Find the area of the resulting quadrangle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ffb7821a73cdc483680bc85e32a88d7eb5c496f1a1a15da729719ce1f9e6d908 | Two jokers are added to a 52-card deck, and the entire stack of 54 cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers? | [
"\\( \\frac{52}{3} \\)"
] | [
"Two jokers are added to a 52-card deck, and the entire stack of 54 cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-4336d62e4173fc9f3a5839612173e54a4f2d1580dc26a11a591f49a7e5903f71 | Two jokers are added to a 52-card deck, and the entire stack of 54 cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers? (HMMT) | [
"\\(\\frac{52}{3}\\)"
] | [
"Two jokers are added to a 52-card deck, and the entire stack of 54 cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers? (HMMT)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-970bf7c3cfde055c03b2633b0d84461d969663ccc361a4e9c0647a807c1c779d | Two kids $A$ and $B$ play a game as follows: From a box containing $n$ marbles ($n > 1$), they alternately take some marbles for themselves, such that: 1. $A$ goes first. 2. The number of marbles taken by $A$ in his first turn, denoted by $k$, must be between $1$ and $n$, inclusive. 3. The number of marbles taken in a ... | [
"0"
] | [
"Two kids $A$ and $B$ play a game as follows: From a box containing $n$ marbles ($n > 1$), they alternately take some marbles for themselves, such that:\n1. $A$ goes first.\n2. The number of marbles taken by $A$ in his first turn, denoted by $k$, must be between $1$ and $n$, inclusive.\n3. The number of marbles tak... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-255e98bca1a27aa4e38b4393994482e19bea9281d4fc75aa4f13d1ee1e911e45 | Two kids have x and y candies respectively. They play a game where one randomly gives the other one piece of candy until one kid has all the candies. What is the expected value of the number of exchanges that occur? | [
"\\( xy \\)"
] | [
"Two kids have x and y candies respectively. They play a game where one randomly gives the other one piece of candy until one kid has all the candies. What is the expected value of the number of exchanges that occur?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-e6d8f980ba22964063403bff4f801d1a7ce6d05797ec2ca96dc274c4e61f447d | Two kinds of cast iron with different percentages of chromium were alloyed. If we take five times as much of one kind of cast iron as the other, then the percentage of chromium in the alloy will be twice that in the smaller of the alloyed parts. Now if we take the same amount of both kinds, then the alloy will contain ... | [
"5\\% and 11\\%"
] | [
"Two kinds of cast iron with different percentages of chromium were alloyed. If we take five times as much of one kind of cast iron as the other, then the percentage of chromium in the alloy will be twice that in the smaller of the alloyed parts. Now if we take the same amount of both kinds, then the alloy will co... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-081a961957169a1168ffd88f3f7818fbf22b880ee6db6952e88ccfe992f0fb64 | Two kung-fu fighters, Pai and Hai, start at opposite corners of a 2013 by 2013 grid. Pai starts at the southwest (SW) corner and can move either north (N) or east (E). Hai starts at the northeast (NE) corner and can move either south (S) or west (W). Both fighters move simultaneously. What is the probability that Pai a... | [
"\\(\\frac{1}{\\binom{2012}{1006}}\\)"
] | [
"Two kung-fu fighters, Pai and Hai, start at opposite corners of a 2013 by 2013 grid. Pai starts at the southwest (SW) corner and can move either north (N) or east (E). Hai starts at the northeast (NE) corner and can move either south (S) or west (W). Both fighters move simultaneously. What is the probability that ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3da89ffc7b2ebf756b01bba11a28bb3ecb3130902efd70f87d124c93f953ba03 | Two ladders, one 20 ft long and the other 30 ft long, cross each other 10 ft above the ground in an alley. How wide is the alley? | [
"12"
] | [
"Two ladders, one 20 ft long and the other 30 ft long, cross each other 10 ft above the ground in an alley. How wide is the alley?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-0cb80e992f04d4959e25870a5dde4f5a098ea2c883f3b3bd54b01cb7f049adc0 | Two large and 1 small pump can fill a swimming pool in 4 hours. One large and 3 small pumps can also fill the same swimming pool in 4 hours. How many hours will it take 4 large and 4 small pumps to fill the swimming pool? (Assume all large pumps are similar and all small pumps are also similar.) | [
"\\(\\frac{5}{4}\\)"
] | [
"Two large and 1 small pump can fill a swimming pool in 4 hours. One large and 3 small pumps can also fill the same swimming pool in 4 hours. How many hours will it take 4 large and 4 small pumps to fill the swimming pool? (Assume all large pumps are similar and all small pumps are also similar.)"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42fb694d42f3b296f47a59be9dad8d01607feb5ebbd16e8fb5692f627b74e4f3 | Two law enforcement officers are 1000 meters apart. One officer heard an explosion 1.5 seconds after the other officer. Given that the speed of sound is approximately 340 meters per second, write an equation for the possible locations of the explosion relative to the two officers. | [
