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-94dce8b8ff7643e00f29641b5c3ab70367de6aa2bf22a83b85f42e60c140de38 | A parallelogram has sides of length $5$ and $6$ and one diagonal of length $7$. Compute the length of the other diagonal. | [
"\\sqrt{73}"
] | [
"A parallelogram has sides of length $5$ and $6$ and one diagonal of length $7$. Compute the length of the other diagonal."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-461faea3a4e693a750e9f985e51ed3daaef8f5d3c96bea4a338e846662da1614 | A parallelogram has sides of length 10 inches and 9 inches, and the shorter diagonal is 7 inches. Calculate the size of one of the acute angles of the parallelogram. | [
"\\(\\arccos\\left(\\frac{11}{15}\\right)\\)"
] | [
"A parallelogram has sides of length 10 inches and 9 inches, and the shorter diagonal is 7 inches. Calculate the size of one of the acute angles of the parallelogram."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-3c555aca4164dcecb7b99820337e6b7bcfc7c00f57db401111ec3b42b520a255 | A parallelogram has the same area as a rhombus with diagonals of length 10 and 12. If the sides of the parallelogram are 8 and 15, then what are the lengths of the altitudes of the parallelogram? | [
"\\(\\frac{15}{2}\\) and \\(4\\)"
] | [
"A parallelogram has the same area as a rhombus with diagonals of length 10 and 12. If the sides of the parallelogram are 8 and 15, then what are the lengths of the altitudes of the parallelogram?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 25
} | 28 | 0 | 0 |
omr-ws-baced23d820579650d00230189eed3207c4283c84c68af39f69357fb58b59c07 | A parallelogram has three of its vertices at $(1,2)$, $(3,8)$, and $(4,1)$. Compute the sum of all possible $x$ coordinates of the fourth vertex. | [
"8"
] | [
"A parallelogram has three of its vertices at $(1,2)$, $(3,8)$, and $(4,1)$. Compute the sum of all possible $x$ coordinates of the fourth vertex."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f67b6e1d8f497b956120cfe8d4e0e220c241eccaecbb2e149906170e36bb2e2 | A parallelogram is circumscribed about a circle of radius $R$. The area of the quadrilateral with vertices at the points of tangency of the circle and the parallelogram is $S$. Find the lengths of the sides of the parallelogram. | [
"$\\frac{4R^3}{S}$"
] | [
"A parallelogram is circumscribed about a circle of radius $R$. The area of the quadrilateral with vertices at the points of tangency of the circle and the parallelogram is $S$. Find the lengths of the sides of the parallelogram."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e41d53fc4f898619a1902a4140d70b8417b2d6e8f4df1701bd698aa5c53b882 | A parallelogram is cut by two sets of \( m \) lines parallel to the sides. Find the number of parallelograms formed. | [
"\\(\\frac{(m+2)^2(m+1)^2}{4}\\)"
] | [
"A parallelogram is cut by two sets of \\( m \\) lines parallel to the sides. Find the number of parallelograms formed."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-d18c851325ba69dd72a7427fd0bf78fd6c229ecd555d846c88a7bda3f48a2d34 | A parallelogram is divided into several convex polygons. Let $N$ be the total number of these convex polygons. Determine the minimum value of $N$ such that there is at least one polygon among them that can be completely covered by the union of the other polygons. | [
"5"
] | [
"A parallelogram is divided into several convex polygons. Let $N$ be the total number of these convex polygons. Determine the minimum value of $N$ such that there is at least one polygon among them that can be completely covered by the union of the other polygons."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-5c111ae32407112b604c15b8547e8d0713dd29f0a67e3e4a3054ac0df5d1d93c | A parallelogram is divided into several triangles. Let $S$ be the area of the parallelogram and $A_1, A_2, \ldots, A_n$ be the areas of the triangles formed. Find the minimum value of $\frac{\sum_{i=1}^{n-1} A_i}{S}$, where the sum is taken over all possible ways of dividing the parallelogram into triangles. | [
"\\dfrac{1}{2}"
] | [
"A parallelogram is divided into several triangles. Let $S$ be the area of the parallelogram and $A_1, A_2, \\ldots, A_n$ be the areas of the triangles formed. Find the minimum value of $\\frac{\\sum_{i=1}^{n-1} A_i}{S}$, where the sum is taken over all possible ways of dividing the parallelogram into triangles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b31f54a1ed078c71f10fbf8c1938f9dcd4a1f5441dc598fc8513f7029dcbed42 | A parallelogram is generated by the vectors $\begin{pmatrix} 2 \\ 1\\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ -1 \\ - 1 \end{pmatrix}.$ [asy] unitsize(0.4 cm); pair A, B, C, D; A = (0,0); B = (7,2); C = (1,3); D = B + C; draw(A--B,Arrow(6)); draw(A--C,Arrow(6)); draw(B--D--C); draw(B--C,dashed); draw(A--D,dashed); [... | [
"\\frac{1}{3}"
] | [
"A parallelogram is generated by the vectors $\\begin{pmatrix} 2 \\\\ 1\\\\ 1 \\end{pmatrix}$ and $\\begin{pmatrix} 1 \\\\ -1 \\\\ - 1 \\end{pmatrix}.$\n\n[asy]\nunitsize(0.4 cm);\n\npair A, B, C, D;\n\nA = (0,0);\nB = (7,2);\nC = (1,3);\nD = B + C;\n\ndraw(A--B,Arrow(6));\ndraw(A--C,Arrow(6));\ndraw(B--D--C);\ndra... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-18b95a0dfc9488b1d5a45dd24361c4db6f5e992a8277a5c90ebd2d7fdcd755e6 | A parallelogram of paper with sides 25 and 10 is given, and the distance between the longer sides is 6. The paper is cut into exactly two parts in such a way that one can stick both pieces together and fold it in a suitable manner to form a cube of edge length $a$. Calculate the value of $a$. | [
"5"
] | [
"A parallelogram of paper with sides 25 and 10 is given, and the distance between the longer sides is 6. The paper is cut into exactly two parts in such a way that one can stick both pieces together and fold it in a suitable manner to form a cube of edge length $a$. Calculate the value of $a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-86a5ca0159bb527de5e979c64cd47eed68e1f776a01307160325c07f8e076905 | A parallelogram with a vertex at $(0, 0)$ has its other three vertices as lattice points in the first quadrant. Given that the parallelogram has an area of $2021$, compute the maximum number of lattice points in the interior of the parallelogram. | [
"2020"
] | [
"A parallelogram with a vertex at $(0, 0)$ has its other three vertices as lattice points in the first quadrant. Given that the parallelogram has an area of $2021$, compute the maximum number of lattice points in the interior of the parallelogram."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-d5ea7ef25c525bca6f5b11c228bf8a00cb85cd0ace1aa0312c9e82a36c747585 | A parallelogram with an angle of $60^\circ$ has sides $a$ (the longest side) and $b$ (the shortest side). The perpendiculars from the vertices of the obtuse angles to the longest diagonal divide it into three equal parts. Determine the ratio $\frac{a}{b}$. | [
"\\(\\frac{\\sqrt{33} + 1}{4}\\)"
] | [
"A parallelogram with an angle of $60^\\circ$ has sides $a$ (the longest side) and $b$ (the shortest side). The perpendiculars from the vertices of the obtuse angles to the longest diagonal divide it into three equal parts. Determine the ratio $\\frac{a}{b}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-9f4fe0ba6bb82b88cc386d76ff43ea6a9340a6515a7becb089e3ef9994b6123f | A parallelogram with diagonals of length $\sqrt{21}$ and $\sqrt{11}$ is sheared into a rectangle. What is the length of a diagonal of this rectangle? | [
"4"
] | [
"A parallelogram with diagonals of length $\\sqrt{21}$ and $\\sqrt{11}$ is sheared into a rectangle. What is the length of a diagonal of this rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-367d435ab10d86ec65e5bab1581993a4105cd12f6ff2f9456df0b9f58b1b62d9 | A parallelogram with sides lengths $16$ and $36$ has positive integer area $A$. What is the minimum possible value of $A$? | [
"1"
] | [
"A parallelogram with sides lengths $16$ and $36$ has positive integer area $A$. What is the minimum possible value of $A$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-208e759de630b3fb4de57ebbc518d889f9928237ba9523ac8bb2eac6c7239025 | A parametric graph is given by \[ \begin{cases} y = \sin t \\ x = \cos t + \frac{1}{2} t \end{cases} \] How many times does the graph intersect itself between \( x = 1 \) and \( x = 40 \)? | [
"24"
] | [
"A parametric graph is given by \n\\[ \\begin{cases} y = \\sin t \\\\ x = \\cos t + \\frac{1}{2} t \\end{cases} \\]\nHow many times does the graph intersect itself between \\( x = 1 \\) and \\( x = 40 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-313652fe5ab31b88d751be9be3d54f982caf7e116b13911b501f26e0be83e01a | A parametric graph is given by \begin{align*} x &= \cos t + \frac{t}{2}, \\ y &= \sin t. \end{align*}How many times does the graph intersect itself between $x = 1$ and $x = 40$? | [
"12"
] | [
