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-2e9d3237767acc83f3e998b9618fc82002378e8e9d4e70e42f644c7387b01f20 | A quadrilateral with sides of lengths 25, 39, 52, and 60 is inscribed in a circle. What is the length of the diameter of this circle? | [
"65"
] | [
"A quadrilateral with sides of lengths 25, 39, 52, and 60 is inscribed in a circle. What is the length of the diameter of this circle?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 17
} | 48 | 0 | 0 |
omr-ws-68fc2083358a54f7d252e973dd8817721689181dcc5011e8bc28e320a79f3dd9 | A quadruple $(A,B,C,D)$ is called "dobarulho" when $A,B,C$ are non-zero digits and $D$ is a positive integer such that: 1. $A \leq 8$ 2. $D > 1$ 3. $D$ divides the six numbers $\overline{ABC}$, $\overline{BCA}$, $\overline{CAB}$, $\overline{(A+1)CB}$, $\overline{CB(A+1)}$, $\overline{B(A+1)C}$. Find all such quadruples... | [
"\\((4, 8, 1, 37)\\) and \\((5, 9, 2, 37)\\)"
] | [
"A quadruple $(A,B,C,D)$ is called \"dobarulho\" when $A,B,C$ are non-zero digits and $D$ is a positive integer such that:\n1. $A \\leq 8$\n2. $D > 1$\n3. $D$ divides the six numbers $\\overline{ABC}$, $\\overline{BCA}$, $\\overline{CAB}$, $\\overline{(A+1)CB}$, $\\overline{CB(A+1)}$, $\\overline{B(A+1)C}$.\nFind a... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 19
} | 19 | 0 | 0 |
omr-ws-ba38245fb2a5c62ca6e035467724c02cfce595f0a4e2b9175af2ad629f31a6b2 | A quadruple $(a, b, c, d)$ of positive integers is called a quixotic quadruple if it satisfies the following conditions: 1. $a \leq b \leq c \leq d$ 2. $a, b, c, d$ are consecutive terms in an arithmetic sequence 3. $a^3 + b^3 + c^3 = d^3$ 4. At least one of $a, b, c, d$ is between 1867 and 2008 inclusive. How many qui... | [
"134"
] | [
"A quadruple $(a, b, c, d)$ of positive integers is called a quixotic quadruple if it satisfies the following conditions:\n1. $a \\leq b \\leq c \\leq d$\n2. $a, b, c, d$ are consecutive terms in an arithmetic sequence\n3. $a^3 + b^3 + c^3 = d^3$\n4. At least one of $a, b, c, d$ is between 1867 and 2008 inclusive.\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-26153fec9b1b7000fa9cf218ca8b8b880f7cd127eb29da1cb90857edbee83d6b | A quadruple $(a, b, c, d)$ of positive integers such that: - $a \le b \le c \le d$, - $a, b, c, d$ are consecutive terms in an arithmetic sequence, - $a^3 + b^3 + c^3 = d^3$, - at least one of $a, b, c, d$ is between 1867 and 2008 inclusive. is called a quixotic quadruple. How many quixotic quadruples are there? | [
"358"
] | [
"A quadruple $(a, b, c, d)$ of positive integers such that:\n- $a \\le b \\le c \\le d$,\n- $a, b, c, d$ are consecutive terms in an arithmetic sequence,\n- $a^3 + b^3 + c^3 = d^3$,\n- at least one of $a, b, c, d$ is between 1867 and 2008 inclusive.\nis called a quixotic quadruple. How many quixotic quadruples are ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6295b32041c0b1bf19f19299521d07da08877f36e5bc058f07fd27b9125ee32d | A quadruple \((a, b, c, d)\) of positive integers is called a quixotic quadruple if it satisfies the following conditions: 1. \(a \leq b \leq c \leq d\), 2. \(a, b, c, d\) are consecutive terms in an arithmetic sequence, 3. \(a^3 + b^3 + c^3 = d^3\), 4. At least one of \(a, b, c, d\) is between 1867 and 2008. How many ... | [
"134"
] | [
"A quadruple \\((a, b, c, d)\\) of positive integers is called a quixotic quadruple if it satisfies the following conditions:\n1. \\(a \\leq b \\leq c \\leq d\\),\n2. \\(a, b, c, d\\) are consecutive terms in an arithmetic sequence,\n3. \\(a^3 + b^3 + c^3 = d^3\\),\n4. At least one of \\(a, b, c, d\\) is between 18... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 9
} | 57 | 0 | 0 |
omr-ws-fa9559ccf0e89ac5c41fb7001c219c1b0f7ec3ad48ac62e0861722cccdbc49ce | A quadruple of integers \((a, b, c, d)\) is called "cool" if \(b = a^2 + 1\), \(c = b^2 + 1\), \(d = c^2 + 1\), and \(\tau(a) + \tau(b) + \tau(c) + \tau(d)\) is an odd number, where \(\tau(k)\) denotes the number of positive divisors of \(k\). Find the number of "cool" quadruples \((a, b, c, d)\) such that \(a, b, c, d... | [
"2"
] | [
"A quadruple of integers \\((a, b, c, d)\\) is called \"cool\" if \\(b = a^2 + 1\\), \\(c = b^2 + 1\\), \\(d = c^2 + 1\\), and \\(\\tau(a) + \\tau(b) + \\tau(c) + \\tau(d)\\) is an odd number, where \\(\\tau(k)\\) denotes the number of positive divisors of \\(k\\). Find the number of \"cool\" quadruples \\((a, b, c... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 1
} | 8 | 0 | 0 |
omr-ws-3905c226ae8324f5871f40f9be854d0c7835a51ef99d181470f81414feed11ed | A quarter circle is folded to form a cone. If \( x \) is the angle between the axis of symmetry and the slant height of the cone, find \(\sin x\). | [
"\\(\\frac{1}{4}\\)"
] | [
"A quarter circle is folded to form a cone. If \\( x \\) is the angle between the axis of symmetry and the slant height of the cone, find \\(\\sin x\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-ccedd097d7fffc22373149fc7a7fa435c89dcc83c82bad27ecbb5717fe9c7ca3 | A quarter circular sector is removed from a circle, and the remainder is folded into a cone by connecting the cut edges. What is the angle at the apex of the cone when viewed from the side? | [
"\\(\\cos^{-1}\\left(-\\frac{1}{8}\\right)\\)"
] | [
"A quarter circular sector is removed from a circle, and the remainder is folded into a cone by connecting the cut edges. What is the angle at the apex of the cone when viewed from the side?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 19
} | 33 | 0 | 0 |
omr-ws-faa017ff705a171963440bde895773ea30aa80aa217ce9c78838f1924f2e3b78 | A quarter of a checkered plane is given, infinite to the right and up, with all rows and columns numbered starting from $0$. Cells with coordinates $(2n, n)$ for $n \geq 1$ are removed. In each remaining cell, the number of paths from the corner cell $(0,0)$ to that cell (moving only up and to the right, and avoiding r... | [
"832394"
] | [
"A quarter of a checkered plane is given, infinite to the right and up, with all rows and columns numbered starting from $0$. Cells with coordinates $(2n, n)$ for $n \\geq 1$ are removed. In each remaining cell, the number of paths from the corner cell $(0,0)$ to that cell (moving only up and to the right, and avoi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-812bb19ae1fe8e2d714c34b70c9fd173666421abfad96a42cac740779aac0f66 | A quarter of the participants in the chess tournament were grandmasters and the rest were masters. Every pair of players played against each other once. 1 point was awarded for a win, 1/2 points for a draw, and 0 points for a loss. The masters accumulated 0.2 times as many points as grandmasters. How many masters and h... | [
"9 masters and 3 grandmasters"
] | [
"A quarter of the participants in the chess tournament were grandmasters and the rest were masters. Every pair of players played against each other once. 1 point was awarded for a win, 1/2 points for a draw, and 0 points for a loss. The masters accumulated 0.2 times as many points as grandmasters. How many masters ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-26ae54435c852ad1b915a5c9aead93be885090710300cf7b74766453122f6124 | A quarter-circle centered at the origin \( O \) is drawn in the first quadrant, meeting the \( x \)-axis at \( A \) and the \( y \)-axis at \( B \). Semi-circles on \( OA \) and \( OB \) as diameters are drawn, and the region where they overlap is shaded grey. The region outside the two semi-circles and inside the quar... | [
"1:1"
] | [
"A quarter-circle centered at the origin \\( O \\) is drawn in the first quadrant, meeting the \\( x \\)-axis at \\( A \\) and the \\( y \\)-axis at \\( B \\). Semi-circles on \\( OA \\) and \\( OB \\) as diameters are drawn, and the region where they overlap is shaded grey. The region outside the two semi-circles ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-018c2558f9187b50dc0df133b77c82803bf71d40aeede6ca38e2e1e399f3c52a | A quarterback releases a pass at a height of 7 feet, and the ball is caught by a receiver 90 feet directly downfield at a height of 4 feet. The pass is released at an angle of 35 degrees. Find the speed of the ball when it is released. | [
"54.1"
] | [
"A quarterback releases a pass at a height of 7 feet, and the ball is caught by a receiver 90 feet directly downfield at a height of 4 feet. The pass is released at an angle of 35 degrees. Find the speed of the ball when it is released."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-4ad6b6c0c64e81ec88679085059b6d864bd914cfd2e581e31ab6cb5e73b2a9fb | A quartic function has the equation \( y = (x - 2)(x - 4)(3x^2 + kx + 8) \), where \( k \) is any real number. How many stationary points can the quartic have? | [
"3"
] | [
"A quartic function has the equation \\( y = (x - 2)(x - 4)(3x^2 + kx + 8) \\), where \\( k \\) is any real number. How many stationary points can the quartic have?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-fcd180618a36d8c815751cc8d0928265c818a7c8daa9e8865d50c7f7cd88b2a8 | A quartic polynomial \( p(x) \) has a double root at \( x = -\frac{21}{4} \), and \( p(x) - 1344x \) has two double roots each \(\frac{1}{4}\) less than an integer. What are these two double roots? | [