"\\(\\frac{(x-500)^2}{65025} - \\frac{y^2}{184975} = 1\\)"
] | [
"Two law enforcement officers are 1000 meters apart. One officer heard an explosion 1.5 seconds after the other officer. Given that the speed of sound is approximately 340 meters per second, write an equation for the possible locations of the explosion relative to the two officers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 2,
"tir": 12
} | 29 | 0 | 0 |
omr-ws-2194094d81f258d121171498b2cff1fbfddd621f013b7e32855a4e5219d58582 | Two letters are randomly picked from the word "ABSCOND." What is the probability that the letters are in dictionary order? | [
"\\(\\frac{1}{2}\\)"
] | [
"Two letters are randomly picked from the word \"ABSCOND.\" What is the probability that the letters are in dictionary order?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 26
} | 46 | 0 | 0 |
omr-ws-06f83015f007aff02fb391e80ea2c2b6bd4588fd120f7c8d25065e6b07ca5ce7 | Two lighthouses are located on a north-south line. From lighthouse A, the bearing of a ship 3742 meters away is 129 degrees and 43 minutes. From lighthouse B, the bearing is 39 degrees and 43 minutes. Find the distance between the lighthouses. | [
"5852"
] | [
"Two lighthouses are located on a north-south line. From lighthouse A, the bearing of a ship 3742 meters away is 129 degrees and 43 minutes. From lighthouse B, the bearing is 39 degrees and 43 minutes. Find the distance between the lighthouses."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-207bfc8dfc451c6fab76b8c09e6c73812d027484f0d7b05b5a1b5a27f624748b | Two line segments $AB$ and $CD$ are taken in the plane. Find the locus of points $M$ for which the sum of the areas of the triangles $AMB$ and $CMD$ is equal to a constant $a^2$. | [
"The locus of points \\( M \\) forms a parallelogram \\( PQRS \\) with diagonals \\( l \\) and \\( m \\)."
] | [
"Two line segments $AB$ and $CD$ are taken in the plane. Find the locus of points $M$ for which the sum of the areas of the triangles $AMB$ and $CMD$ is equal to a constant $a^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-942df05baf2da473320b743194e396044859349aa09d66b05cae13d94ffbafa7 | Two line segments OA and OB are inclined to each other at $120^\circ$ at point O. OA = OB = 20 cm, and P is a fixed point on the internal angle bisector of $\angle AOB$ such that OP = 10 cm. Two points are chosen on the line segments OA and OB such that the two points, and P form an equilateral triangle. The two points... | [
"100"
] | [
"Two line segments OA and OB are inclined to each other at $120^\\circ$ at point O. OA = OB = 20 cm, and P is a fixed point on the internal angle bisector of $\\angle AOB$ such that OP = 10 cm. Two points are chosen on the line segments OA and OB such that the two points, and P form an equilateral triangle. The two... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-b74e69689c184a2206c26bab4a70ffa0c6d4b791e09941397b6f83e82137dbd4 | Two line segments OA and OB are inclined to each other at $120^\circ$ at point O. P is a fixed point on the internal angle bisector of $\angle AOB$ with $OA = OB = 20 \text{ cm}$ and $OP = 10 \text{ cm}$. Two points are chosen on the line segments OA and OB such that the two points and P form an equilateral triangle. T... | [
"100"
] | [
"Two line segments OA and OB are inclined to each other at $120^\\circ$ at point O. P is a fixed point on the internal angle bisector of $\\angle AOB$ with $OA = OB = 20 \\text{ cm}$ and $OP = 10 \\text{ cm}$. Two points are chosen on the line segments OA and OB such that the two points and P form an equilateral tr... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-9efc0cfd122f93a06eb4f2e784627bff716f09fe6a8a6826b6fbaa03852476fb | Two line segments \(\overline{AB}\) and \(\overline{CD}\) are parallel and line segment \(\overline{AD}\) is perpendicular to both these lines. In addition, line \(\overline{AD}\) intersects line \(\overline{BC}\) at point \(E\). It is also known that line segment \(\overline{AE}\) is 6 units long and that line segment... | [
"30"
] | [
"Two line segments \\(\\overline{AB}\\) and \\(\\overline{CD}\\) are parallel and line segment \\(\\overline{AD}\\) is perpendicular to both these lines. In addition, line \\(\\overline{AD}\\) intersects line \\(\\overline{BC}\\) at point \\(E\\). It is also known that line segment \\(\\overline{AE}\\) is 6 units l... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d3ffbdf626b25c055ecef7c9c41792a6edd5ce2750170a20244ec894e38aa0ff | Two line segments are randomly spun within a circle. Let $\theta$ be the smallest angle between them. Calculate the expected value of $\frac{\theta}{180 - \theta}$. | [
"2 \\ln 2 - 1"
] | [