"A parametric graph is given by\n\\begin{align*}\nx &= \\cos t + \\frac{t}{2}, \\\\\ny &= \\sin t.\n\\end{align*}How many times does the graph intersect itself between $x = 1$ and $x = 40$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-433396aaee275af1db50c513ccc443dc643406719dd25107de5ef7b447d20bb1 | A parcel of land has dimensions \( n \times n \) meters and is divided into strips (rectangles \( 1 \times m \) or \( m \times 1 \), where \( m \) is an integer). A man can choose an integer \( k \) and take the land on all the strips that have length \( k \). Let \( l(n) \) be the largest integer such that he may take... | [
"\\(\\sqrt{\\frac{3}{12-\\pi^2}}\\)"
] | [
"A parcel of land has dimensions \\( n \\times n \\) meters and is divided into strips (rectangles \\( 1 \\times m \\) or \\( m \\times 1 \\), where \\( m \\) is an integer). A man can choose an integer \\( k \\) and take the land on all the strips that have length \\( k \\). Let \\( l(n) \\) be the largest integer... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-47bdf20a834c411d791633ab13e499d26e107ca8ee2e69a4161b980bf9515791 | A parent particle divides into 0, 1, or 2 particles with probabilities 1/4, 1/2, and 1/4, respectively, and then disappears. Starting with one particle, let \( X_i \) be the number of particles in the \( i \)-th generation. Find \( P(X_2 > 0) \). | [
"\\(\\frac{39}{64}\\)"
] | [
"A parent particle divides into 0, 1, or 2 particles with probabilities 1/4, 1/2, and 1/4, respectively, and then disappears. Starting with one particle, let \\( X_i \\) be the number of particles in the \\( i \\)-th generation. Find \\( P(X_2 > 0) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-59d5f389bb2c4469fc74b7278bdf5382aed77d92665fa6a9dd5809b54d63d739 | A parent particle divides into 0, 1, or 2 particles with probabilities 1/4, 1/2, and 1/4, respectively, and then disappears. Starting with one particle, let \( X_i \) be the number of particles in the \( i \)-th generation. Find the probability that \( X_1 = 2 \) given \( X_2 = 1 \). | [
"\\(\\frac{1}{5}\\)"
] | [
"A parent particle divides into 0, 1, or 2 particles with probabilities 1/4, 1/2, and 1/4, respectively, and then disappears. Starting with one particle, let \\( X_i \\) be the number of particles in the \\( i \\)-th generation. Find the probability that \\( X_1 = 2 \\) given \\( X_2 = 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-6a2dd58357127413109cff400e331feba6a9f238b2ebd38ba8613300fd1eba2f | A park has the shape of a convex pentagon with an area of $50000\sqrt{3} \, m^2$. A man standing at an interior point $O$ of the park is at a distance of at most $200 \, m$ from each vertex of the pentagon. Calculate the minimum possible sum of the distances from point $O$ to each of the five sides of the pentagon. | [
"500\\sqrt{3}"
] | [
"A park has the shape of a convex pentagon with an area of $50000\\sqrt{3} \\, m^2$. A man standing at an interior point $O$ of the park is at a distance of at most $200 \\, m$ from each vertex of the pentagon. Calculate the minimum possible sum of the distances from point $O$ to each of the five sides of the penta... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-cebf62b5e851f0de30fbf9ccaec6bb14f189d3f51955fc078db1fd6e36336875 | A park is in the shape of a regular hexagon with each side measuring 2 km. Starting at a corner, Alice walks along the perimeter of the park for a distance of 5 km. How many kilometers is she from her starting point? | [
"\\(\\sqrt{13}\\)"
] | [
"A park is in the shape of a regular hexagon with each side measuring 2 km. Starting at a corner, Alice walks along the perimeter of the park for a distance of 5 km. How many kilometers is she from her starting point?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 63,
"tir": 62
} | 125 | 0 | 0 |
omr-ws-512075f88128fedefb830a1e1d76a18c3e94fb12a9b335df340e97c142030122 | A park is in the shape of a regular hexagon with sides $2$ km. Alicia walked $5$ km, starting at a vertex and following the perimeter of the park. How many kilometers, measured in a straight line, is it from the starting point? | [
"$\\sqrt{13}$ km"
] | [
"A park is in the shape of a regular hexagon with sides $2$ km. Alicia walked $5$ km, starting at a vertex and following the perimeter of the park. How many kilometers, measured in a straight line, is it from the starting point?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-bee85aa059a0f1e6cfef405ddcb1287c55b8724cce8ef4b8c81be0c1ec727c58 | A parking lot consists of $2012$ parking spots equally spaced in a line, numbered $1$ through $2012.$ One by one, $2012$ cars park in these spots under the following procedure: the first car picks from the $2012$ spots uniformly randomly, and each following car picks uniformly randomly among all possible choices which ... | [
"\\dfrac{1}{2012}"
] | [
"A parking lot consists of $2012$ parking spots equally spaced in a line, numbered $1$ through $2012.$ One by one, $2012$ cars park in these spots under the following procedure: the first car picks from the $2012$ spots uniformly randomly, and each following car picks uniformly randomly among all possible choices w... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 3
} | 3 | 0 | 0 |
omr-ws-20ac67a1ff11303190ee625979546dd6b1b0a63dba6ebdcbdbb0a69c7fa67312 | A parking lot for compact cars has 12 adjacent spaces, 8 of which are occupied. A large sport-utility vehicle arrives, needing 2 adjacent open spaces. What is the probability that it will be able to park? | [
"\\(\\frac{41}{55}\\)"
] | [
"A parking lot for compact cars has 12 adjacent spaces, 8 of which are occupied. A large sport-utility vehicle arrives, needing 2 adjacent open spaces. What is the probability that it will be able to park?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-6da2bf0763f3c1e51a96434bf06a67a6dfb7e338a9f81c229699c333e08388c8 | A parking lot for compact cars has 12 adjacent spaces, of which 8 are already occupied. A large SUV arrives, needing two adjacent open spaces. What is the probability that it will be able to park? | [
"\\(\\frac{41}{55}\\)"
] | [
"A parking lot for compact cars has 12 adjacent spaces, of which 8 are already occupied. A large SUV arrives, needing two adjacent open spaces. What is the probability that it will be able to park?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-26082b446719c7c3167850aaaa1533a07063b606236255a03de8e60f82a6bec8 | A parking lot has 16 spaces in a row. Twelve cars arrive and park in randomly chosen available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park? | [
"\\(\\frac{17}{28}\\)",
"\\dfrac{17}{28}"
] | [
"A parking lot has 16 spaces in a row. Twelve cars arrive and park in randomly chosen available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 28
} | 60 | 0 | 0 |
omr-ws-e539cfafcb64fa67d07d6f89de9ee2401542914e275a593ae28f55213baa45ee | A parking lot has 16 spaces in a row. Twelve cars arrive and park in randomly chosen spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park? | [
"\\dfrac{17}{28}"
] | [
"A parking lot has 16 spaces in a row. Twelve cars arrive and park in randomly chosen spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-47201f75b5e07f231a5599b57a6d3571e5312bdee6fd882ad636fe657a46fbc5 | A parking lot has 16 spaces in a row. Twelve cars arrive and park randomly in 12 of these spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park? | [
"\\(\\frac{17}{28}\\)",
"\\dfrac{17}{28}"
] | [
"A parking lot has 16 spaces in a row. Twelve cars arrive and park randomly in 12 of these spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 30
} | 62 | 0 | 0 |
omr-ws-24cb07a1ad8c2c58f48c96d739d96ad18cd9604b4e1b81eedfee848369bcb37b | A parking lot has 16 spaces in a row. Twelve cars arrive and park randomly in available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park? | [
"\\(\\frac{17}{28}\\)"
] | [
"A parking lot has 16 spaces in a row. Twelve cars arrive and park randomly in available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-4b4b3b8bd5a780ba811a4cbdc74183c92af7aee4f328f862c802a719561faee7 | A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park? | [
"$\\frac{17}{28}$",
"\\(\\frac{17}{28}\\)"
] | [
"A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 58
} | 90 | 0 | 0 |
omr-ws-968ceb4322ae225b9de6267ad732c8aa7bd18826c4b76e99d75fcb3cdff44867 | A parking lot has 5 spots remaining and 5 different cars in line waiting to be parked. If one of the cars is too big to fit in the two outermost spots, in how many different ways can the five cars be arranged? | [
"72"
] | [
"A parking lot has 5 spots remaining and 5 different cars in line waiting to be parked. If one of the cars is too big to fit in the two outermost spots, in how many different ways can the five cars be arranged?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-39b64ef04816999358288a1a494178fce0a5ca7226cac06b256bab0258533c08 | A parking lot in a workplace has 8 slots. There are smaller cars that only take up 1 slot, and larger vans that take up two slots. There are three possible types of cars and two types of larger vans. How many possible ways can the parking lot be arranged if it is full? | [