"\\( x = -\\dfrac{77}{4} \\) and \\( x = \\dfrac{3}{4} \\)"
] | [
"A quartic polynomial \\( p(x) \\) has a double root at \\( x = -\\frac{21}{4} \\), and \\( p(x) - 1344x \\) has two double roots each \\(\\frac{1}{4}\\) less than an integer. What are these two double roots?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 7
} | 11 | 0 | 0 |
omr-ws-3328efdde987375601a6fdaac8dde67581ec3bc061c7c66ea22b1e30cb2ef1fc | A quartic polynomial \(p(x)\) has a double root at \(x = -\frac{21}{4}\), and \(p(x) - 1344x\) has two double roots each \(\frac{1}{4}\) less than an integer. What are these two double roots? | [
"\\(\\frac{19}{4}, \\frac{23}{4}\\)"
] | [
"A quartic polynomial \\(p(x)\\) has a double root at \\(x = -\\frac{21}{4}\\), and \\(p(x) - 1344x\\) has two double roots each \\(\\frac{1}{4}\\) less than an integer. What are these two double roots?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f58ff592918f3b66589147a7327dc74fabc9a68459b34afbcc37cb9df27175e | A queen has an ice sphere with a diameter of 2 meters that melts into her hexagonal bath. The hexagonal bath has sides of 1.5 meters and a depth of 0.8 meters. The ice melts at a rate of 1.5 mm of depth per minute. How many minutes will it take for the bath to be completely full, assuming the volume does not change bet... | [
"infinity minutes"
] | [
"A queen has an ice sphere with a diameter of 2 meters that melts into her hexagonal bath. The hexagonal bath has sides of 1.5 meters and a depth of 0.8 meters. The ice melts at a rate of 1.5 mm of depth per minute. How many minutes will it take for the bath to be completely full, assuming the volume does not chang... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-45fbea9729e76e768732f9c1ee1a705844bbcb944926672df37f2fa4c7fd0a11 | A question on Joe’s math test asked him to compute \( \frac{a}{b} + \frac{3}{4} \), where \( a \) and \( b \) were both integers. Because he didn’t know how to add fractions, he submitted \( \frac{a+3}{b+4} \) as his answer. But it turns out that he was right for these particular values of \( a \) and \( b \)! What is ... | [
"-3"
] | [
"A question on Joe’s math test asked him to compute \\( \\frac{a}{b} + \\frac{3}{4} \\), where \\( a \\) and \\( b \\) were both integers. Because he didn’t know how to add fractions, he submitted \\( \\frac{a+3}{b+4} \\) as his answer. But it turns out that he was right for these particular values of \\( a \\) and... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7aafd95f39dcaaed4eb952b7339756c4dae8d8f325fc36b5307518b6b9790e06 | A question paper consists of two parts, A and B. Part A has 5 questions, each with 1 alternate, and Part B has 4 questions without any alternates. Find the number of ways one can attempt the paper such that at least one question is attempted from each part. | [
"3630"
] | [
"A question paper consists of two parts, A and B. Part A has 5 questions, each with 1 alternate, and Part B has 4 questions without any alternates. Find the number of ways one can attempt the paper such that at least one question is attempted from each part."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 13
} | 32 | 0 | 0 |
omr-ws-795f03cb31cb74855821865f93f527e63739df6817fb50428a46eb1788e00970 | A quilt is made out of solid green, solid blue, and floral squares. The quilt is 8 by 12 squares. There must be 15 times as many solid squares as floral squares. The price is $3 for each solid square and $5 for each floral square. Exactly $300 is to be spent. How many of each type of square is required? | [
"6"
] | [
"A quilt is made out of solid green, solid blue, and floral squares. The quilt is 8 by 12 squares. There must be 15 times as many solid squares as floral squares. The price is $3 for each solid square and $5 for each floral square. Exactly $300 is to be spent. How many of each type of square is required?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-f2f55647a25b447bdd3dda4660c6cb033228d71ea9073bb18f9a8738afaf5582 | A rabbit begins at the origin and runs up the y-axis with speed \(a\). At the same time, a dog runs at speed \(b\) from the point \((c,0)\) in pursuit of the rabbit. Determine the path of the dog. Assume the arc length of a curve is given by \[ L = \int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}\ dx. \] | [
"y = \\frac{c}{2} \\left( \\frac{\\left( \\frac{x}{c} \\right)^{1 + \\frac{a}{b}}}{1 + \\frac{a}{b}} - \\frac{\\left( \\frac{x}{c} \\right)^{1 - \\frac{a}{b}}}{1 - \\frac{a}{b}} \\right) + \\frac{a b c}{b^2 - a^2}"
] | [
"A rabbit begins at the origin and runs up the y-axis with speed \\(a\\). At the same time, a dog runs at speed \\(b\\) from the point \\((c,0)\\) in pursuit of the rabbit. Determine the path of the dog. Assume the arc length of a curve is given by \n\\[ L = \\int_a^b \\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2}\\ dx.... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-74a6267983303858f753cf3e1786ebf8f71c7190b4603f0dc418c37d66fd6805 | A rabbit has an initial advantage of 50 jumps over a dog. The rabbit takes 5 jumps for every 2 jumps the dog takes, but the dog advances as much in 3 jumps as the rabbit does in 8 jumps. How many jumps must the dog take to catch up to the rabbit? | [
"300"
] | [
"A rabbit has an initial advantage of 50 jumps over a dog. The rabbit takes 5 jumps for every 2 jumps the dog takes, but the dog advances as much in 3 jumps as the rabbit does in 8 jumps. How many jumps must the dog take to catch up to the rabbit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-ae4e78e06a8ce9b3a5168800586b79977854258e39428b55be32690afc54cb0d | A rabbit initially stands at position $0$ on the real line and can jump to any integer position but cannot stand still. Let $N(a)$ be the number of ways the rabbit can jump a total distance of $2019$ and stop at position $a$. Determine all integers $a$ such that $N(a)$ is odd. | [
"\\(\\{1, -1\\}\\)"
] | [
"A rabbit initially stands at position $0$ on the real line and can jump to any integer position but cannot stand still. Let $N(a)$ be the number of ways the rabbit can jump a total distance of $2019$ and stop at position $a$. Determine all integers $a$ such that $N(a)$ is odd."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 5
} | 16 | 0 | 0 |
omr-ws-af0986bff0038ad1bf3b7fa0dc4410c0c5b2558a9bdf8d871c89d9118b0ed027 | A rabbit is 60 of her own leaps ahead of a dog. The rabbit takes three leaps while the dog takes two, but the dog covers the distance of three rabbit leaps in the same time the rabbit covers seven of its own leaps. In how many leaps does the dog catch the rabbit? | [
"72"
] | [
"A rabbit is 60 of her own leaps ahead of a dog. The rabbit takes three leaps while the dog takes two, but the dog covers the distance of three rabbit leaps in the same time the rabbit covers seven of its own leaps. In how many leaps does the dog catch the rabbit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-dcbe4e9f3c71965a94b51314fd1e629c660b3ca35f9445f5c63a9d2378d8e63b | A rabbit is placed on a $2n \times 2n$ chessboard. Every time the rabbit moves to one of the adjacent squares (adjacent means sharing an edge). It is known that the rabbit went through every square and came back to the place where the rabbit started, and the path of the rabbit forms a polygon $\mathcal{P}$. Find the ma... | [
"$4n^2 - 2n$ for even $n$, $4n^2 - 2n - 2$ for odd $n$"
] | [
"A rabbit is placed on a $2n \\times 2n$ chessboard. Every time the rabbit moves to one of the adjacent squares (adjacent means sharing an edge). It is known that the rabbit went through every square and came back to the place where the rabbit started, and the path of the rabbit forms a polygon $\\mathcal{P}$. Find... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-390e6f8bfaf6293de2d3800634542815b27440eb49504e785734cea4213a41c5 | A rabbit starts at position 1 on a number line. Each move, it randomly jumps 1 unit to the right or left. If it jumps from 1 to 0, it is immediately sent back to 1 without any extra moves. What is the expected number of moves for the rabbit to reach position 4 on the number line? | [
"12"
] | [
"A rabbit starts at position 1 on a number line. Each move, it randomly jumps 1 unit to the right or left. If it jumps from 1 to 0, it is immediately sent back to 1 without any extra moves. What is the expected number of moves for the rabbit to reach position 4 on the number line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-905ededb09af964c4803c1b70ac6db379d02f733fd859b3a5c78932598b4df56 | A rabid dog is running between points $A$ and $B$, which are 2 kilometers apart, at a speed of 5 m/s. Every 10 minutes, a bus departs from point $A$ heading for point $B$ at 45 m/s, and every 15 minutes, a bus departs from point $B$ heading for point $A$ at 50 m/s. The dog starts at the midpoint of $A$ and $B$ and runs... | [