"Two line segments are randomly spun within a circle. Let $\\theta$ be the smallest angle between them. Calculate the expected value of $\\frac{\\theta}{180 - \\theta}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-6323380163867277f237bca8fffa42389df8dba4a6ae3abb8f9e82cafa946737 | Two line segments, $AB$ of length $a$ and $CD$ of length $b$, lie on skew lines the angle between which is $\alpha $. The feet $O$ and $O'$ of a common perpendicular (of length $c$) to these lines divide $AB$ and $CD$ so that $$OA:OB=2:3 \mbox{and} CO':O'D=3:2.$$ Find the lengths of the line segments $BD$ and $BC$. | [
"$BD=\\frac{1}{5}\\sqrt {9a^2+4b^2+25c^2-12ab\\cos \\alpha }$, $BC=\\frac{1}{5}\\sqrt {9a^2+9b^2+25c^2+18ab\\cos \\alpha }$"
] | [
"Two line segments, $AB$ of length $a$ and $CD$ of length $b$, lie on skew lines the angle between which is $\\alpha $. The feet $O$ and $O'$ of a common perpendicular (of length $c$) to these lines divide $AB$ and $CD$ so that $$OA:OB=2:3 \\mbox{and} CO':O'D=3:2.$$ Find the lengths of the line segments $BD$ and $B... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4ac6254842228ce536d27e6b3ad0a9928fe42c885b527d32110c68e5dd31c683 | Two lines $\ell_1$ and $\ell_2$ are perpendicular and intersect at point $(2, 4)$ in the $xy$ coordinate plane. A third line $\ell_3$ intersects $\ell_1$ at $(a,a)$ and intersects $\ell_2$ at $(a + 10, 2a + 9)$. Suppose that $a < 0$. Compute the area of a triangle whose vertices are the points that lie on exactly two l... | [
"25"
] | [
"Two lines $\\ell_1$ and $\\ell_2$ are perpendicular and intersect at point $(2, 4)$ in the $xy$ coordinate plane. A third line $\\ell_3$ intersects $\\ell_1$ at $(a,a)$ and intersects $\\ell_2$ at $(a + 10, 2a + 9)$. Suppose that $a < 0$. Compute the area of a triangle whose vertices are the points that lie on exa... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7d5d54b03c243707bc14169ed4d7e5c07ac8f07c1be7770b19a54ca4ef850f97 | Two lines \( \ell_1 \) and \( \ell_2 \) meet at \( O \). Points \( A \) and \( B \) are on \( \ell_1 \), and points \( A' \) and \( B' \) are on \( \ell_2 \). The lines \( AB' \) and \( A'B \) intersect at \( O' \). Let \( M \), \( N \), and \( P \) be the midpoints of \( AA' \), \( BB' \), and \( OO' \) respectively. ... | [
"1"
] | [
"Two lines \\( \\ell_1 \\) and \\( \\ell_2 \\) meet at \\( O \\). Points \\( A \\) and \\( B \\) are on \\( \\ell_1 \\), and points \\( A' \\) and \\( B' \\) are on \\( \\ell_2 \\). The lines \\( AB' \\) and \\( A'B \\) intersect at \\( O' \\). Let \\( M \\), \\( N \\), and \\( P \\) be the midpoints of \\( AA' \\)... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f62385b78cc22ed138840342ff5ed0fdeaf64538dddf7eedf0be618845d09a96 | Two lines \( l_1 \) and \( l_2 \) intersect at a point \( A \). Circles \( \omega_1 \) and \( \omega_2 \) are tangent to both \( l_1 \) and \( l_2 \) and intersect at points \( P \) and \( Q \). Given that line \( AP \) intersects \( \omega_1 \) again at \( B \) and \( \omega_2 \) again at \( C \), calculate the value ... | [
"\\dfrac{1}{3}"
] | [
"Two lines \\( l_1 \\) and \\( l_2 \\) intersect at a point \\( A \\). Circles \\( \\omega_1 \\) and \\( \\omega_2 \\) are tangent to both \\( l_1 \\) and \\( l_2 \\) and intersect at points \\( P \\) and \\( Q \\). Given that line \\( AP \\) intersects \\( \\omega_1 \\) again at \\( B \\) and \\( \\omega_2 \\) aga... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3e45e83aaec4fafb7b6a7e2c9958f7ea9c82ec2ecd43ee52dcb1619b321861e | Two lines \( l_{1} \) and \( l_{2} \) intersect at point \( P \). Points \( A \), \( B \), and \( C \) are on \( l_{1} \), and points \( D \), \( E \), and \( F \) are on \( l_{2} \) such that \( \frac{AB}{BC} = \frac{DE}{EF} \). Let \( Q \) be the fixed point (not \( P \)) through which the circumcircles of \( \triang... | [
"\\dfrac{21}{25}"
] | [
"Two lines \\( l_{1} \\) and \\( l_{2} \\) intersect at point \\( P \\). Points \\( A \\), \\( B \\), and \\( C \\) are on \\( l_{1} \\), and points \\( D \\), \\( E \\), and \\( F \\) are on \\( l_{2} \\) such that \\( \\frac{AB}{BC} = \\frac{DE}{EF} \\). Let \\( Q \\) be the fixed point (not \\( P \\)) through wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-c7bcc1aaf2b021d0f463de8674b28d8319f2ea96217589d5ba308d6720d7fd12 | Two lines \( m \) and \( n \) intersect at a unique point \( P \). A point \( M \) moves along \( m \) with constant speed, while another point \( N \) moves along \( n \) with the same speed. They both pass through the point \( P \), but not at the same time. Let \( Q \) be a fixed point such that \( P, Q, M, \) and \... | [
"no_valid_answer_found"
] | [
"Two lines \\( m \\) and \\( n \\) intersect at a unique point \\( P \\). A point \\( M \\) moves along \\( m \\) with constant speed, while another point \\( N \\) moves along \\( n \\) with the same speed. They both pass through the point \\( P \\), but not at the same time. Let \\( Q \\) be a fixed point such th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-34b6c81fb046e22f777f069368dd73e60e65dffbdc01fab915b1d088605bce6a | Two lines \(BC\) and \(EF\) are parallel. Let \(D\) be a point on segment \(BC\) different from \(B\) and \(C\). Let \(I\) be the intersection of \(BF\) and \(CE\). Denote the circumcircles of \(\triangle CDE\) and \(\triangle BDF\) as \(K\) and \(L\), respectively. Circles \(K\) and \(L\) are tangent to \(EF\) at \(E\... | [