"22363"
] | [
"A parking lot in a workplace has 8 slots. There are smaller cars that only take up 1 slot, and larger vans that take up two slots. There are three possible types of cars and two types of larger vans. How many possible ways can the parking lot be arranged if it is full?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 19
} | 39 | 0 | 0 |
omr-ws-1da5eb4ec6bd21854a42fb17301c2b597c5e78f935ff849ab07da3bcdba65df7 | A parliament has $n$ senators. The senators form 10 parties and 10 committees, such that any senator belongs to exactly one party and one committee. Find the least possible $n$ for which it is possible to label the parties and the committees with numbers from 1 to 10, such that there are at least 11 senators for which ... | [
"101"
] | [
"A parliament has $n$ senators. The senators form 10 parties and 10 committees, such that any senator belongs to exactly one party and one committee. Find the least possible $n$ for which it is possible to label the parties and the committees with numbers from 1 to 10, such that there are at least 11 senators for w... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-6e696797cb533f23cd13a5688490638c989e990467a0abaa8e9a7733e46ab1bb | A part of a forest park which is located between two parallel roads has caught fire. The fire is spreading at a speed of $10$ kilometers per hour. If the distance between the starting point of the fire and both roads is $10$ kilometers, what is the area of the burned region after two hours in kilometers squared? (Assum... | [
"\\(200\\sqrt{3} + 400 \\frac{\\pi}{3}\\)"
] | [
"A part of a forest park which is located between two parallel roads has caught fire. The fire is spreading at a speed of $10$ kilometers per hour. If the distance between the starting point of the fire and both roads is $10$ kilometers, what is the area of the burned region after two hours in kilometers squared? (... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-525d25db17e5abd82f555afd9334fa6eb95d7071eeb2b2be3e04a3210ad13045 | A particle P of mass \( m \) kg is projected vertically upwards with speed \( U \) m/s in a medium where the resistance to motion has magnitude \( \frac{1}{10}mv^2 \) when the speed of the particle is \( v \) m/s. After \( t \) seconds, the particle has height \( x \) meters, velocity \( v \) m/s, and acceleration \( a... | [
"a = -10 - \\frac{1}{10}v^2"
] | [
"A particle P of mass \\( m \\) kg is projected vertically upwards with speed \\( U \\) m/s in a medium where the resistance to motion has magnitude \\( \\frac{1}{10}mv^2 \\) when the speed of the particle is \\( v \\) m/s. After \\( t \\) seconds, the particle has height \\( x \\) meters, velocity \\( v \\) m/s, a... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-527bf4e58666a4e4ffa0d2106945437a509174abe3e94f05fe54e9f33a763cfb | A particle \((x,y)\) moves so that its angular velocities about \((1,0)\) and \((-1,0)\) are equal in magnitude but opposite in sign. Let the path of the particle be described by the differential equation \( y(x^2 + y^2 + 1) \, dx = x(x^2 + y^2 - 1) \, dy \). Determine the constant \(C\) such that the rectangular hyper... | [
"0"
] | [
"A particle \\((x,y)\\) moves so that its angular velocities about \\((1,0)\\) and \\((-1,0)\\) are equal in magnitude but opposite in sign. Let the path of the particle be described by the differential equation \\( y(x^2 + y^2 + 1) \\, dx = x(x^2 + y^2 - 1) \\, dy \\). Determine the constant \\(C\\) such that the ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f6caeae2c144d15091a5322cad9006cff6ee40b2c001ff0d3e8e3750f16d1733 | A particle begins at a point in the Cartesian plane and travels horizontally to the right for a distance of one unit. From there it moves up and to the right along a line of positive slope $m$ for a distance of one unit. From there it returns to the original starting point along a line of positive slope $M$. Express $M... | [
"\\dfrac{\\sqrt{1 + m^2} - 1}{m}"
] | [
"A particle begins at a point in the Cartesian plane and travels horizontally to the right for a distance of one unit. From there it moves up and to the right along a line of positive slope $m$ for a distance of one unit. From there it returns to the original starting point along a line of positive slope $M$. Expre... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a8ab5d624cff63c45091fb0b51af5e479ae0a0e585c040a7f6850dd2f18d9f5a | A particle can travel at a speed of $2$ meters/sec along the $x$-axis and $1$ meter/sec elsewhere. Starting at the origin, which regions of the plane can the particle reach within $1$ second? | [
"(x^2 + y^2 \\leq 1) \\text{ or } ((x - 2)^2 \\geq 3y^2 \\text{ and } x \\geq \\frac{1}{2}) \\text{ or } ((x + 2)^2 \\geq 3y^2 \\text{ and } x \\leq -\\frac{1}{2})"
] | [
"A particle can travel at a speed of $2$ meters/sec along the $x$-axis and $1$ meter/sec elsewhere. Starting at the origin, which regions of the plane can the particle reach within $1$ second?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-4a5fc45c9e511422478e725b81ccbe012b8c7c335b0f1bb6d3c34db5c7642fcc | A particle can travel at speeds up to $ \frac{2m}{s}$ along the $ x$-axis, and up to $ \frac{1m}{s}$ elsewhere in the plane. Provide a labelled sketch of the region which can be reached within one second by the particle starting at the origin. | [
"\n\\begin{tikzpicture}\n\\draw[->] (-3,0) -- (3,0) node[right] {$x$};\n\\draw[->] (0,-2) -- (0,2) node[above] {$y$};\n\\draw (0,0) circle (1);\n\\draw (0.5,0.866) -- (2,0) (-0.5,0.866) -- (-2,0);\n\\draw (0.5,-0.866) -- (2,0) (-0.5,-0.866) -- (-2,0);\n\\end{tikzpicture}\n"
] | [
"A particle can travel at speeds up to $ \\frac{2m}{s}$ along the $ x$-axis, and up to $ \\frac{1m}{s}$ elsewhere in the plane. Provide a labelled sketch of the region which can be reached within one second by the particle starting at the origin."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-7600470bb6ea78047088e07008fab4fd0941213455e752860ee84b09592d7387 | A particle falls in a vertical plane from rest under the influence of gravity and a force perpendicular to and proportional to its velocity. Obtain the equations of the trajectory and identify the curve. | [
"\\text{The trajectory is a cycloid.}"
] | [
"A particle falls in a vertical plane from rest under the influence of gravity and a force perpendicular to and proportional to its velocity. Obtain the equations of the trajectory and identify the curve."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-55974d2e5db015143233f8b4b06f0ac1650c0d18a28b2a59adeedd7eaa11b3f6 | A particle initially at rest moves along a straight line with an acceleration of \(-18\cos(\omega t)\) m/s\(^2\), where \(\omega\) is a positive constant and \(t\) is time in seconds. The initial displacement of the particle from a point \(P\) is 1 meter. The particle has a maximum speed of 6 m/s. Find the value of \(\... | [
"2\\cos(3t) - 1"
] | [
"A particle initially at rest moves along a straight line with an acceleration of \\(-18\\cos(\\omega t)\\) m/s\\(^2\\), where \\(\\omega\\) is a positive constant and \\(t\\) is time in seconds. The initial displacement of the particle from a point \\(P\\) is 1 meter. The particle has a maximum speed of 6 m/s. Fin... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8d4267d2097481b071e00e233e30d8bd239ab0bcaed835f3dacc7676210df20e | A particle is acted upon by a force which varies in inverse proportion to the square of the distance to a certain object. It is known to equal 1 N when the distance to the object was 2 m. Calculate the work done by the force in transferring the particle from the point which is at the distance of 10 m from the object to... | [
"14/15 (J)"
] | [
"A particle is acted upon by a force which varies in inverse proportion to the square of the distance to a certain object. It is known to equal 1 N when the distance to the object was 2 m. Calculate the work done by the force in transferring the particle from the point which is at the distance of 10 m from the obje... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-018ac1fb4baec67e40dc489a8ee429080e1431cc18d8bfca23c9c98e9235bc2d | A particle is at the origin $(0, 0)$ of the Cartesian plane. In each step, the particle can move 1 unit in any of the directions: left, right, up, or down. Find the number of ways to go from $(0, 0)$ to $(0, 2)$ in 6 steps. (Note: Two paths are considered different if the order of traversal of each point is different, ... | [
"225"
] | [