"10"
] | [
"A rabid dog is running between points $A$ and $B$, which are 2 kilometers apart, at a speed of 5 m/s. Every 10 minutes, a bus departs from point $A$ heading for point $B$ at 45 m/s, and every 15 minutes, a bus departs from point $B$ heading for point $A$ at 50 m/s. The dog starts at the midpoint of $A$ and $B$ and... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4626d506d4c82c00751b15f1a434b0a6072e15c2a25a1496268b4bb45f46420 | A race track is in the form of a right-angle triangle. The longer leg is 2 km more than the shorter leg (both legs being on a highway). The start and end points are also connected by a side road. An escort vehicle took the side road at 30 km/h and then covered the two intervals along the highway during the same time at... | [
"24"
] | [
"A race track is in the form of a right-angle triangle. The longer leg is 2 km more than the shorter leg (both legs being on a highway). The start and end points are also connected by a side road. An escort vehicle took the side road at 30 km/h and then covered the two intervals along the highway during the same ti... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-98a986dfcc9aba9a30c15a93ad78e1986e91a64d3c3ecbd7bbeeb5982ed7b8eb | A racetrack is divided into two sections of equal length. Farley can drive the first section at 18 miles per hour and the second section at a faster \( k \) miles per hour. By the time she is 50% done with the race, Farley has only driven 35% of the length of the track. What is the value of \( k \)? | [
"45"
] | [
"A racetrack is divided into two sections of equal length. Farley can drive the first section at 18 miles per hour and the second section at a faster \\( k \\) miles per hour. By the time she is 50% done with the race, Farley has only driven 35% of the length of the track. What is the value of \\( k \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74cad5be3be59c661e67384de0e57a4b6baac673e0b8fb888f6fd8d4976b6df3 | A radiator contains 8 quarts of a mixture that is 40% antifreeze. How much of this mixture should be drained and replaced with pure antifreeze so that the resultant mixture will contain 60% antifreeze? | [
"\\dfrac{8}{3}"
] | [
"A radiator contains 8 quarts of a mixture that is 40% antifreeze. How much of this mixture should be drained and replaced with pure antifreeze so that the resultant mixture will contain 60% antifreeze?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7c684b18922b16e525360baf063c3ccfc0b3356f3b451b31cb4659ae0c3a8839 | A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly $17^\circ$ (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again? | [
"360"
] | [
"A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly $17^\\circ$ (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-58924d3813e7c58e60472cd2aa1442c7525ec366e96700adee5001b605bc1b35 | A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly 17° (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again? | [
"181"
] | [
"A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly 17° (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5fcf7c521a2295dc4de04f8ab407560c5b58bd215af37f03940e2ef7575a4fd6 | A raft starts from pier $A$ down the river which flows at speed $v$ km/h. An hour later a motor-launch whose speed in dead water is 10 km/h follows the raft. Having overtaken the raft, the motor-launch comes back. Find all the values of v for which the raft covers more than 15 km by the moment the motor-launch returns ... | [
"$5<v<10$"
] | [
"A raft starts from pier $A$ down the river which flows at speed $v$ km/h. An hour later a motor-launch whose speed in dead water is 10 km/h follows the raft. Having overtaken the raft, the motor-launch comes back. Find all the values of v for which the raft covers more than 15 km by the moment the motor-launch re... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1b4e87581ffb5166c64d328973bf63e1e5bbcd4d548c30222c05ea7d20f11bb | A railroad track is laid along the arc of a circle with a radius of 1800 ft. The circular part of the track subtends a central angle of 40 degrees. How long (in seconds) will it take a point on the front of a train traveling at 30 mph to go around this portion of the track? | [
"\\(\\frac{100\\pi}{11}\\) or approximately 28.56 seconds."
] | [
"A railroad track is laid along the arc of a circle with a radius of 1800 ft. The circular part of the track subtends a central angle of 40 degrees. How long (in seconds) will it take a point on the front of a train traveling at 30 mph to go around this portion of the track?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 25
} | 45 | 0 | 0 |
omr-ws-154064936a315b733e4d3f0e6d7a7e6c4cc81f1e4d98faa4e55b915e91948053 | A railroad track is one mile long and fixed at both ends. If one inch is added to the middle of the track, causing it to buckle into a perfect arc of a circle, how high does the middle of the track get above the ground? | [
"12.83 feet"
] | [
"A railroad track is one mile long and fixed at both ends. If one inch is added to the middle of the track, causing it to buckle into a perfect arc of a circle, how high does the middle of the track get above the ground?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 18
} | 29 | 0 | 0 |
omr-ws-d14246c527a8985bce153414bb6b4a0242f6cf42dd7cbc013484d60b22b492d9 | A railway line connects station A to station K, passing through stations B, C, D, E, F, G, H, I, and J in that order. The total distance between station A and K is 56 km. The distance between any three successive stations does not exceed 12 km, and the distance between any four successive stations is at least 17 km. Wh... | [
"29"
] | [
"A railway line connects station A to station K, passing through stations B, C, D, E, F, G, H, I, and J in that order. The total distance between station A and K is 56 km. The distance between any three successive stations does not exceed 12 km, and the distance between any four successive stations is at least 17 k... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-a05b2d0c3e959382a1344c075d46b4b20d01f66f3abded57457efcd6c90164b2 | A railway line is divided into 10 sections by the stations $A, B, C, D, E, F, G, H, I, J,$ and $K$. The distance between $A$ and $K$ is $56$ km. A trip along two successive sections never exceeds $12$ km. A trip along three successive sections is at least $17$ km. What is the distance between $B$ and $G$? | [
"29"
] | [
"A railway line is divided into 10 sections by the stations $A, B, C, D, E, F, G, H, I, J,$ and $K$. The distance between $A$ and $K$ is $56$ km. A trip along two successive sections never exceeds $12$ km. A trip along three successive sections is at least $17$ km. What is the distance between $B$ and $G$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5d6dea10891a8839b111c4ee5bf4c033361640e92c5ba152db9fd793cbb9809c | A railway line is divided into ten sections by stations $E_1, E_2, \ldots, E_{11}$. The distance between the first and the last station is $56$ km. A trip through two consecutive stations never exceeds $12$ km, and a trip through three consecutive stations is at least $17$ km. Calculate the distance between $E_2$ and $... | [
"28"
] | [
"A railway line is divided into ten sections by stations $E_1, E_2, \\ldots, E_{11}$. The distance between the first and the last station is $56$ km. A trip through two consecutive stations never exceeds $12$ km, and a trip through three consecutive stations is at least $17$ km. Calculate the distance between $E_2$... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-1a050e8bb40e585da068372a591b5f806695bcff7d547ac65c9c1873661bd40c | A railway was being repaired by two teams of workers. Each repaired 10 km of the track despite the fact that the second team worked one day less than did the first. How many kilometers of the track did each team repair per day if both teams together repaired 4.5 km daily? | [
"The first team repaired 2 km, and the second, 2.5 km daily"
] | [
"A railway was being repaired by two teams of workers. Each repaired 10 km of the track despite the fact that the second team worked one day less than did the first. How many kilometers of the track did each team repair per day if both teams together repaired 4.5 km daily?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d4dbbebd38655d6bf2b771d9b63df034702b575753c112cd2812a830813e025 | A rain drop with an initial mass $m_0$ falls under gravity, evaporating uniformly as it does, the decrease in the mass being proportional to time (the proportionality factor is $k$). How many seconds after the drop starts falling is the kinetic energy of the drop the greatest and what is it equal to? (The air resistanc... | [
"$T(t_0)=\\frac{2}{27}\\cdot \\frac{m_0^3 g^2}{k^2}\\left(\\frac{g\\mbox{ cm}^2}{s^2}\\right)$ in time $t_0=\\frac{2m_0}{3k}$ (s)"
] | [