"\\dfrac{5}{7}"
] | [
"Two lines \\(BC\\) and \\(EF\\) are parallel. Let \\(D\\) be a point on segment \\(BC\\) different from \\(B\\) and \\(C\\). Let \\(I\\) be the intersection of \\(BF\\) and \\(CE\\). Denote the circumcircles of \\(\\triangle CDE\\) and \\(\\triangle BDF\\) as \\(K\\) and \\(L\\), respectively. Circles \\(K\\) and ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-e2ba6b751e5f33e5d96faca8c658a643c8acdaa91fe0b8a91888903797ce3bfc | Two lines are perpendicular. One line has a direction vector of $\begin{pmatrix} 3 \\ -7 \end{pmatrix}.$ The other line has a direction vector of $\begin{pmatrix} a \\ 2 \end{pmatrix}.$ Find $a.$ | [
"\\frac{14}{3}"
] | [
"Two lines are perpendicular. One line has a direction vector of $\\begin{pmatrix} 3 \\\\ -7 \\end{pmatrix}.$ The other line has a direction vector of $\\begin{pmatrix} a \\\\ 2 \\end{pmatrix}.$ Find $a.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fc9db58a04a73c3c9ab0ac24ab327a8e65b1c85d741dfc1a1d6b243223d267e1 | Two lines defined by the equations $y = mx + 4$ and $y = 3x + b$, where $m$ and $b$ are constants, intersect at the point $(6, 10)$. What is the value of $b + m$? | [
"-7"
] | [
"Two lines defined by the equations $y = mx + 4$ and $y = 3x + b$, where $m$ and $b$ are constants, intersect at the point $(6, 10)$. What is the value of $b + m$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-83378a26415b5de010f6821c2ff2efe3e4ed5a4dffea5f06d1b7e4bcfa65aa5d | Two lines have the same non-zero $y$-intercept. The first line has a slope of 10 and an $x$-intercept of $(s, 0)$. The second line has a slope of 6 and an $x$-intercept of $(t, 0)$. What is the ratio of $s $ to $t$? Express your answer as a common fraction. | [
"\\frac 35"
] | [
"Two lines have the same non-zero $y$-intercept. The first line has a slope of 10 and an $x$-intercept of $(s, 0)$. The second line has a slope of 6 and an $x$-intercept of $(t, 0)$. What is the ratio of $s $ to $t$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ac677d35855d2bf29fe1ad6b074f42c9a7a47d36ed4d1c69e3c3350ab9d85b1 | Two lines intersect at $R$ within a circle with center $O$. The points of intersection on the circle are $P, Q, S$, and $T$ as illustrated. Suppose $\angle SOT = \theta$ and $\angle QOP = \phi$. Find $\angle SRT$ in terms of $\theta$ and $\phi$. | [
"\\(\\frac{1}{2}\\theta + \\frac{1}{2}\\phi\\)"
] | [
"Two lines intersect at $R$ within a circle with center $O$. The points of intersection on the circle are $P, Q, S$, and $T$ as illustrated. Suppose $\\angle SOT = \\theta$ and $\\angle QOP = \\phi$. Find $\\angle SRT$ in terms of $\\theta$ and $\\phi$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-b488616424cc8109c6c33246e6ea3a623b3a2a074c782887d92900c5c343b691 | Two lines intersect at a point $A_0$ and are separated by an angle of $6^\circ$. Points $A_2, A_4, A_6, \ldots$ are taken on one line and points $A_1, A_3, A_5, \ldots$ on the other line such that the segments $A_0 A_1, A_1 A_2, A_2 A_3, A_3 A_4, \ldots$ all have the same length. What is the maximum number of distinct ... | [
"15"
] | [
"Two lines intersect at a point $A_0$ and are separated by an angle of $6^\\circ$. Points $A_2, A_4, A_6, \\ldots$ are taken on one line and points $A_1, A_3, A_5, \\ldots$ on the other line such that the segments $A_0 A_1, A_1 A_2, A_2 A_3, A_3 A_4, \\ldots$ all have the same length. What is the maximum number of ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-7deaa3779ae88cff264d70ecdb3573403e859e3987b47fe342e9096d1d84830a | Two lines of length 3 units are mutually perpendicular at their ends, forming an L shape. A point is chosen randomly and independently on each line and fixed as the center of a circle with a unit radius. What is the probability that the circles intersect? | [
"\\(\\frac{\\pi}{9}\\)"
] | [
"Two lines of length 3 units are mutually perpendicular at their ends, forming an L shape. A point is chosen randomly and independently on each line and fixed as the center of a circle with a unit radius. What is the probability that the circles intersect?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-4ddf7b0ac01ec78ebcbf8f5dc5cd711eeb242a7007463dfb5253c9be0ea8fa31 | Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ at points $A, C, E$ and $B, D, F$ respectively, in that order from left to right. If the $x$-coordinates of $A, B, C, D, E, F$ are $a_1, a_2, a_3, a_4, a_5, a_6$ respectively, find the value of $\frac{(a_4 - a_3) - (a_2 - a_1) - (a_6 - a_5)}{a_4 - a... | [
"2"
] | [
"Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ at points $A, C, E$ and $B, D, F$ respectively, in that order from left to right. If the $x$-coordinates of $A, B, C, D, E, F$ are $a_1, a_2, a_3, a_4, a_5, a_6$ respectively, find the value of $\\frac{(a_4 - a_3) - (a_2 - a_1) - (a_6 - a_5)}{a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f16ce3e634c81366abcf7a28eaec57879a36c9dbbbfbe6333a369f6bed8243f2 | Two lines, $\ell$ and $m$, form an angle of $30^\circ$. Let $P$ be a variable point on line $\ell$, and $Q$ be a point on line $\ell$ such that $PQ=1$. Point $Q'$ is formed by reflecting $Q$ with respect to line $m$, and line $n$ is formed by reflecting line $m$ with respect to line $PQ'$. Let $S$ be the union of all l... | [
"\\dfrac{\\pi}{4}"
] | [
"Two lines, $\\ell$ and $m$, form an angle of $30^\\circ$. Let $P$ be a variable point on line $\\ell$, and $Q$ be a point on line $\\ell$ such that $PQ=1$. Point $Q'$ is formed by reflecting $Q$ with respect to line $m$, and line $n$ is formed by reflecting line $m$ with respect to line $PQ'$. Let $S$ be the union... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-9dd965d89cbd43fa0c1b8dfdb3c75f4dde96efcd5201887e701aa292023a4542 | Two lines, $l_1$ and $l_2$, are tangent to the parabola $x^2 - 4(x + y) + y^2 = 2xy + 8$ such that they intersect at a point whose coordinates sum to $-32$. The minimum possible sum of the slopes of $l_1$ and $l_2$ can be written as $\frac{m}{n}$ for relatively prime integers $m$ and $n$. Find $m+n$. | [
"63"
] | [
"Two lines, $l_1$ and $l_2$, are tangent to the parabola $x^2 - 4(x + y) + y^2 = 2xy + 8$ such that they intersect at a point whose coordinates sum to $-32$. The minimum possible sum of the slopes of $l_1$ and $l_2$ can be written as $\\frac{m}{n}$ for relatively prime integers $m$ and $n$. Find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-d5d411a0e886b27103ebdbf33699377ed7e4ad3a4df9794fa135bc156d553111 | Two lines, each of length 10 units, are spaced 5 units apart. Two points are chosen at random, one on each line. What is the probability that the length of the line segment connecting these two points is greater than $5\sqrt{2}$? | [
"0.25"
] | [
"Two lines, each of length 10 units, are spaced 5 units apart. Two points are chosen at random, one on each line. What is the probability that the length of the line segment connecting these two points is greater than $5\\sqrt{2}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-526977b93f4abe997839f40c57a5deef89526e7297cbf30c9a0d26422dacca24 | Two liters of glycerin was poured from a full vessel, the vessel topped with two liters of water. After stirring, two liters of the mixture was poured from the vessel and another two liters of water was added instead. The mixture thus obtained was stirred up and again, two liters of the mixture was replaced with two li... | [
"3.5 liters of glycerin and 0.5 liter of water"
] | [
"Two liters of glycerin was poured from a full vessel, the vessel topped with two liters of water. After stirring, two liters of the mixture was poured from the vessel and another two liters of water was added instead. The mixture thus obtained was stirred up and again, two liters of the mixture was replaced with t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d6d5a65c32de36fd540235986a750e9b0ce9e845836fd7d7e21b91ab536f2cb | Two logicians, S and P, are given the information that \( x, y \in \mathbb{N} \), \( 1 < x < y \), and \( x + y < 100 \). P is given the value of \( xy \) and S is given the value of \( x + y \). They have the following conversation: - P: I don't know the two numbers. - S: I know that you don't know the two numbers. - ... | [
"\\( x = 4 \\) and \\( y = 13 \\)"
] | [
"Two logicians, S and P, are given the information that \\( x, y \\in \\mathbb{N} \\), \\( 1 < x < y \\), and \\( x + y < 100 \\). P is given the value of \\( xy \\) and S is given the value of \\( x + y \\). They have the following conversation:\n\n- P: I don't know the two numbers.\n- S: I know that you don't kno... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 13
} | 17 | 0 | 0 |
omr-ws-ca443a63bf9b75cc6e528f45f5f0992f61a7a01c93edbc88e51e0ccbdfa682f3 | Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling along the flat surface until it comes to re... | [
"179"
] | [
"Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling along the flat surface until it c... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 8,
"tir": 4
} | 34 | 0 | 0 |
omr-ws-0c931ac4e2ddf702d752aa6ed5e9d721f4a27c09edd5cf63440eee7144253757 | Two lost lovers, Laxe and Kaan, are both standing on the equators of planets with radius $13$ miles. The center of the planets are $170$ miles apart. At some time, both of them are as close to each other as possible. The planets rotate in opposite directions of each other at the same rate. What is the maximum possible ... | [
"168"
] | [
"Two lost lovers, Laxe and Kaan, are both standing on the equators of planets with radius $13$ miles. The center of the planets are $170$ miles apart. At some time, both of them are as close to each other as possible. The planets rotate in opposite directions of each other at the same rate. What is the maximum poss... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-6f1d91cb3b2cb9027b6fb11c6c747bafc341933a98d25efa4897ff6cba4e4aa5 | Two magicians are performing a game with a chessboard of size $n \times n$. Initially, a spectator places a black or white token on each square. The first magician then asks the spectator to choose a square $C$. The first magician then chooses a square $D$ and flips the token on $D$. The second magician, who has been i... | [