"A particle is at the origin $(0, 0)$ of the Cartesian plane. In each step, the particle can move 1 unit in any of the directions: left, right, up, or down. Find the number of ways to go from $(0, 0)$ to $(0, 2)$ in 6 steps. (Note: Two paths are considered different if the order of traversal of each point is differ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-c02ed6ad5453608dea2bc03e3b54900ce79845feb00514cc4f2c1b18872136b1 | A particle is constrained to move on a surface defined by \( f(x,y,z) = 0 \) and is subject to no forces. Let \( \gamma(t) \) be the path of the particle parameterized by time \( t \). If the particle moves along a geodesic, find the expression for the acceleration vector \( \mathbf{a}(t) \) of the particle in terms of... | [
" \\mathbf{a}(t) = - \\Gamma^k_{ij} \\frac{dx^i}{dt} \\frac{dx^j}{dt} "
] | [
"A particle is constrained to move on a surface defined by \\( f(x,y,z) = 0 \\) and is subject to no forces. Let \\( \\gamma(t) \\) be the path of the particle parameterized by time \\( t \\). If the particle moves along a geodesic, find the expression for the acceleration vector \\( \\mathbf{a}(t) \\) of the parti... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-06f4d0c802d5622226096a95242dda38074047aeabb44fae09eed1a39de42b8e | A particle is in a $5 \times 5$ grid. Each second, it moves to an adjacent cell. The particle starts at cell $A$, travels to cell $B$ in $3$ seconds, to cell $C$ in $4$ seconds, and finally back to cell $A$ in $5$ seconds. How many possible triples $\{A,B,C\}$ exist? | [
"136"
] | [
"A particle is in a $5 \\times 5$ grid. Each second, it moves to an adjacent cell. The particle starts at cell $A$, travels to cell $B$ in $3$ seconds, to cell $C$ in $4$ seconds, and finally back to cell $A$ in $5$ seconds. How many possible triples $\\{A,B,C\\}$ exist?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-1888302f4c2d5071a4b5be02523910ef6d3ab8d853e447de30128e3e11456207 | A particle is located at $(5,0)$ on the coordinate plane. Define a move as a counterclockwise rotation of $\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Given that the particle's position after $150$ moves is $(p,q)$, find the greatest integer less than or equal ... | [
"19"
] | [
"A particle is located at $(5,0)$ on the coordinate plane. Define a move as a counterclockwise rotation of $\\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Given that the particle's position after $150$ moves is $(p,q)$, find the greatest integer less than or ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-648b870f09e4ce18b1fba59b7fc2c73b05982ca07844117bedcce2eaa9c2cbac | A particle is located at $(5,0)$ on the coordinate plane. Define a move for the particle as a counterclockwise rotation of $\pi/4$ radians about the origin followed by a translation of 10 units in the positive $x$-direction. Given that the particle's position after 150 moves is $(p,q)$, find the greatest integer less t... | [
"19"
] | [
"A particle is located at $(5,0)$ on the coordinate plane. Define a move for the particle as a counterclockwise rotation of $\\pi/4$ radians about the origin followed by a translation of 10 units in the positive $x$-direction. Given that the particle's position after 150 moves is $(p,q)$, find the greatest integer ... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-65bf68476ea7cd86232390d85dad0217e016ccde168908189402a806c00f3dd5 | A particle is located at \( (0, 0) \), initially facing the +x direction. A move consists of moving 5 in the direction the particle is facing and then rotating the particle \( 45^\circ \) counterclockwise. What will the greatest integer less than the product of its coordinates be after the particle makes 50 of these mo... | [
"30"
] | [
"A particle is located at \\( (0, 0) \\), initially facing the +x direction. A move consists of moving 5 in the direction the particle is facing and then rotating the particle \\( 45^\\circ \\) counterclockwise. What will the greatest integer less than the product of its coordinates be after the particle makes 50 o... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-332a644ecb60796a2634860a4f563896574bb0c4dcd884afeb6707c0f6c50132 | A particle is located at \((5,0)\) on the coordinate plane. Define a move for the particle as a counterclockwise rotation of \(\frac{\pi}{4}\) radians about the origin followed by a translation of 10 units in the positive \(x\)-direction. Given that the particle's position after 150 moves is \((p,q)\), find the greates... | [
"19"
] | [
"A particle is located at \\((5,0)\\) on the coordinate plane. Define a move for the particle as a counterclockwise rotation of \\(\\frac{\\pi}{4}\\) radians about the origin followed by a translation of 10 units in the positive \\(x\\)-direction. Given that the particle's position after 150 moves is \\((p,q)\\), f... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-20ec9597050a037887a86915d3895102a0c1d226b0e0d317a54c50a1fb116e30 | A particle is located on the coordinate plane at $(5,0)$. Define a ''move'' for the particle as a counterclockwise rotation of $\frac{\pi}{4}$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Find the particle's position after $150$ moves. | [
"(-5 \\sqrt{2}, 5 + 5 \\sqrt{2})"
] | [
"A particle is located on the coordinate plane at $(5,0)$. Define a ''move'' for the particle as a counterclockwise rotation of $\\frac{\\pi}{4}$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Find the particle's position after $150$ moves."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-99e8b468d85db1db7a8320eec5f6d7b8ad0790e0fd6cbacb0b70e02e44d12f22 | A particle is located on the coordinate plane at $(5,0)$. Define a move for the particle as a counterclockwise rotation of $\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Given that the particle's position after $150$ moves is $(p,q)$, find the greatest integer le... | [
"19"
] | [
"A particle is located on the coordinate plane at $(5,0)$. Define a move for the particle as a counterclockwise rotation of $\\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Given that the particle's position after $150$ moves is $(p,q)$, find the greatest inte... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61ecab3c3c33b561d3e42cd19e229bd2cd1c7ec724615c79507fcfc3243308db | A particle is moving along the $x$-axis. It starts with an initial velocity of $1\frac{\text{m}}{\text{s}}$ and has acceleration that varies with position given by $a(x) = 2x^3$, where $x$ is the position on the $x$-axis in meters. Find the position as a function of time. | [
"t = \\int_{0}^{x} \\frac{1}{\\sqrt{s^4 + 1}} \\, ds"
] | [
"A particle is moving along the $x$-axis. It starts with an initial velocity of $1\\frac{\\text{m}}{\\text{s}}$ and has acceleration that varies with position given by $a(x) = 2x^3$, where $x$ is the position on the $x$-axis in meters. Find the position as a function of time."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-842be35dcb3a91716cf207511963298546a084280d671531cfaa633252db315f | A particle is moving along the curve \( y = 5x^{1/2} \). As the particle passes through the point \((16, 20)\), its \( x \)-coordinate increases at a rate of 4 cm/s. How fast is the distance from the particle to the origin changing at this instant? | [
"\\dfrac{57\\sqrt{41}}{82}"
] | [
"A particle is moving along the curve \\( y = 5x^{1/2} \\). As the particle passes through the point \\((16, 20)\\), its \\( x \\)-coordinate increases at a rate of 4 cm/s. How fast is the distance from the particle to the origin changing at this instant?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f625fdf61473e7ad4ce1217a7cf448f97449b404af55ed7b32b80e4507ad45c | A particle is moving along the curve \( y = \sqrt{x} \). As the particle passes through the point \( (4, 2) \), its x-coordinate increases at a rate of \( 3 \text{cm/s} \). How fast is the distance from the particle to the origin changing at this point? | [
"\\( \\frac{27}{4\\sqrt{5}} \\)"
] | [
"A particle is moving along the curve \\( y = \\sqrt{x} \\). As the particle passes through the point \\( (4, 2) \\), its x-coordinate increases at a rate of \\( 3 \\text{cm/s} \\). How fast is the distance from the particle to the origin changing at this point?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-8cfd7e5d09361785e54c504a6ced55bf81efa0f4ed82649bd6447cb80915df3c | A particle is moving clockwise around a circle of 1 meter radius centered at the origin at a rate of 2 meters per second. How fast is the temperature experienced by the particle changing in degrees Celsius per meter at the point (0.5, √3/2)? | [
"0"
] | [
"A particle is moving clockwise around a circle of 1 meter radius centered at the origin at a rate of 2 meters per second. How fast is the temperature experienced by the particle changing in degrees Celsius per meter at the point (0.5, √3/2)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-57e44b3e16a864d8d56e1a558ea8f61d39892169bd18d55629ad13c48cc57f24 | A particle is moving on the $x$-axis such that $\frac{dx}{dt} = (x-1)(x+2)(x-3)$. The particle is initially positioned so that it does not wander off to infinity. Which point of equilibrium will the particle be close to after a sufficiently long time? | [