"A rain drop with an initial mass $m_0$ falls under gravity, evaporating uniformly as it does, the decrease in the mass being proportional to time (the proportionality factor is $k$). How many seconds after the drop starts falling is the kinetic energy of the drop the greatest and what is it equal to? (The air resi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06e486db488c3af8ce904d099fdf559e0663dd12382677af06fc7b177ad87f4f | A raindrop is a perfect sphere with radius \( r \) cm and surface area \( S \) cm\(^2\). Condensation accumulates on the raindrop at a rate equal to \( kS \), where \( k = 2 \) cm/sec. If the volume of the raindrop is increasing at a constant rate, find the rate at which the radius of the raindrop is increasing when th... | [
"2"
] | [
"A raindrop is a perfect sphere with radius \\( r \\) cm and surface area \\( S \\) cm\\(^2\\). Condensation accumulates on the raindrop at a rate equal to \\( kS \\), where \\( k = 2 \\) cm/sec. If the volume of the raindrop is increasing at a constant rate, find the rate at which the radius of the raindrop is inc... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75d142501e625def42af62d19031eae40ecb77f7f6a06e0161c805ed7b990646 | A rancher has $1000 to spend on buying steers at $25 each and cows at $26 each. If the number of steers \( s \) and the number of cows \( c \) are both positive integers, determine the number of solutions and which quantity exceeds the other. | [
"\\(\\textbf{(E)}\\)"
] | [
"A rancher has $1000 to spend on buying steers at $25 each and cows at $26 each. If the number of steers \\( s \\) and the number of cows \\( c \\) are both positive integers, determine the number of solutions and which quantity exceeds the other."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-395dbdff80366467999f3d09e53ba702e5316f82e799c8fcc4926df068d9283d | A rancher has 1600 feet of fencing and wants to fence off a rectangular lot at the water's edge of a river bend with a radius of 300 feet. One side of the rectangle is along the river, and the other two sides are equal. The corners are right angles, and the river bends into the lot, creating a concave side. What should... | [
"800"
] | [
"A rancher has 1600 feet of fencing and wants to fence off a rectangular lot at the water's edge of a river bend with a radius of 300 feet. One side of the rectangle is along the river, and the other two sides are equal. The corners are right angles, and the river bends into the lot, creating a concave side. What s... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-e35453165419b2e593a6392c2999c7aca4d2a54556d6334888c86a4c0724b14e | A rancher has 195 feet of fencing to enclose a corral adjacent to his barn. What is the greatest possible area for the corral? | [
"4753.125"
] | [
"A rancher has 195 feet of fencing to enclose a corral adjacent to his barn. What is the greatest possible area for the corral?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-919ed65fe8ce33958dcff9e7df11a59930af7c50b87c60683da893e4cb3e4085 | A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals. Write the area \( A \) of the corral as a function of \( x \), where \( x \) is the width of each corral and \( y \) is the length of each corral. | [
"\\( A(x) = 200x - 3x^2 \\)"
] | [
"A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals. Write the area \\( A \\) of the corral as a function of \\( x \\), where \\( x \\) is the width of each corral and \\( y \\) is the length of each corral."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 7
} | 8 | 0 | 0 |
omr-ws-aa1669f07d28a0a8b2032ea32b742330e342408c65836a5278c4d5c16d593891 | A rancher plans to use 160 yards of fencing to enclose a rectangular corral and to divide it into two parts by a fence parallel to the shorter sides of the corral. Find the dimensions of the corral if the area is 1000 square yards. | [
"20 yards by 50 yards"
] | [
"A rancher plans to use 160 yards of fencing to enclose a rectangular corral and to divide it into two parts by a fence parallel to the shorter sides of the corral. Find the dimensions of the corral if the area is 1000 square yards."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 31
} | 38 | 0 | 0 |
omr-ws-18eba412e920f6196d286779528df032be65d4859d97f927d7fcf6a95b76933b | A rancher plans to use 600 ft of fencing to enclose a rectangular corral and to divide it into two parts by a fence parallel to the shorter sides of the corral. Find the dimensions of the corral if its area is 8000 ft². | [
"\\( 31.67 \\) ft by \\( 252.495 \\) ft"
] | [
"A rancher plans to use 600 ft of fencing to enclose a rectangular corral and to divide it into two parts by a fence parallel to the shorter sides of the corral. Find the dimensions of the corral if its area is 8000 ft²."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 3,
"tir": 16
} | 21 | 0 | 0 |
omr-ws-dce83328c8bd965073e3e3bc8898b155d527b970aa6954fd9abb74dc43c5319b | A random chord of three different notes is played. The chord is called dissonant if the distance between any two of those notes are $1$, $2$, $6$, $10$, or $11$ semitones apart. Given that the longest distance between two notes is at most $12$ semitones, what is the probability that the chord will be dissonant? | [
"\\dfrac{245}{286}"
] | [
"A random chord of three different notes is played. The chord is called dissonant if the distance between any two of those notes are $1$, $2$, $6$, $10$, or $11$ semitones apart. Given that the longest distance between two notes is at most $12$ semitones, what is the probability that the chord will be dissonant?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-e4a943a74a15151feb6ea5c4b979474184e9244abc23fc24055e6f5c73abe3a1 | A random decimal between 0 and 1 is selected and added to a sum. This process is repeated until the sum is greater than 7. What is the probability that the process was repeated 12 or 13 times? | [
"\\frac{1}{11!} \\sum_{k=0}^{7} (-1)^k \\binom{11}{k} (7 - k)^{11} - \\frac{1}{13!} \\sum_{k=0}^{7} (-1)^k \\binom{13}{k} (7 - k)^{13}"
] | [
"A random decimal between 0 and 1 is selected and added to a sum. This process is repeated until the sum is greater than 7. What is the probability that the process was repeated 12 or 13 times?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-515b571102d81ad016cfc4e9cdffb359fe7cd5b0dd72b84fac2ac3bad112c4d9 | A random integer $n \ge 4$ is chosen, and then four distinct integers $0 < a, b, c, d \le n$ are chosen at random. Given that ${a, b, c, d} = {6, 19, 27, 40}$, what is the expected value of $n$? | [
"\\(\\frac{117}{2}\\)"
] | [
"A random integer $n \\ge 4$ is chosen, and then four distinct integers $0 < a, b, c, d \\le n$ are chosen at random. Given that ${a, b, c, d} = {6, 19, 27, 40}$, what is the expected value of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-c120d7bd9c28664be424e1fb0b959ff06923356f191a8f017b433d339029a3fb | A random integer \( X \) between 1 and 35, inclusive, is selected uniformly. Two players, A and B, play a game where Player A first picks an integer between 1 and 35. Knowing A's number, Player B then picks a different integer between 1 and 35. The player who picked the number closest to \( X \) but not exceeding \( X ... | [
"\\(\\frac{1}{6}\\)"
] | [
"A random integer \\( X \\) between 1 and 35, inclusive, is selected uniformly. Two players, A and B, play a game where Player A first picks an integer between 1 and 35. Knowing A's number, Player B then picks a different integer between 1 and 35. The player who picked the number closest to \\( X \\) but not exceed... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 3
} | 11 | 0 | 0 |
omr-ws-f7137f50f8c2626fd225795dd9f3d33fd2818d1ee23e1fd2259cf5c0701d6208 | A random integer \( x \) is chosen between \( 1 \) and \( 2^n \) (inclusive). You make a sequence of integer guesses \( a_1, a_2, \ldots, a_m \) to determine \( x \). After each guess, beginning with the second, you are told whether \( |a_i - x| \) or \( |a_{i-1} - x| \) is greater (or neither, if they are equal). Find... | [
"n + 1"
] | [
"A random integer \\( x \\) is chosen between \\( 1 \\) and \\( 2^n \\) (inclusive). You make a sequence of integer guesses \\( a_1, a_2, \\ldots, a_m \\) to determine \\( x \\). After each guess, beginning with the second, you are told whether \\( |a_i - x| \\) or \\( |a_{i-1} - x| \\) is greater (or neither, if t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36ecc823a022ee4ff5be7ab01a34ab35fcf589cf9d6cbcd013953448a380f3ad | A random integer in the range $[1, 1000]$, inclusive, is chosen. The probability that the integer is triangular can be expressed as $\frac{m}{n}$, where $m$ and $n$ share a common factor of $2$. Find $m + n$. | [
"1044"
] | [
"A random integer in the range $[1, 1000]$, inclusive, is chosen. The probability that the integer is triangular can be expressed as $\\frac{m}{n}$, where $m$ and $n$ share a common factor of $2$. Find $m + n$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-f858c64a32c29adb4b895d1b8b7ccb5f94e001030750ce84ba1998f049bf3044 | A random line passes through the center of an equilateral triangle with side length $\frac{2\sqrt{3}}{3}$. What is the expected length of the line segment inside the triangle? | [