"\\( n = 2^\\alpha \\)"
] | [
"Two magicians are performing a game with a chessboard of size $n \\times n$. Initially, a spectator places a black or white token on each square. The first magician then asks the spectator to choose a square $C$. The first magician then chooses a square $D$ and flips the token on $D$. The second magician, who has ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 37,
"tir": 2
} | 39 | 0 | 0 |
omr-ws-6e7e0230eb2e7ffef0600d64c2063e0dc01cdb9fa8d594cbb05e5302210e7ba8 | Two magicians perform a game where the first magician encloses the second magician in a cabin. The first magician asks an audience member, Daniel, to place tokens (black or white) on each square of an $n \times n$ chessboard. Daniel then chooses a square $C$. The first magician chooses a square $D$ and flips the token ... | [
"n \\text{ is a power of } 2"
] | [
"Two magicians perform a game where the first magician encloses the second magician in a cabin. The first magician asks an audience member, Daniel, to place tokens (black or white) on each square of an $n \\times n$ chessboard. Daniel then chooses a square $C$. The first magician chooses a square $D$ and flips the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2bed6fbeeda0d83eed69fd2fd9c37cbff40df8022c8983b161072666f84875b6 | Two magicians perform a trick with a circle marked with one semicircle. Viewers mark 100 points on the circle, and the first magician erases one of these points. The second magician then determines whether the erased point was on the marked semicircle based on the remaining 99 points. Let $N$ be the number of ways to m... | [
"374"
] | [
"Two magicians perform a trick with a circle marked with one semicircle. Viewers mark 100 points on the circle, and the first magician erases one of these points. The second magician then determines whether the erased point was on the marked semicircle based on the remaining 99 points. Let $N$ be the number of ways... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-d7424925591f70be0a0dcb8779cc05a11d073f9b58423ce10f6b864d8725f2e9 | Two magicians perform a trick with a deck of 100 cards labeled from 1 to 100. The first magician leaves the room. The second magician asks three spectators to each choose one card. The second magician then adds one more card from the remaining deck, and the four cards are shuffled. The first magician, upon returning, l... | [
"67"
] | [
"Two magicians perform a trick with a deck of 100 cards labeled from 1 to 100. The first magician leaves the room. The second magician asks three spectators to each choose one card. The second magician then adds one more card from the remaining deck, and the four cards are shuffled. The first magician, upon returni... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-2e20e078d6ad38b117033eeda7632a8ab51ae3f23ce8621bd9958ec9312c23a3 | Two mathematicians are on the corner of Barbarossa street and Martin Luther street and need to reach the corner of Meininger street and Martin Luther street. They do not know the direction or distance to Meininger Street along Martin Luther Street. What is the smallest positive integer $k$ such that they can be sure to... | [
"8"
] | [
"Two mathematicians are on the corner of Barbarossa street and Martin Luther street and need to reach the corner of Meininger street and Martin Luther street. They do not know the direction or distance to Meininger Street along Martin Luther Street. What is the smallest positive integer $k$ such that they can be su... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-088c08bdda2e0d897e3b8c6bbf310a0c91c76e0ff061a705f3c7a161f6152365 | Two mathematicians are surveying the damage done in a local park by a hurricane. Less than 1/3 of the trees were lost. If you multiply by 10 the number formed by taking the last two digits of the number of trees that there used to be, and add to this the number formed by removing the last two digits of the number of tr... | [
"243"
] | [
"Two mathematicians are surveying the damage done in a local park by a hurricane. Less than 1/3 of the trees were lost. If you multiply by 10 the number formed by taking the last two digits of the number of trees that there used to be, and add to this the number formed by removing the last two digits of the number ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff38e338e1cd3f9abb728a343fb1a9ae2e4f51e816cebbca44debf3a3058fee6 | Two mathematicians arrive at a cafeteria independently at random times between 9 AM and 10 AM and each stays for exactly \( m \) minutes. Find \( m \) if the probability that either one arrives while the other is in the cafeteria is 40%. | [
"\\( 60 - 12\\sqrt{15} \\)"
] | [