"1"
] | [
"A particle is moving on the $x$-axis such that $\\frac{dx}{dt} = (x-1)(x+2)(x-3)$. The particle is initially positioned so that it does not wander off to infinity. Which point of equilibrium will the particle be close to after a sufficiently long time?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cc89c5674e65c4fd7d275a840ca2dd18aec9395087c0d9d4fec509776d8eb99c | A particle is moving on the $x$-axis, where $x$ is in centimeters. Its velocity, $v$, in cm/s, when it is at the point with coordinate $x$ is given by $v = x^2 + 5x - 4$. Find the acceleration of the particle when it is at the point $x = 6$. | [
"17"
] | [
"A particle is moving on the $x$-axis, where $x$ is in centimeters. Its velocity, $v$, in cm/s, when it is at the point with coordinate $x$ is given by $v = x^2 + 5x - 4$. Find the acceleration of the particle when it is at the point $x = 6$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2215670f4c250874b88c44d9fb076cb166c00a9ca946b2dba15b49dbf688d4e1 | A particle is placed on the parabola \( y = x^2 - x - 6 \) at a point \( P \) whose \( y \)-coordinate is 6. It is allowed to roll along the parabola until it reaches the nearest point \( Q \) whose \( y \)-coordinate is \(-6\). What is the horizontal distance traveled by the particle (the numerical value of the differ... | [
"3"
] | [
"A particle is placed on the parabola \\( y = x^2 - x - 6 \\) at a point \\( P \\) whose \\( y \\)-coordinate is 6. It is allowed to roll along the parabola until it reaches the nearest point \\( Q \\) whose \\( y \\)-coordinate is \\(-6\\). What is the horizontal distance traveled by the particle (the numerical va... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-074f153464ce3dd4aeefcb736c76bcc864273dee09b63e6f87955b59c368a94e | A particle is projected from a distance \( c \) from a fixed point with an initial velocity \( \frac{\sqrt{17\mu}}{\sqrt{2}c^2} \) in a direction making an angle \( \tan^{-1}\left(\frac{2\sqrt{2}}{3}\right) \) with the initial distance. The force of attraction to the fixed point is given by \( \frac{\mu}{r^5} \). Let \... | [
"\\dfrac{3e^{\\frac{\\pi}{2\\sqrt{2}}} + 1}{2\\left(3e^{\\frac{\\pi}{2\\sqrt{2}}} - 1\\right)}"
] | [
"A particle is projected from a distance \\( c \\) from a fixed point with an initial velocity \\( \\frac{\\sqrt{17\\mu}}{\\sqrt{2}c^2} \\) in a direction making an angle \\( \\tan^{-1}\\left(\\frac{2\\sqrt{2}}{3}\\right) \\) with the initial distance. The force of attraction to the fixed point is given by \\( \\fr... | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-177f313a3e8ecd069898db71963d3257dd3f472ec61abd5d52cc8766173c8017 | A particle is projected vertically upwards in the medium with speed equal to the terminal velocity \( V \). Find the speed of the particle, in terms of \( V \), when it returns to the point of projection. | [
"\\dfrac{V}{\\sqrt{2}}"
] | [
"A particle is projected vertically upwards in the medium with speed equal to the terminal velocity \\( V \\). Find the speed of the particle, in terms of \\( V \\), when it returns to the point of projection."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-251a5539b24252df8b7ff7c829447726b184c3c8e74efdcabd3435af0a7609bb | A particle is released at a velocity \( V \) m/s at an angle \( A \) degrees from a point 15 meters above ground level. It just clears a fence 30 meters away and 26.25 meters high. Determine the values of \( V \) and \( A \). | [
"36.87^\\circ"
] | [
"A particle is released at a velocity \\( V \\) m/s at an angle \\( A \\) degrees from a point 15 meters above ground level. It just clears a fence 30 meters away and 26.25 meters high. Determine the values of \\( V \\) and \\( A \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f7492af7c8509ad41b59d38b205e92fe83c8e7158c2204254d92706883070442 | A particle located at a vertex of a square $ABCD$ moves to one of the neighbouring vertices with equal probability. Suppose that the particle starts moving from the vertex $A$. Find the probability that it will visit each of the other three vertices at least once before returning to $A$. | [
"\\(\\frac{1}{3}\\)"
] | [
"A particle located at a vertex of a square $ABCD$ moves to one of the neighbouring vertices with equal probability. Suppose that the particle starts moving from the vertex $A$. Find the probability that it will visit each of the other three vertices at least once before returning to $A$."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 16
} | 38 | 0 | 0 |
omr-ws-9725908db95f7685a0fc1cf358652d1825ba9788c2636abcfd26d1cf0718795a | A particle moves according to the following rules: if a move added $(p, q)$ to get to the current position, the next move will add either $(p-1, q)$ or $(p, q-1)$ to move to the next position. Neither $p$ nor $q$ can ever be negative. Particle $A$ starts at $(0, 0)$ and moves to $(3, 4)$. The particle continues to move... | [
"14"
] | [
"A particle moves according to the following rules: if a move added $(p, q)$ to get to the current position, the next move will add either $(p-1, q)$ or $(p, q-1)$ to move to the next position. Neither $p$ nor $q$ can ever be negative. Particle $A$ starts at $(0, 0)$ and moves to $(3, 4)$. The particle continues to... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-8f999036b8162e62d4e8ee23878c27e7be7c78ea4c7f162afd88ee1d58ddab1d | A particle moves along a curve in the $xy$ plane with constant speed $2$. The motion starts at the origin when $t = 0$ and the initial velocity is $v(0) = 2\vec{i}$. The curvature at each instant $t$ is given by $\kappa(t) = 4t$. Find the velocity of the particle when $t=\frac{\sqrt{\pi}}{4}$, given that the curve neve... | [
"\\(\\sqrt{2}\\hat{i} + \\sqrt{2}\\hat{j}\\)"
] | [
"A particle moves along a curve in the $xy$ plane with constant speed $2$. The motion starts at the origin when $t = 0$ and the initial velocity is $v(0) = 2\\vec{i}$. The curvature at each instant $t$ is given by $\\kappa(t) = 4t$. Find the velocity of the particle when $t=\\frac{\\sqrt{\\pi}}{4}$, given that the ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-c53570c3410f8b1be91c3d82ddcb249033d901124195d5808bd2f81a3ec4bae6 | A particle moves along a line with its position at time \( t \) given by \( s(t) = t^3 - 6t^2 + 9t + 2 \). Calculate the sum of all times \( t \) in the open interval \( 0 < t < 3 \) when the particle changes direction. | [
"1"
] | [
"A particle moves along a line with its position at time \\( t \\) given by \\( s(t) = t^3 - 6t^2 + 9t + 2 \\). Calculate the sum of all times \\( t \\) in the open interval \\( 0 < t < 3 \\) when the particle changes direction."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-059831b6c2ded2a9286473ed15f17c5daf45f7905f26d2886ad02bc6af0a4d12 | A particle moves along a path described by \( y = x^2 \). At what point along the curve are \( x \) and \( y \) changing at the same rate? Find this rate if at that time \( t \) we have \( x = \sin t \) and \( y = \sin^2 t \). | [
"\\(\\left(\\frac{1}{2}, \\frac{1}{4}\\right)\\)"
] | [
"A particle moves along a path described by \\( y = x^2 \\). At what point along the curve are \\( x \\) and \\( y \\) changing at the same rate? Find this rate if at that time \\( t \\) we have \\( x = \\sin t \\) and \\( y = \\sin^2 t \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc9da5e743b460fa4333d67433f6245b889a825c4dc566551d0ce73a62ca8b87 | A particle moves along a path with constant horizontal speed \( v \) and its vertical acceleration's absolute value does not exceed a constant \( k \). If the particle travels a horizontal distance \( l \) and reaches a height \( h \) at the end of this distance, calculate the maximum possible value of \( k \) given \(... | [
"\\dfrac{3}{2}"
] | [
"A particle moves along a path with constant horizontal speed \\( v \\) and its vertical acceleration's absolute value does not exceed a constant \\( k \\). If the particle travels a horizontal distance \\( l \\) and reaches a height \\( h \\) at the end of this distance, calculate the maximum possible value of \\(... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3fdfca39ca957ea7dd2f703b3910eeb79a4f238950811cd8e20d17305b4e8880 | A particle moves along a straight line under gravity from a point $P$ to a point $Q$ on a circle in the same vertical plane. The distance travelled from $P$ in time $t$ is given by $\dfrac{1}{2}gt^2 \sin{\alpha}$, where $\alpha$ is the angle of inclination of the line $PQ$ to the horizontal. Determine the geometrical c... | [
"The point \\( Q \\) is the intersection of the line \\( PB \\) with the circle, where \\( B \\) is the lowest point of the circle."