"\\(\\frac{6\\ln(\\sqrt{3}+2)}{\\pi}\\)"
] | [
"A random line passes through the center of an equilateral triangle with side length $\\frac{2\\sqrt{3}}{3}$. What is the expected length of the line segment inside the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-853b9fff99e0efac4590479dcda94e11a334ef7f2cdf9f189ffde2b434459459 | A random number generator generates a number between 0 and 1 (including irrational numbers). You can choose to generate one or two numbers. If the total is less than 1, you get to keep your total multiplied by 1000 dollars; otherwise, you get nothing. You can choose whether to spin a second time after the first spin. H... | [
"\\( -1 + \\sqrt{2} \\)"
] | [
"A random number generator generates a number between 0 and 1 (including irrational numbers). You can choose to generate one or two numbers. If the total is less than 1, you get to keep your total multiplied by 1000 dollars; otherwise, you get nothing. You can choose whether to spin a second time after the first sp... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 8
} | 30 | 0 | 0 |
omr-ws-6857db8129e3338ad283e909881af2c558a4168bb3d208fa111f87c626cbae21 | A random number generator generates a random number each second. On the \( n^{\text{th}} \) second, the generator randomly outputs an integer from the set \(\{1, 2, \cdots, n\}\). What is the expected number of seconds it will take before the sum of the outputted numbers is at least 280? | [
"32"
] | [
"A random number generator generates a random number each second. On the \\( n^{\\text{th}} \\) second, the generator randomly outputs an integer from the set \\(\\{1, 2, \\cdots, n\\}\\). What is the expected number of seconds it will take before the sum of the outputted numbers is at least 280?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-b75dcad899fa5b93c40f8b140802afffcc5684e93f19126c8f1232e88351a20b | A random number generator generates one of the positive integers $1, 2, \ldots, n$ at random, with any number $i$ having probability $p_i$ of being generated. The generator is repeatedly used and each time, the resulting number is written down on a list. For a positive integer $k$, find the expected value of the number... | [
"\\sum_{S \\subseteq [n]} (-1)^{|S|+1} \\frac{1 - \\left(1 - \\sum_{i \\in S} p_i\\right)^k}{\\sum_{i \\in S} p_i}"
] | [
"A random number generator generates one of the positive integers $1, 2, \\ldots, n$ at random, with any number $i$ having probability $p_i$ of being generated. The generator is repeatedly used and each time, the resulting number is written down on a list. For a positive integer $k$, find the expected value of the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-d0379d95cabceed03789ecc5e948defb17be5c5526b59a48c11ea5e22a22fece | A random number generator generates two numbers $a, b$ uniformly and independently in the interval $[-\pi, \pi]$. What is the probability that the function $$f(x) = \ln(\ln(\ln(a \sin x + b \cos x)))$$ is defined for at least one real value of $x$? | [
"0"
] | [
"A random number generator generates two numbers $a, b$ uniformly and independently in the interval $[-\\pi, \\pi]$. What is the probability that the function $$f(x) = \\ln(\\ln(\\ln(a \\sin x + b \\cos x)))$$ is defined for at least one real value of $x$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93927de1e27f17d9bd4b27f6fe9a8ba33a2a017c8933714e9e7a3a83fa0b872e | A random number generator generates two numbers $a, b$ uniformly and independently in the interval $[0, 10]$. What is the probability that the line with equation $x + ay = b$ intersects the ellipse with equation $x^2 + ay^2 = b$? | [
"\\dfrac{119}{200}"
] | [
"A random number generator generates two numbers $a, b$ uniformly and independently in the interval $[0, 10]$. What is the probability that the line with equation $x + ay = b$ intersects the ellipse with equation $x^2 + ay^2 = b$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee5c859d78f5615c4ff2a4ba7a80c385b17d19f8a65d9df5e893631f605372b3 | A random number generator generates two numbers \( a, b \) uniformly and independently in the interval \([0, 10]\). What is the probability that the line with equation \( ax + by = 1 \) intersects two adjacent sides of the unit square \(\{x, y : 0 \le x, y \le 1\}\)? | [
"\\dfrac{163}{200}"
] | [
"A random number generator generates two numbers \\( a, b \\) uniformly and independently in the interval \\([0, 10]\\). What is the probability that the line with equation \\( ax + by = 1 \\) intersects two adjacent sides of the unit square \\(\\{x, y : 0 \\le x, y \\le 1\\}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-598bde3d9bcb54d3a98d94d23c97049181fe73bc275ca75175de1819c986215a | A random number generator generates two numbers \(a\) and \(b\) uniformly and independently in the interval \([-t, t]\), where \(t > 0\). Find the probability, as a function of \(t\), that the graph of the function \(y = x^2 + ax + b\) intersects the graph of its inverse function. | [
"\n\\begin{cases}\n1 & \\text{if } t \\leq 1, \\\\\n1 - \\dfrac{2}{3} \\dfrac{(t - 1)^{3/2}}{t^2} & \\text{if } t > 1.\n\\end{cases}\n"
] | [
"A random number generator generates two numbers \\(a\\) and \\(b\\) uniformly and independently in the interval \\([-t, t]\\), where \\(t > 0\\). Find the probability, as a function of \\(t\\), that the graph of the function \\(y = x^2 + ax + b\\) intersects the graph of its inverse function."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe8e16191843549667851da7f8ffb18d5ced1b782f6e32626d64724fb46b9425 | A random number generator outputs integers from the set {1, 2, 3, 4, 5, 6, 7, 8}, with each of the eight choices equally likely. If ten such random integers are created, what is the probability that their product is one more than a multiple of 8? | [
"\\dfrac{1}{4096}"
] | [
"A random number generator outputs integers from the set {1, 2, 3, 4, 5, 6, 7, 8}, with each of the eight choices equally likely. If ten such random integers are created, what is the probability that their product is one more than a multiple of 8?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-44970e2bc966cda4fc423b1ceded3ca0bbdc56a4864feeb11fa514a4a196d1bb | A random number generator randomly generates the integers 1 through 9 with equal probability. Find the probability that after \( n \) numbers are generated, the product of these numbers is a multiple of 10. | [
"\\( 1 - \\left( \\frac{4}{9} \\right)^n \\)"
] | [
"A random number generator randomly generates the integers 1 through 9 with equal probability. Find the probability that after \\( n \\) numbers are generated, the product of these numbers is a multiple of 10."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe06f840183558d9107d600f3ec783f8858ff010bf63cafd56b0cc28c24d9ed2 | A random number generator randomly generates the integers 1, 2, ..., 9 with equal probability. Find the probability that after \( n \) numbers are generated, the product of these numbers is a multiple of \( n \). | [
"\\(\\frac{9^n - 8^n - 5^n + 4^n}{9^n}\\)"
] | [
"A random number generator randomly generates the integers 1, 2, ..., 9 with equal probability. Find the probability that after \\( n \\) numbers are generated, the product of these numbers is a multiple of \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-8b42fa090c523fa7f071b790d306135b665db9c66c151e55c11514c14a3fbf8f | A random number generator spits out a 1, 2, 3, 4, or 5 randomly, with each number being equally likely. If three numbers are output, what is the probability that their product is a multiple of 6? | [
"42/125"
] | [
"A random number generator spits out a 1, 2, 3, 4, or 5 randomly, with each number being equally likely. If three numbers are output, what is the probability that their product is a multiple of 6?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-08bea539a62fca3ca3d4dd2c32b76506c4b0c67e2571954e9a80fc95931e4ad8 | A random number is selected from the set of Fibonacci numbers. What is the probability that it is divisible by 42 but not by 13? Express your answer as a common fraction. | [
"\\(\\frac{1}{28}\\)"
] | [
"A random number is selected from the set of Fibonacci numbers. What is the probability that it is divisible by 42 but not by 13? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 19
} | 37 | 0 | 0 |
omr-ws-94376fafb18189b15132d5ef1db557699bb4e81f20e8f315bf4a8f6197ce2229 | A random number selector can only select one of the nine integers 1, 2,..., 9, and it makes these selections with equal probability. Determine the probability that after $n$ selections ( $n>1$ ), the product of the $n$ numbers selected will be divisible by 10. | [
"1-(8/9)^n-(5/9)^n+(4/9)^n"
] | [
"A random number selector can only select one of the nine integers 1, 2,..., 9, and it makes these selections with equal probability. Determine the probability that after $n$ selections ( $n>1$ ), the product of the $n$ numbers selected will be divisible by 10."
] | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 6
} | 28 | 0 | 0 |
omr-ws-34bc18e0dd5ccb8dc92910c10ab59961316384360dfdc93b942cbc0a4e138756 | A random number selector can only select one of the ten integers \(0, 1, 2, \ldots, 9\) with equal probability. Determine the probability that after \(n\) selections (\(n > 1\)), the product of the \(n\) numbers selected will be divisible by 10. | [
"\\( 1 - \\left(\\frac{4}{5}\\right)^n - \\left(\\frac{5}{9}\\right)^n + \\left(\\frac{2}{5}\\right)^n \\)"
] | [
"A random number selector can only select one of the ten integers \\(0, 1, 2, \\ldots, 9\\) with equal probability. Determine the probability that after \\(n\\) selections (\\(n > 1\\)), the product of the \\(n\\) numbers selected will be divisible by 10."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 2,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-26471e8ff890891636f800dda64c91058578778f5b7695227429699038ed33b5 | A random permutation of the word "caltop" is chosen. The probability that there is at least one string of 4 consecutive letters that is alphabetically ordered can be expressed as $\frac{a}{720}$. Find $a$. | [
"78"
] | [
"A random permutation of the word \"caltop\" is chosen. The probability that there is at least one string of 4 consecutive letters that is alphabetically ordered can be expressed as $\\frac{a}{720}$. Find $a$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 17
} | 34 | 0 | 0 |
omr-ws-65a27c6335903df1e9115c04a50391f442ea011c287d6376289f3ad210dde936 | A random point \((X,Y)\) has a normal distribution on a plane with \(E[X]=E[Y]=0\) and \(\text{var}[X]=\text{var}[Y]=\sigma^2\). The distance of the point \((X,Y)\) from the origin is \(R\). Find \(E(R)\). | [
"\\(\\sigma \\sqrt{\\frac{\\pi}{2}}\\)"
] | [
"A random point \\((X,Y)\\) has a normal distribution on a plane with \\(E[X]=E[Y]=0\\) and \\(\\text{var}[X]=\\text{var}[Y]=\\sigma^2\\). The distance of the point \\((X,Y)\\) from the origin is \\(R\\). Find \\(E(R)\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-ca172449d560ef5b02e7b6e04f85e57ea97517c16d6b3a9b75beb5bf8d9e7ed6 | A random point is selected on the $x$-axis between $x = -4$ and $x = 4$. What is the expected value of the distance between this point and the point $(0,3)$? | [
"$\\frac{1}{8}(9\\text{arsinh}(4/3)+20)$"
] | [
"A random point is selected on the $x$-axis between $x = -4$ and $x = 4$. What is the expected value of the distance between this point and the point $(0,3)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-64963180bb7b45f5223211e1b388baf4c80c265069b751e587be5209aec931bc | A random positive integer is chosen modulo 14. If the integer is congruent to 1, 3, 5, 7, 9, 11, or 13 modulo 14, then a piece of shiny stuff is given. Given that no piece of shiny stuff was received, what is the probability that the integer is not congruent to 2 or 10 modulo 7? | [
"\\dfrac{5}{7}"
] | [
"A random positive integer is chosen modulo 14. If the integer is congruent to 1, 3, 5, 7, 9, 11, or 13 modulo 14, then a piece of shiny stuff is given. Given that no piece of shiny stuff was received, what is the probability that the integer is not congruent to 2 or 10 modulo 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb306af95011aeed2f5a624ea3485fb9b9c5121812d77aec9ecb529582da18ee | A random positive integer is selected from 1 to 100, and then one of its divisors is randomly selected. What is the probability that this divisor is divisible by 11? | [
"9/200"
] | [
"A random positive integer is selected from 1 to 100, and then one of its divisors is randomly selected. What is the probability that this divisor is divisible by 11?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 44,
"tir": 27
} | 71 | 0 | 0 |
omr-ws-d80811b22b4f7046b7712e56cdcd3c2c61f01c3f2cc3368f42c0b0701f67b618 | A random rectangle is selected from the set of all distinct rectangles that fit within the boundaries $|x|=2020$ and $|y|=2020$, with positive area, lattice points as vertices, and sides parallel to the boundary lines. The expected value of the area of this rectangle can be expressed as $\frac{m}{n}$ for relatively pri... | [
"64"
] | [
"A random rectangle is selected from the set of all distinct rectangles that fit within the boundaries $|x|=2020$ and $|y|=2020$, with positive area, lattice points as vertices, and sides parallel to the boundary lines. The expected value of the area of this rectangle can be expressed as $\\frac{m}{n}$ for relative... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2698e76cba5e3d65f0573d9dbbb2e18753d7395148b0198bb3691379190169f4 | A random sample $X_{1}, X_{2}, \ldots, X_{n}$ is taken from a normal population with mean $\mu$ and variance $14$. Determine the smallest sample size $n$ such that the probability that the sample mean $\bar{X}$ lies within $0.2$ of $\mu$ is at least $0.90$. | [
"948"
] | [
"A random sample $X_{1}, X_{2}, \\ldots, X_{n}$ is taken from a normal population with mean $\\mu$ and variance $14$. Determine the smallest sample size $n$ such that the probability that the sample mean $\\bar{X}$ lies within $0.2$ of $\\mu$ is at least $0.90$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 5
} | 8 | 0 | 0 |
omr-ws-d60147e200ca65ae0c2929d3452d1a4937b36e4c564c5583e3520adb5a95f05e | A random sample of size $2n+1$ is taken from the uniform distribution on $[0,1]$. Find the distribution of the sample median. | [
"\\( (2n + 1) \\binom{2n}{n} x^n (1 - x)^n \\)"
] | [
"A random sample of size $2n+1$ is taken from the uniform distribution on $[0,1]$. Find the distribution of the sample median."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-f70bf078ca0da7b144d04788b173e73bd1b2b75c0bd7a770cfd5dab9d9dbe1a0 | A random variable $X$ is normally distributed with a mean of $6.7$ and a standard deviation of $0.7$. Five independent values $x_1, x_2, x_3, x_4, x_5$ are generated from $X$. The smallest and largest of these values are removed. Find the distribution of the mean of the remaining three values. | [
"\\( N(6.7, 0.7^2) \\)"
] | [
"A random variable $X$ is normally distributed with a mean of $6.7$ and a standard deviation of $0.7$. Five independent values $x_1, x_2, x_3, x_4, x_5$ are generated from $X$. The smallest and largest of these values are removed. Find the distribution of the mean of the remaining three values."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f86e72b2cce42c5d5ed5a5549d3f739ef728463f278c2cbed66624c30a1894a6 | A random variable $\xi$ has a symmetric distribution with $\textsf{E}|\xi| < \infty$ and $\textsf{P}(\xi = 0) < 1$. Let $F$ be the cumulative distribution function of $\xi$. Calculate the value of $\textsf{E}(\xi F(\xi))$ given that $\textsf{E}(\xi^2) = 2$. | [
"no_valid_answer_found"
] | [