"Two mathematicians arrive at a cafeteria independently at random times between 9 AM and 10 AM and each stays for exactly \\( m \\) minutes. Find \\( m \\) if the probability that either one arrives while the other is in the cafeteria is 40%."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-56a96a8de1e8a8c0620778b1a939e858fba11dbe79b763e57a7e7a53ab920062 | Two mathematicians arrive at a cafeteria independently at random times between 9 AM and 10 AM and stay for exactly \( m \) minutes. Find the value of \( m \) if the probability that either one arrives while the other is in the cafeteria is 40%. | [
"13.524"
] | [
"Two mathematicians arrive at a cafeteria independently at random times between 9 AM and 10 AM and stay for exactly \\( m \\) minutes. Find the value of \\( m \\) if the probability that either one arrives while the other is in the cafeteria is 40%."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-1b05dc5d6a69fb8df9c3ebf64d62988384b9cab71e795fca7fa5b59f10aae4bd | Two mathematicians arrive at a cafeteria independently at random times between 9 a.m. and 10 a.m. and each stays for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40\%$. Given that $m = a - b\sqrt{c}$, where $a, b,$ and $c$ are positive integers and $c$ is not divi... | [
"87"
] | [
"Two mathematicians arrive at a cafeteria independently at random times between 9 a.m. and 10 a.m. and each stays for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40\\%$. Given that $m = a - b\\sqrt{c}$, where $a, b,$ and $c$ are positive integers and $c$ is n... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 8
} | 30 | 0 | 0 |
omr-ws-22c65bb52c446ba404dbae7e33698e1fa6f77f6a26a592c49d09c14ad4f33006 | Two mathematicians arrive at a cafeteria independently at random times between 9 and 10 a.m. and each stays for exactly \( m \) minutes. The probability that either one arrives while the other is in the cafeteria is 40%. Given that \( m = a - b\sqrt{c} \), where \( a \), \( b \), and \( c \) are positive integers and \... | [
"87"
] | [
"Two mathematicians arrive at a cafeteria independently at random times between 9 and 10 a.m. and each stays for exactly \\( m \\) minutes. The probability that either one arrives while the other is in the cafeteria is 40%. Given that \\( m = a - b\\sqrt{c} \\), where \\( a \\), \\( b \\), and \\( c \\) are positiv... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c80689ff1332189b83cf389c2d8b1f42cbd541424cfe24145c24e3a858a0ec08 | Two mathematicians take a lunch break separately between 12:00 and 1:00 and stay for $\frac{m}{n}$ minutes. If the probability of them meeting each other is 0.4, compute $m+n$. | [
"37"
] | [
"Two mathematicians take a lunch break separately between 12:00 and 1:00 and stay for $\\frac{m}{n}$ minutes. If the probability of them meeting each other is 0.4, compute $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-408c3c515cfed86fcbb96c942eaab7e81eb2d92086d280b53ab8ae498dbd3d6a | Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m = a - b\sqrt {c},$ where $a, b,$ and $c$ are p... | [
"87"
] | [
"Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \\%,$ and $m = a - b\\sqrt {c},$ where $a, b,$ and $c... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1b9b63b7787e2ec7436c4181b6e1a4d754e7a51e6c37a72a9d77b771d7ebf925 | Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m = a - b\sqrt {c},$ where $a, b,$ and $c$ are p... | [
"087"
] | [
"Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \\%,$ and $m = a - b\\sqrt {c},$ where $a, b,$ and... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 8
} | 31 | 0 | 0 |
omr-ws-f9b09b3f4c29d1f10818c9214432ead044f5f558c7bbfb64ee7e9fdbe976e0b0 | Two mathematicians visit a coffee house at a random time during a two-hour time frame, stay for exactly 20 minutes, and immediately leave. What is the probability that they do not run into each other? | [
"\\(\\frac{25}{36}\\)"
] | [
"Two mathematicians visit a coffee house at a random time during a two-hour time frame, stay for exactly 20 minutes, and immediately leave. What is the probability that they do not run into each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5f239a1634049dfc3a44516fb6b6c4c462db4fb5e18a10e50c7203df6b139ed6 | Two mathematicians were both born in the last 500 years. Each lives (or will live) to be 100 years old, then dies. Each mathematician is equally likely to be born at any point during those 500 years. What is the probability that they were contemporaries for any length of time? | [
"\\frac{9}{25}"
] | [
"Two mathematicians were both born in the last 500 years. Each lives (or will live) to be 100 years old, then dies. Each mathematician is equally likely to be born at any point during those 500 years. What is the probability that they were contemporaries for any length of time?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-92388a0305cf5531ebcd45c2dd2b024642560cc0529fec650a3a49accc5e1ba4 | Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects a secret positive integer \( n < 60 \). The computer tells Kelly the unit digit of \( n \) and tells Jason the number of divisors of \( n \). Kelly and Jason have the following dialogue: 1. Kelly: I don't know what \( n \) is, and I'm su... | [