] | [
"A particle moves along a straight line under gravity from a point $P$ to a point $Q$ on a circle in the same vertical plane. The distance travelled from $P$ in time $t$ is given by $\\dfrac{1}{2}gt^2 \\sin{\\alpha}$, where $\\alpha$ is the angle of inclination of the line $PQ$ to the horizontal. Determine the geom... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 3,
"tir": 4
} | 9 | 0 | 0 |
omr-ws-057df23fc900859c5260c287dde7da49faaee6aabecc25c6cb73d33a919f5f43 | A particle moves along the $x$-axis with velocity $v(t) = \frac{t^2}{\sin^2(t) + 2}$. The particle is at position $x = 3$ at time $t = 2$. What is the particle's position at time $t = 7$? | [
"51"
] | [
"A particle moves along the $x$-axis with velocity $v(t) = \\frac{t^2}{\\sin^2(t) + 2}$. The particle is at position $x = 3$ at time $t = 2$. What is the particle's position at time $t = 7$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-8c24515cd919c3c7d7843adde1d7174182d0567fdf898872cfcced4950de1826 | A particle moves along the graph \( y = \cos x \) such that the x-component of acceleration is always 2. At time \( t = 0 \), the particle is at the point \( (\pi, -1) \) and the velocity vector is \( (0, 0) \). Find the speed of the particle when its position is \( (4, \cos 4) \). | [
"2.3238"
] | [
"A particle moves along the graph \\( y = \\cos x \\) such that the x-component of acceleration is always 2. At time \\( t = 0 \\), the particle is at the point \\( (\\pi, -1) \\) and the velocity vector is \\( (0, 0) \\). Find the speed of the particle when its position is \\( (4, \\cos 4) \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 19
} | 19 | 0 | 0 |
omr-ws-dffab669939a66cda79881a8997bace808aebcdc7d810d181d6c66c68b6b5b7c | A particle moves along the graph \( y = \cos x \) such that the x-component of acceleration is always 2. At time \( t = 0 \), the particle is at the point \( (\pi, -1) \) and the velocity vector is \( (0, 0) \). Find the x and y coordinates of the position of the particle in terms of \( t \). | [
"\\( x(t) = t^2 + \\pi \\), \\( y(t) = \\cos(t^2 + \\pi) \\)"
] | [
"A particle moves along the graph \\( y = \\cos x \\) such that the x-component of acceleration is always 2. At time \\( t = 0 \\), the particle is at the point \\( (\\pi, -1) \\) and the velocity vector is \\( (0, 0) \\). Find the x and y coordinates of the position of the particle in terms of \\( t \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 18
} | 26 | 0 | 0 |
omr-ws-5403d19b1a6d7746117818de94f90e8c45a83a34a8a37629b940c3d20bdfa38b | A particle moves along the hyperbola given by the equation \( x^2 - 18y^2 = 9 \) such that the y-coordinate increases at a rate of 9 units per second. Calculate the rate at which the x-coordinate is changing when \( x = 9 \). | [
"\\( \\pm 36 \\)"
] | [
"A particle moves along the hyperbola given by the equation \\( x^2 - 18y^2 = 9 \\) such that the y-coordinate increases at a rate of 9 units per second. Calculate the rate at which the x-coordinate is changing when \\( x = 9 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 11
} | 19 | 0 | 0 |
omr-ws-2f9292c32f4cead2960684722bbf540ac032535e96d0abe5adf442c17b3285e7 | A particle moves along the x-axis so that at any time \( t > 0 \), its acceleration is given by \( a(t) = \ln(1 + 2^t) \). If the velocity of the particle is 2 at time \( t = 1 \), then what is the velocity of the particle at time \( t = 2 \)? | [
"2.886"
] | [
"A particle moves along the x-axis so that at any time \\( t > 0 \\), its acceleration is given by \\( a(t) = \\ln(1 + 2^t) \\). If the velocity of the particle is 2 at time \\( t = 1 \\), then what is the velocity of the particle at time \\( t = 2 \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f51b01984c1ea97adae27ceb86f3a106afe8ebfa8d3cd6463bfed847e28705cf | A particle moves along the x-axis so that at any time \(t > 0\), its acceleration is given by \(a(t) = \ln(1 + 2^t)\). If the velocity of the particle is 2 at time \(t = 1\), what is the velocity of the particle at time \(t = 2\)? | [
"2 + \\frac{3}{2} \\ln 2"
] | [
"A particle moves along the x-axis so that at any time \\(t > 0\\), its acceleration is given by \\(a(t) = \\ln(1 + 2^t)\\). If the velocity of the particle is 2 at time \\(t = 1\\), what is the velocity of the particle at time \\(t = 2\\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-78f2329d9d3c4af45d170c9c2dbe3881308ec8b4f79c86f7f99ae415aa1db231 | A particle moves along the x-axis so that its position is given by the function \( x(t) \) for any time \( t \) in the interval \([0, 5]\). A: Find its average velocity. B: Find the minimum acceleration of the particle. C: What is the total distance traveled by the particle in the given interval? | [
"A: 4, B: 0, C: 28"
] | [
"A particle moves along the x-axis so that its position is given by the function \\( x(t) \\) for any time \\( t \\) in the interval \\([0, 5]\\).\n\nA: Find its average velocity.\nB: Find the minimum acceleration of the particle.\nC: What is the total distance traveled by the particle in the given interval?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d704c5c0de761f4762c33a817b73ce5cd5ffa510b69e7d82b5c781b04350cd4 | A particle moves along the x-axis such that its acceleration at any time \( t \) is given by \( a(t) = 2t - 7 \). If the initial velocity of the particle is 6, at what time \( t \) during the interval \( 0 \leq t \leq 4 \) is the particle farthest to the right? | [
"1"
] | [
"A particle moves along the x-axis such that its acceleration at any time \\( t \\) is given by \\( a(t) = 2t - 7 \\). If the initial velocity of the particle is 6, at what time \\( t \\) during the interval \\( 0 \\leq t \\leq 4 \\) is the particle farthest to the right?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-85d2ae9be14fd78c632319a2876923fa37dcf81e34a76190f90a75692aacb957 | A particle moves along the x-axis such that its position at time \( t \) for \( t \geq 0 \) is given by \( x = \frac{1}{3}t^3 - 3t^2 + 8t \). What is the total distance the particle has traveled when \( t = 3 \)? | [
"\\(\\frac{22}{3}\\)"
] | [
"A particle moves along the x-axis such that its position at time \\( t \\) for \\( t \\geq 0 \\) is given by \\( x = \\frac{1}{3}t^3 - 3t^2 + 8t \\). What is the total distance the particle has traveled when \\( t = 3 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-6aad852fa774e0838524c9273ccaca8cc1ab2b12ff85eca68329c1cdcc8b7d87 | A particle moves along the x-axis with velocity at time \( t \) given by \( v = 5t^2 + 2t \). Find the distance it moves in the 2nd second. | [
"\\(\\frac{44}{3}\\)"
] | [
"A particle moves along the x-axis with velocity at time \\( t \\) given by \\( v = 5t^2 + 2t \\). Find the distance it moves in the 2nd second."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-d296bfba185ebc67411f41c975960873cb6d5b935925c0ce032c59eae71056a3 | A particle moves along the x-axis with velocity at time \( t \) given by \( v(t) = -1 + e^{1 - t} \). Find the total distance traveled by the particle over the time interval \([0,3]\). | [
"\\(-1 + e + \\frac{1}{e^2}\\)"
] | [