"A random variable $\\xi$ has a symmetric distribution with $\\textsf{E}|\\xi| < \\infty$ and $\\textsf{P}(\\xi = 0) < 1$. Let $F$ be the cumulative distribution function of $\\xi$. Calculate the value of $\\textsf{E}(\\xi F(\\xi))$ given that $\\textsf{E}(\\xi^2) = 2$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-27060b4b332ca56414cf2e657766ed130a3ce7825a00393821a1f8ebde870217 | A random variable \( X \) is defined as the larger of the two values when two dice are thrown, or the value if the values are the same. Find the probability distribution for \( X \). | [
"\\[\n\\begin{array}{|c|c|c|c|c|c|c|}\n\\hline\nX & 1 & 2 & 3 & 4 & 5 & 6 \\\\\n\\hline\nP(X) & \\frac{1}{36} & \\frac{1}{12} & \\frac{5}{36} & \\frac{7}{36} & \\frac{1}{4} & \\frac{11}{36} \\\\\n\\hline\n\\end{array}\n\\]"
] | [
"A random variable \\( X \\) is defined as the larger of the two values when two dice are thrown, or the value if the values are the same. Find the probability distribution for \\( X \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 24
} | 35 | 0 | 0 |
omr-ws-ffe84741e888404b186d779a45302570d9e3035edd6a895c0011bc3db7cc0b07 | A random variable \( X \) is uniformly distributed on \( (0,1] \). Another random variable \( Y \) is defined as \( Y = \frac{1}{X} - 1 \). Find the probability density function of \( Y \), and calculate the value of the integral \( \int_{0}^{\infty} y f_Y(y) \, dy \). | [
"\\infty"
] | [
"A random variable \\( X \\) is uniformly distributed on \\( (0,1] \\). Another random variable \\( Y \\) is defined as \\( Y = \\frac{1}{X} - 1 \\). Find the probability density function of \\( Y \\), and calculate the value of the integral \\( \\int_{0}^{\\infty} y f_Y(y) \\, dy \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aaa6dc49d105aba9577d22119f60646fce0a35423ceb07e690fa0a6fe8ff0191 | A random variable \( x \) represents the number of coins received or paid based on the outcome of rolling a fair six-sided die. If an even number comes up, we receive that number of coins, and if an odd number comes up, we pay that number of coins. Calculate the variance of \( x \). | [
"\\(\\frac{179}{12}\\)"
] | [
"A random variable \\( x \\) represents the number of coins received or paid based on the outcome of rolling a fair six-sided die. If an even number comes up, we receive that number of coins, and if an odd number comes up, we pay that number of coins. Calculate the variance of \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-161dc7cb234578c5ab12f86ed848b2c648adbe35a176cc94674db080b827d79a | A random walk on the three-dimensional integer lattice starts at $(0, 0, 0)$. A standard six-sided die is rolled six times, and after each roll, the walker moves one unit in the $+x, -x, +y, -y, +z, -z$ direction, respectively, based on the die roll. Find the probability that after the sixth roll, the walker is back at... | [
"$\\frac{155}{3888}$"
] | [
"A random walk on the three-dimensional integer lattice starts at $(0, 0, 0)$. A standard six-sided die is rolled six times, and after each roll, the walker moves one unit in the $+x, -x, +y, -y, +z, -z$ direction, respectively, based on the die roll. Find the probability that after the sixth roll, the walker is ba... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 30
} | 52 | 0 | 0 |
omr-ws-113cf3d80219c6c4c3af9ad823a1e6ac1b0a13f51b10de069640d0d082f5031b | A randomly shuffled standard deck of cards has 52 cards, 13 of each of the four suits. There are 4 Aces and 4 Kings, one of each of the four suits. One repeatedly draws cards from the deck until one draws an Ace. Given that the first King appears before the first Ace, the expected number of cards one draws after the fi... | [
"52"
] | [
"A randomly shuffled standard deck of cards has 52 cards, 13 of each of the four suits. There are 4 Aces and 4 Kings, one of each of the four suits. One repeatedly draws cards from the deck until one draws an Ace. Given that the first King appears before the first Ace, the expected number of cards one draws after t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-4905453b9b88c95bb5fc189755558a1ef13e21c26c58297cc5ef382de0ca9b49 | A rare genetic condition affects one in a lakh of the population. A newly developed method detects the condition with an accuracy of 99.9%. Suppose a person tests positive for this condition by this new method twice consecutively. What is the probability that the person actually has the genetic condition? | [
"\\( 0.909 \\)"
] | [
"A rare genetic condition affects one in a lakh of the population. A newly developed method detects the condition with an accuracy of 99.9%. Suppose a person tests positive for this condition by this new method twice consecutively. What is the probability that the person actually has the genetic condition?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e2a075afcfd219d264718603d63c23d197f62471a7455497e97dad953ed2410a | A rational $\frac{1}{k}$, where $k$ is a positive integer, is said to be [i]n-unsound[/i] if its base $N$ representation terminates. Let $S_n$ be the set of all [i]n-unsound[/i] rationals. The sum of all the elements in the union set $S_2 \cup S_3 \cup \cdots \cup S_{14}$ is $\frac{m}{n}$, where $m$ and $n$ are relativ... | [
"1193"
] | [
"A rational $\\frac{1}{k}$, where $k$ is a positive integer, is said to be [i]n-unsound[/i] if its base $N$ representation terminates. Let $S_n$ be the set of all [i]n-unsound[/i] rationals. The sum of all the elements in the union set $S_2 \\cup S_3 \\cup \\cdots \\cup S_{14}$ is $\\frac{m}{n}$, where $m$ and $n$ ... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-72b17827029f8877e0ec688b9df86cf67b95c10adff9ac2fa000576bda521c21 | A rational number \( r_0 \) is written on the board. Every minute, if the number on the board is \( r_n \), it is replaced by \( r_{n+1} = \sqrt{r_n + 1} \). Find the smallest positive integer \( k \) such that \( r_k \) is guaranteed to be irrational, given that \( r_0 = \frac{1}{2} \). | [
"1"
] | [
"A rational number \\( r_0 \\) is written on the board. Every minute, if the number on the board is \\( r_n \\), it is replaced by \\( r_{n+1} = \\sqrt{r_n + 1} \\). Find the smallest positive integer \\( k \\) such that \\( r_k \\) is guaranteed to be irrational, given that \\( r_0 = \\frac{1}{2} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc2d3c911d227b04f47fa013efab10e0fe8f441e7758f2b983207cea7a7608d3 | A rational number \( x \) is called good if it satisfies \( x = \frac{p}{q} > 1 \) with \( p \) and \( q \) being positive integers, \(\gcd(p, q) = 1\), and there exist constant numbers \( \alpha \) and \( N \) such that for any integer \( n \geq N \), \[ |\{x^n\} - \alpha| \leq \frac{1}{2(p + q)}. \] Find all the good... | [
"\\( x \\) is an integer greater than 1."