"10"
] | [
"Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects a secret positive integer \\( n < 60 \\). The computer tells Kelly the unit digit of \\( n \\) and tells Jason the number of divisors of \\( n \\). Kelly and Jason have the following dialogue:\n\n1. Kelly: I don't know what \\( n \\... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-2147e30bee58a3b4f5d3440c4bad6b77220f23a9fc0d43485746fc48cc6521bf | Two mathematicians, Lagrangia and Galois, play a game where they alternately remove numbers from the set \(\{0, 1, 2, \ldots, 1024\}\). Lagrangia starts by removing 512 numbers, then Galois removes 256 numbers, and so on, until 2 numbers remain. The remaining numbers are \(a\) and \(b\) with \(a < b\). Galois pays Lagr... | [
"32"
] | [
"Two mathematicians, Lagrangia and Galois, play a game where they alternately remove numbers from the set \\(\\{0, 1, 2, \\ldots, 1024\\}\\). Lagrangia starts by removing 512 numbers, then Galois removes 256 numbers, and so on, until 2 numbers remain. The remaining numbers are \\(a\\) and \\(b\\) with \\(a < b\\). ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-bb98e415b730cddad0fdb9c8df1626d51cde2c1da4d7b82cdfe71a58f5bbee9d | Two mathematicians, X and Y, are given positive natural numbers A and B, respectively. They are told that A = M * N and B = M + N, where M and N are natural numbers greater than 1 and their sum is less than 100. The following conversation occurs: - X: "I really don't know your number." - Y: "Well, I knew this before yo... | [
"13"
] | [
"Two mathematicians, X and Y, are given positive natural numbers A and B, respectively. They are told that A = M * N and B = M + N, where M and N are natural numbers greater than 1 and their sum is less than 100. The following conversation occurs:\n- X: \"I really don't know your number.\"\n- Y: \"Well, I knew this... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-369fb3977b20adb1bb872f8b5f5b648e4e3a373743db0ae56f40d97c4f644905 | Two measuring sticks use the same unit, but the scale on one stick has 635 marks per unit and the scale on the other stick has 465 marks per unit. Suppose the sticks are placed together so that the 0 marks on each scale line up. What is the number of the next mark on the 465 scale that lines up exactly with the mark on... | [
"93"
] | [
"Two measuring sticks use the same unit, but the scale on one stick has 635 marks per unit and the scale on the other stick has 465 marks per unit. Suppose the sticks are placed together so that the 0 marks on each scale line up. What is the number of the next mark on the 465 scale that lines up exactly with the ma... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-5cb520858a6e680b3026c82f6f61069e7f6c73b1995c2ac9bc9de8c63a48da91 | Two medians of a triangle with unequal sides are 3 inches and 6 inches. Its area is \(3 \sqrt{15}\) square inches. What is the length of the third median in inches? | [
"\\(3 \\sqrt{6}\\)"
] | [
"Two medians of a triangle with unequal sides are 3 inches and 6 inches. Its area is \\(3 \\sqrt{15}\\) square inches. What is the length of the third median in inches?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-de80f37a210b3701ac7082b255960ffa2573e0a0ec3a2afc65c3df476efb096c | Two medians of a triangle with unequal sides are 3 inches and 6 inches. The area of the triangle is $3 \sqrt{15}$ square inches. What is the length of the third median in inches? | [
"\\(3\\sqrt{6}\\)"
] | [
"Two medians of a triangle with unequal sides are 3 inches and 6 inches. The area of the triangle is $3 \\sqrt{15}$ square inches. What is the length of the third median in inches?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-2e668d0ed551126882c2a08a9fb30a2d7285b6b755d1e5290f736870d83f139f | Two medians of an acute triangle divide it into regions with areas $4, 5, 7, x$. Find all possible values of $x$. | [
"\\( x = 2, 6, 8 \\)"
] | [
"Two medians of an acute triangle divide it into regions with areas $4, 5, 7, x$. Find all possible values of $x$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8203f653afeef933be9aae072001df0a972854f7b05cddd1d2b93f12a96d1d6 | Two men (Rusczyk and Crawford) and two boys (Rep123max and Syntax Error) need to cross a lake. Their boat will either carry one man or two boys. What is the fewest number of one-way trips that it will take to get everybody to the other side of the lake? Assume that one person is required to row the boat. | [
"9"
] | [
"Two men (Rusczyk and Crawford) and two boys (Rep123max and Syntax Error) need to cross a lake. Their boat will either carry one man or two boys. What is the fewest number of one-way trips that it will take to get everybody to the other side of the lake? Assume that one person is required to row the boat."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 1
} | 16 | 0 | 0 |
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