"A particle moves along the x-axis with velocity at time \\( t \\) given by \\( v(t) = -1 + e^{1 - t} \\). Find the total distance traveled by the particle over the time interval \\([0,3]\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-b547f86818ed7325941a0e7ba2294a433ffa95679472558b12c94d72eca7a0db | A particle moves along the x-axis with velocity given by \( v(t) = 3t^2 - 2t - 1 \) for \( t \leq 0 \). The position \( x(t) \) is 5 when \( t = 2 \). A. Write a polynomial expression for the position of the particle at any time \( t \leq 0 \). B. For what values of \( t \) in the interval \( 0 \leq t \leq 3 \) is the ... | [
"17"
] | [
"A particle moves along the x-axis with velocity given by \\( v(t) = 3t^2 - 2t - 1 \\) for \\( t \\leq 0 \\). The position \\( x(t) \\) is 5 when \\( t = 2 \\).\n\nA. Write a polynomial expression for the position of the particle at any time \\( t \\leq 0 \\).\n\nB. For what values of \\( t \\) in the interval \\( ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b2dfd02694a2e2bcbf5e257449cc46de2ded6e4641de74970b5efaa8998cb93 | A particle moves along the x-axis with velocity given by \( v(t) = 5 t e^{-t} - 1 \) for \( t \geq 0 \). At \( t = 0 \), the particle is at position \( x = 1 \). What is the total distance traveled by the particle from \( t = 0 \) to \( t = 4 \)? | [
"1.823"
] | [
"A particle moves along the x-axis with velocity given by \\( v(t) = 5 t e^{-t} - 1 \\) for \\( t \\geq 0 \\). At \\( t = 0 \\), the particle is at position \\( x = 1 \\). What is the total distance traveled by the particle from \\( t = 0 \\) to \\( t = 4 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-fca8f46a2912056153efa225b787fe6617bb96cc6369d136b00fdb517bf2d3c1 | A particle moves along the x-axis with velocity given by \( v(t) = 5te^{-t} - 1 \) for \( t \geq 0 \). At \( t = 0 \), the particle is at position \( x = 1 \). What is the total distance traveled by the particle from \( t = 0 \) to \( t = 4 \)? | [
"1.82"
] | [
"A particle moves along the x-axis with velocity given by \\( v(t) = 5te^{-t} - 1 \\) for \\( t \\geq 0 \\). At \\( t = 0 \\), the particle is at position \\( x = 1 \\). What is the total distance traveled by the particle from \\( t = 0 \\) to \\( t = 4 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-2bc718865a453a7f0bf01723cf639c6c160c3f206c5e23dccd3eb6343b58f38e | A particle moves along the x-axis with velocity given by \( v(t) = \cos(2 - t^2) \). The position of the particle is 3 at time \( t = 0 \). What is the position of the particle when its velocity is first equal to zero? | [
"2.817"
] | [
"A particle moves along the x-axis with velocity given by \\( v(t) = \\cos(2 - t^2) \\). The position of the particle is 3 at time \\( t = 0 \\). What is the position of the particle when its velocity is first equal to zero?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-220d362577ac8d96ab2a9a053d6d1f06b4fd8079adbe338877b3d76452abd90c | A particle moves along the x-axis, starting from 0 at \( t = 0 \), such that its speed is either 10 or -10 at any point in time. If it reaches 20 before it reaches -10, then Bob wins. If it reaches -10 before it reaches 20, then Alice wins. Find the probability that Alice wins. | [
"\\(\\frac{2}{3}\\)"
] | [
"A particle moves along the x-axis, starting from 0 at \\( t = 0 \\), such that its speed is either 10 or -10 at any point in time. If it reaches 20 before it reaches -10, then Bob wins. If it reaches -10 before it reaches 20, then Alice wins. Find the probability that Alice wins."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f970039b83c2c392158718516334ba29803d9dc7daa73b9767a318e471e91a25 | A particle moves along the y-axis with velocity \( v(t) = 1 - \arctan(e^t) \). Is the speed of the particle increasing or decreasing at \( t = 2 \)? | [
"Speed is increasing."
] | [
"A particle moves along the y-axis with velocity \\( v(t) = 1 - \\arctan(e^t) \\). Is the speed of the particle increasing or decreasing at \\( t = 2 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-ade98a25f9cfb1c07b213b1d23e8f24f49f7e73a35ba09abfedd52a6a9e78e06 | A particle moves freely in a straight line except for a resistive force proportional to its speed. Its speed falls from $1,000 \dfrac{ft}{s}$ to $900 \dfrac{ft}{s}$ over $1200 ft.$ Find the time taken to the nearest $0.01 s.$ [No calculators or log tables allowed!] | [
"1.26"
] | [
"A particle moves freely in a straight line except for a resistive force proportional to its speed. Its speed falls from $1,000 \\dfrac{ft}{s}$ to $900 \\dfrac{ft}{s}$ over $1200 ft.$ Find the time taken to the nearest $0.01 s.$ [No calculators or log tables allowed!]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5e2a5531a4316194a8cb7ef3502b1c021851c7934a31127c568d24268e800aa0 | A particle moves from $(0, 0)$ to $(n, n)$ directed by a fair coin. For each head, it moves one step east, and for each tail, it moves one step north. At $(n, y)$ where $y < n$, it stays there if a head comes up, and at $(x, n)$ where $x < n$, it stays there if a tail comes up. Let $k$ be a fixed positive integer. Find... | [
"\\(\\frac{2 \\binom{2n+k-1}{n-1}}{2^{2n+k}}\\)"
] | [
"A particle moves from $(0, 0)$ to $(n, n)$ directed by a fair coin. For each head, it moves one step east, and for each tail, it moves one step north. At $(n, y)$ where $y < n$, it stays there if a head comes up, and at $(x, n)$ where $x < n$, it stays there if a tail comes up. Let $k$ be a fixed positive integer.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-bd4b6087a0ad858e9e22e18d01277355e58d6d26ae0e250602d8fc090aa44e94 | A particle moves in 3-space according to the equations: $$ \frac{dx}{dt} = yz, \; \frac{dy}{dt} = xz, \; \frac{dz}{dt} = xy. $$ Suppose $x(0) = 0$ and $y(0) = 0$. Calculate the value of $x(t)^2 + y(t)^2 + z(t)^2$ for any $t > 0$. | [
"0"
] | [
"A particle moves in 3-space according to the equations:\n$$ \\frac{dx}{dt} = yz, \\; \\frac{dy}{dt} = xz, \\; \\frac{dz}{dt} = xy. $$\nSuppose $x(0) = 0$ and $y(0) = 0$. Calculate the value of $x(t)^2 + y(t)^2 + z(t)^2$ for any $t > 0$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-b657db3d3765f91d0b314b1685792d9c978c72c9f18bd710434e7603fa12e741 | A particle moves in a Cartesian plane from (2,0,11) to (20,1,1). All points in its route are in the form (x,y,z) where at least two of x, y, z are integers, and the route it takes is the shortest possible. Determine the number of possible routes this particle takes. | [
"\\(\\frac{29!}{18!1!10!}\\)"
] | [
"A particle moves in a Cartesian plane from (2,0,11) to (20,1,1). All points in its route are in the form (x,y,z) where at least two of x, y, z are integers, and the route it takes is the shortest possible. Determine the number of possible routes this particle takes."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 2,
"tir": 24
} | 41 | 0 | 0 |
omr-ws-b846f81fcb3315128cbd421bd848a0c64c96b91c7130bd569481ef0db0e87e62 | A particle moves in a plane with an acceleration parallel to the y-axis that varies as the distance from the x-axis, given by \( a_y = -k^2y \), where \( k \) is a constant. If the particle starts from the point \( (0, 1) \) with an initial velocity \( v_0 \) along the x-axis, find the amplitude \( A \) of the trajecto... | [
"1"
] | [