] | [
"A rational number \\( x \\) is called good if it satisfies \\( x = \\frac{p}{q} > 1 \\) with \\( p \\) and \\( q \\) being positive integers, \\(\\gcd(p, q) = 1\\), and there exist constant numbers \\( \\alpha \\) and \\( N \\) such that for any integer \\( n \\geq N \\), \\[ |\\{x^n\\} - \\alpha| \\leq \\frac{1}{... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-8ba6c6a8dd64e7d2ef3728a88c30a05ba49a0a87d2f6a3ffe2f1bbabea86fce9 | A rational number \(\frac{1}{k}\), where \(k\) is a positive integer, is said to be \(n\)-unsound if its base \(n\) representation terminates. Let \(S_n\) be the set of all \(n\)-unsound rationals. Find the sum of all elements in \( S=\bigcup_{k=2}^{14}S_k \). | [
"\\dfrac{241}{60}"
] | [
"A rational number \\(\\frac{1}{k}\\), where \\(k\\) is a positive integer, is said to be \\(n\\)-unsound if its base \\(n\\) representation terminates. Let \\(S_n\\) be the set of all \\(n\\)-unsound rationals. Find the sum of all elements in \\( S=\\bigcup_{k=2}^{14}S_k \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-3af792c75cfd4a33b2cbfbe741fd3680832a6f0ff4478d440be517a40f89bc0e | A rational number is called secretive if it has at least three consecutive zeros in its binary representation that are preceded by a 1. For example, \(32_{10} = 100000_2\) and \(\frac{1}{15} = 0.\overline{0001}_2\) are both secretive, but \(\frac{1}{16} = 0.0001_2\) is not. For how many positive integers \(p\) with \(1... | [
"1004"
] | [
"A rational number is called secretive if it has at least three consecutive zeros in its binary representation that are preceded by a 1. For example, \\(32_{10} = 100000_2\\) and \\(\\frac{1}{15} = 0.\\overline{0001}_2\\) are both secretive, but \\(\\frac{1}{16} = 0.0001_2\\) is not. For how many positive integers ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-f9f133069499267494c2cd0de2750ce2888c0e84493c93b48e19e76e65ef348a | A rational number is defined as spheric if it can be expressed as the sum of three squares of rational numbers. Given a spheric rational number $r$, find the sum of all possible values of $r^3$ that can be expressed as the sum of three squares of integers. Provide your answer in the form of $a + b\sqrt{c}$, where $a$, ... | [
"0"
] | [
"A rational number is defined as spheric if it can be expressed as the sum of three squares of rational numbers. Given a spheric rational number $r$, find the sum of all possible values of $r^3$ that can be expressed as the sum of three squares of integers. Provide your answer in the form of $a + b\\sqrt{c}$, where... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-1550fdffe3baba74cdb223e87529ffab719dfa1330cffa3c56a85318013b9633 | A rational number written in base eight is $\underline{ab} . \underline{cd}$, where all digits are nonzero. The same number in base twelve is $\underline{bb} . \underline{ba}$. Find the base-ten number $\underline{abc}$. | [
"321"
] | [
"A rational number written in base eight is $\\underline{ab} . \\underline{cd}$, where all digits are nonzero. The same number in base twelve is $\\underline{bb} . \\underline{ba}$. Find the base-ten number $\\underline{abc}$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-08c7566abb04c9bcc18e69b049e8d405ebfc3b278dd1eac462664d47225db902 | A rational number written in base eight is $\underline{ab}. \underline{cd}$, where all digits are nonzero. The same number in base twelve is $\underline{bb}. \underline{ba}$. Find the base-ten number $\underline{abc}$. | [
"321"
] | [
"A rational number written in base eight is $\\underline{ab}. \\underline{cd}$, where all digits are nonzero. The same number in base twelve is $\\underline{bb}. \\underline{ba}$. Find the base-ten number $\\underline{abc}$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-e1f2b3922907e767d691b66b7ce9c5f7b2a786942355aa132d9a1dbd1e4621e1 | A rational number written in base eight is $\underline{a} \underline{b} . \underline{c} \underline{d}$, where all digits are nonzero. The same number in base twelve is $\underline{b} \underline{b} . \underline{b} \underline{a}$. Find the base-ten number $\underline{a} \underline{b} \underline{c}$. | [
"321"
] | [
"A rational number written in base eight is $\\underline{a} \\underline{b} . \\underline{c} \\underline{d}$, where all digits are nonzero. The same number in base twelve is $\\underline{b} \\underline{b} . \\underline{b} \\underline{a}$. Find the base-ten number $\\underline{a} \\underline{b} \\underline{c}$."
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-2a8ff39cd6196d8fe942c749c3cb7fe918d752a345c5d3183400cf0cfe691ce8 | A rational number written in base-eight is \( AB.CD \) where all digits are non-zero. The same number in base-twelve is \( BB.BA \). Find the base-ten number \( ABC \). | [
"756"
] | [
"A rational number written in base-eight is \\( AB.CD \\) where all digits are non-zero. The same number in base-twelve is \\( BB.BA \\). Find the base-ten number \\( ABC \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b39b6fa5ecfcdab5d5c75f25696aa59fb620aff8b4572301979f7390f048fb4f | A ray $PO$ is drawn inside the right angle $MPN$. In each of the formed acute angles, a circle is inscribed, and both circles touch the ray $PO$ at the point $O$. Find the length of the segment $PO$ if the radii of the circles are $2$ and $3$. | [
"6"
] | [
"A ray $PO$ is drawn inside the right angle $MPN$. In each of the formed acute angles, a circle is inscribed, and both circles touch the ray $PO$ at the point $O$. Find the length of the segment $PO$ if the radii of the circles are $2$ and $3$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 1
} | 14 | 0 | 0 |
omr-ws-04f97fdd99e6c9f6107a50542942bb7f1455e8432133006c66de5c417fd292f6 | A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola $y^2 = x^2 - x + 1$ in the first quadrant. This ray makes an angle of $\theta$ with the positive $x$-axis. Compute $\cos\theta$. | [
"\\(\\frac{2}{\\sqrt{7}}\\)"
] | [
"A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola $y^2 = x^2 - x + 1$ in the first quadrant. This ray makes an angle of $\\theta$ with the positive $x$-axis. Compute $\\cos\\theta$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-a690229c187de4440605ce427c02a722ef969346b100fbd828c345c94951a997 | A ray of light emanates from the point $M (5, 4)$ at the angle\linebreak $\varphi = \arctan 2$ to the axis $Ox$ and is reflected from it. Write the equation of the incident and reflected rays (the equations of the straight lines containing those rays). | [
"$y=2x-6$, $y=-2x+6$"
] | [
"A ray of light emanates from the point $M (5, 4)$ at the angle\\linebreak $\\varphi = \\arctan 2$ to the axis $Ox$ and is reflected from it. Write the equation of the incident and reflected rays (the equations of the straight lines containing those rays)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42d043da878175ed756725d0194ef15e8bbfebb25966d9c789c54953bd07f0b5 | A ray of light enters a $5 \times 7$ rectangle through point $A$ at a $45^\circ$ angle and bounces off the sides of the rectangle. How many times will the ray of light bounce before it reaches any one of the corners $A$, $B$, $C$, or $D$? | [
"10"
] | [
"A ray of light enters a $5 \\times 7$ rectangle through point $A$ at a $45^\\circ$ angle and bounces off the sides of the rectangle. How many times will the ray of light bounce before it reaches any one of the corners $A$, $B$, $C$, or $D$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ebb5e1ba8d8e36f3f783a89348f798e0e2a222c1d043d6d02828d8caa670d17 | A ray of light is issued from a point interior to a given cone. All the internal surface of this cone can reflect the light. If the initial angle of incidence of the light ray with the cone's surface is $\theta$ degrees, find the maximum number of reflections the light can make with the cone's surface before it no long... | [
"\\left\\lfloor \\frac{90^\\circ}{\\theta} \\right\\rfloor"
] | [
"A ray of light is issued from a point interior to a given cone. All the internal surface of this cone can reflect the light. If the initial angle of incidence of the light ray with the cone's surface is $\\theta$ degrees, find the maximum number of reflections the light can make with the cone's surface before it n... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-69cd3db761aac708de49d7b81fcfd0c6792c0d492b3b3bcfaabdb3cc91be0435 | A ray of light is sent in a random direction towards the $x$-axis from a station $Q(0,1)$ on the $y$-axis and the ray meets the $x$-axis at a point $P$. Find the probability density function of the abscissa of $P$. | [
"\\(\\frac{1}{\\pi(x^2 + 1)}\\)"
] | [
"A ray of light is sent in a random direction towards the $x$-axis from a station $Q(0,1)$ on the $y$-axis and the ray meets the $x$-axis at a point $P$. Find the probability density function of the abscissa of $P$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-4b53bfcc4207f73762540d1358444ee867aa5c6911f5b78d2a191e7c114858f7 | A ray of light originates from point \( A \) and travels in a plane, being reflected \( n \) times between lines \( AD \) and \( CD \), before striking a point \( B \) (which may be on \( AD \) or \( CD \)) perpendicularly and retracing its path to \( A \). If \( \measuredangle CDA = 8^\circ \), what is the largest val... | [
"10"
] | [
"A ray of light originates from point \\( A \\) and travels in a plane, being reflected \\( n \\) times between lines \\( AD \\) and \\( CD \\), before striking a point \\( B \\) (which may be on \\( AD \\) or \\( CD \\)) perpendicularly and retracing its path to \\( A \\). If \\( \\measuredangle CDA = 8^\\circ \\)... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
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