"A particle moves in a plane with an acceleration parallel to the y-axis that varies as the distance from the x-axis, given by \\( a_y = -k^2y \\), where \\( k \\) is a constant. If the particle starts from the point \\( (0, 1) \\) with an initial velocity \\( v_0 \\) along the x-axis, find the amplitude \\( A \\) ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40145fe81984716d48b426d3cd505f4dc5d4c8dc74174b3a61f5a7ba21841ea5 | A particle moves in a straight line $OCP$ under the influence of a force $m\mu$ directed towards a point $C$. The point $C$ moves along $OC$ with a constant acceleration $f$. Initially, $C$ is at rest at the origin $O$, and the particle is at a distance $c$ from $O$ with an initial velocity $V$. Determine the distance ... | [
" \\left(c + \\frac{f}{\\mu}\\right) \\cos(\\sqrt{\\mu} t) + \\frac{V}{\\sqrt{\\mu}} \\sin(\\sqrt{\\mu} t) + \\frac{f}{2} t^2 - \\frac{f}{\\mu} "
] | [
"A particle moves in a straight line $OCP$ under the influence of a force $m\\mu$ directed towards a point $C$. The point $C$ moves along $OC$ with a constant acceleration $f$. Initially, $C$ is at rest at the origin $O$, and the particle is at a distance $c$ from $O$ with an initial velocity $V$. Determine the dis... | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-0989eaabc842f381c432d98ea98d7bb1821ea518020c4df4d5ee504b168a9a20 | A particle moves in a straight line and its displacement \(x\) metres from the origin after \(t\) seconds is given by \(x = \cos^2(3t)\), \(t > 0\). i) Calculate the time \(t\) when the particle is first at \(x = \frac{3}{4}\). ii) Determine the velocity of the particle when it is first at \(x = \frac{3}{4}\). iii) Exp... | [
"\\dfrac{\\pi}{3}"
] | [
"A particle moves in a straight line and its displacement \\(x\\) metres from the origin after \\(t\\) seconds is given by \\(x = \\cos^2(3t)\\), \\(t > 0\\).\n\ni) Calculate the time \\(t\\) when the particle is first at \\(x = \\frac{3}{4}\\).\n\nii) Determine the velocity of the particle when it is first at \\(x... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-82636d08b9f38d61d9faf7560d140514c1bc6ef10eeacae97091f287a6fd6d67 | A particle moves in a straight line with monotonically decreasing acceleration. It starts from rest and has velocity $v$ a distance $d$ from the start. What is the maximum time it could have taken to travel the distance $d$? | [
"\\( \\frac{2d}{v} \\)"
] | [
"A particle moves in a straight line with monotonically decreasing acceleration. It starts from rest and has velocity $v$ a distance $d$ from the start. What is the maximum time it could have taken to travel the distance $d$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-6954b699ce312126b5fa7f1d3bd4915a505dbdc80806b73fdb51693f8ab454ad | A particle moves in the Cartesian Plane according to the following rules: 1. From any lattice point $(a,b)$, the particle may only move to $(a + 1,b)$, $(a,b + 1)$, or $(a + 1,b + 1)$. 2. There are no right angle turns in the particle's path. How many different paths can the particle take from $(0,0)$ to $(5,5)$? | [
"83"
] | [
"A particle moves in the Cartesian Plane according to the following rules:\n1. From any lattice point $(a,b)$, the particle may only move to $(a + 1,b)$, $(a,b + 1)$, or $(a + 1,b + 1)$.\n2. There are no right angle turns in the particle's path.\n\nHow many different paths can the particle take from $(0,0)$ to $(5,... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-b2c29df932eb9b542463de5e46696e9156149cec61811ed70586156c5ff8c513 | A particle moves in the Cartesian plane according to the following rules: From any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$ There are no right angle turns in the particle's path. How many different paths can the particle take from $(0,0)$ to $(5,5)$? | [
"83"
] | [
"A particle moves in the Cartesian plane according to the following rules:\nFrom any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$\nThere are no right angle turns in the particle's path.\nHow many different paths can the particle take from $(0,0)$ to $(5,5)$?\n"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-514aa708c567cc59b750174b4f6dcd75fd0f396f17f70c07ab320e6a3175a548 | A particle moves in the Euclidean plane with coordinates given by $x = t^3 - t$ and $y = t^4 + t$. Calculate the value of $t$ at which the magnitude of the particle's velocity vector reaches its maximum. | [
"0"
] | [
"A particle moves in the Euclidean plane with coordinates given by $x = t^3 - t$ and $y = t^4 + t$. Calculate the value of $t$ at which the magnitude of the particle's velocity vector reaches its maximum."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-a5d327907b0fa3ed292b1a79c3b3af2bc0665d4ff84ebf3ca44399cb46e59fbe | A particle moves in the Euclidean plane with coordinates given by \( x = t^3 - t \) and \( y = t^4 + t \). Calculate the value of \( t \) at which the speed of the particle is maximized. | [
"0"
] | [
"A particle moves in the Euclidean plane with coordinates given by \\( x = t^3 - t \\) and \\( y = t^4 + t \\). Calculate the value of \\( t \\) at which the speed of the particle is maximized."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-ce3df2182a662b83b00fa612206e621fdc1d7ef7daf79991544496579698e19a | A particle moves inside a square with vertices A=(0, 0), B=(1, 0), C=(1, 1), and D=(0, 1). It begins at the point (2/3, 0), travels to some point on BC, then travels to some point on CD, then travels to some point on DA, and then travels to the point (1/6, 0). What is the minimum distance that the particle could have t... | [
"\\(\\frac{5}{2}\\)"
] | [
"A particle moves inside a square with vertices A=(0, 0), B=(1, 0), C=(1, 1), and D=(0, 1). It begins at the point (2/3, 0), travels to some point on BC, then travels to some point on CD, then travels to some point on DA, and then travels to the point (1/6, 0). What is the minimum distance that the particle could h... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-0a92717df64ba3f4e736a0d292dd7a20687a86a7a20236c15241e4b1b05d52d1 | A particle moves on a circle with center \( O \), starting from rest at a point \( P \) and coming to rest again at a point \( Q \), without coming to rest at any intermediate point. Let \( R \) be the point between \( P \) and \( Q \) where the acceleration vector of the particle points in along the radius \( RO \). I... | [
"\\dfrac{r\\theta}{2}"
] | [
"A particle moves on a circle with center \\( O \\), starting from rest at a point \\( P \\) and coming to rest again at a point \\( Q \\), without coming to rest at any intermediate point. Let \\( R \\) be the point between \\( P \\) and \\( Q \\) where the acceleration vector of the particle points in along the r... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4ee76a19d2585265939526fbce4ae1494c631a31ced74481936ffae95ec212f5 | A particle moves on the eight vertices of a cube, moving to each of the three adjacent vertices with equal probability at each step. Let \( i \) be the initial vertex occupied by the particle, and let \( o \) be the vertex opposite \( i \). Calculate the expected number of steps until the first visit to \( o \). | [
"10"
] | [
"A particle moves on the eight vertices of a cube, moving to each of the three adjacent vertices with equal probability at each step. Let \\( i \\) be the initial vertex occupied by the particle, and let \\( o \\) be the vertex opposite \\( i \\). Calculate the expected number of steps until the first visit to \\( ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
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