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-0728dad45e5b5b69b89c6e1d6c14298beb948e7fbd941cf9ed5e8ce31dc3c8a5 | Two players, \(A\) and \(B\), play on an \(n \times n\) board (\(n \ge 1\)). Player \(A\) picks an \(a \times b\) board (\(a, b \ge 1\)) that covers the right-down cell, and Player \(B\) picks an \(a \times b\) board that covers the left-up cell. Player \(A\) starts from the right-down cell. The player who makes the la... | [
"2"
] | [
"Two players, \\(A\\) and \\(B\\), play on an \\(n \\times n\\) board (\\(n \\ge 1\\)). Player \\(A\\) picks an \\(a \\times b\\) board (\\(a, b \\ge 1\\)) that covers the right-down cell, and Player \\(B\\) picks an \\(a \\times b\\) board that covers the left-up cell. Player \\(A\\) starts from the right-down cel... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-f4f8b4ffa4758add6a71b49dcc3ad973fe30d8fc3b788e3d1dfb401ca2c13e66 | Two players, liyi and x^4t^2, take turns inserting one of the signs +, -, or × in any of the empty places between the terms 1 2 3 4 ... 99 100 until an arithmetic progression \( S \) is determined at the end of 99 turns. If liyi goes first and aims to make \( S \) even, what is the minimum number of multiplication sign... | [
"0"
] | [
"Two players, liyi and x^4t^2, take turns inserting one of the signs +, -, or × in any of the empty places between the terms 1 2 3 4 ... 99 100 until an arithmetic progression \\( S \\) is determined at the end of 99 turns. If liyi goes first and aims to make \\( S \\) even, what is the minimum number of multiplica... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c420dcb42eed3566cbf9cc72b1618adb133d58dd09ef1817aa5a85d879a2fbc3 | Two players, the builder and the destroyer, play a game where they alternately choose different elements from the set $\{0,1,\ldots,10\}$. The builder wins if any four of the six integers they choose form an arithmetic sequence. The destroyer wins if they can prevent the formation of such an arithmetic four-tuple. Whic... | [
"Destroyer has a winning strategy."
] | [
"Two players, the builder and the destroyer, play a game where they alternately choose different elements from the set $\\{0,1,\\ldots,10\\}$. The builder wins if any four of the six integers they choose form an arithmetic sequence. The destroyer wins if they can prevent the formation of such an arithmetic four-tup... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 19
} | 20 | 0 | 0 |
omr-ws-76525d6c7778771c966403cb0a829edf86cafcd3d1b0771a40ed674b6642e16e | Two players, the builder and the destroyer, play the following game. The builder starts, and players choose alternatively different elements from the set \(\{0, 1, \ldots, 10\}\). The builder wins if some four integers of those six integers he chose form an arithmetic sequence. The destroyer wins if he can prevent the ... | [
"\\text{The builder}"
] | [
"Two players, the builder and the destroyer, play the following game. The builder starts, and players choose alternatively different elements from the set \\(\\{0, 1, \\ldots, 10\\}\\). The builder wins if some four integers of those six integers he chose form an arithmetic sequence. The destroyer wins if he can pr... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-3cdb5f636139b37eb9a827d76a9a1e887f89e1e50e3ca49389a5773fa2caaa3a | Two points $A$ and $B$ and a line $\ell$ are fixed in the plane such that $\ell$ is not perpendicular to $AB$ and does not intersect the segment $AB$. Consider all circles with center $O$ not lying on $\ell$, passing through $A$ and $B$, and meeting $\ell$ at points $C$ and $D$. Let $P$ be the point of tangency of the ... | [
"no_valid_answer_found"
] | [
"Two points $A$ and $B$ and a line $\\ell$ are fixed in the plane such that $\\ell$ is not perpendicular to $AB$ and does not intersect the segment $AB$. Consider all circles with center $O$ not lying on $\\ell$, passing through $A$ and $B$, and meeting $\\ell$ at points $C$ and $D$. Let $P$ be the point of tangenc... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-002b9382f9a95cef651e1f86c243346bfc419518b3373e865805b4ddb4d7a845 | Two points $A$ and $B$ are 105 km apart. A bus left $A$ for $B$ at a speed $v$ km/h. Thirty minutes later a car traveling at 40 km/h followed the bus. Having overtaken the bus, the car returns to $A$ at the same speed. For what range of values of $v$ will the bus reach $B$ before the car arrives at $A$? | [
"$30<v<40$"
] | [
"Two points $A$ and $B$ are 105 km apart. A bus left $A$ for $B$ at a speed $v$ km/h. Thirty minutes later a car traveling at 40 km/h followed the bus. Having overtaken the bus, the car returns to $A$ at the same speed. For what range of values of $v$ will the bus reach $B$ before the car arrives at $A$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44f5727ff73d274f3c0ca80eccced0c3d22d8872b0e912d02cfd22106b90f5d0 | Two points $A$ and $B$ are chosen independently and uniformly at random in the interval $(0, 2\pi)$. For each choice of $A$ and $B$, consider the arc of the unit circle that connects $A$ to $B$ without passing through a fixed point $C$. Repeat this experiment $n$ times. Determine the probability that one of the arcs in... | [
"\\dfrac{2^n}{\\dbinom{2n}{n}}"
] | [
"Two points $A$ and $B$ are chosen independently and uniformly at random in the interval $(0, 2\\pi)$. For each choice of $A$ and $B$, consider the arc of the unit circle that connects $A$ to $B$ without passing through a fixed point $C$. Repeat this experiment $n$ times. Determine the probability that one of the a... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-533846133429ffc2b79cf87a00e118bd6278d37fcc67741f8ca1943086ce721a | Two points $A$ and $B$ are chosen randomly and independently on the perimeter of a regular hexagon. Find the probability that there exists a point $C$ inside the hexagon such that $\triangle ABC$ is equilateral. | [
"\\dfrac{1}{6}"
] | [
"Two points $A$ and $B$ are chosen randomly and independently on the perimeter of a regular hexagon. Find the probability that there exists a point $C$ inside the hexagon such that $\\triangle ABC$ is equilateral."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0e8f56fc15db7d4e7951c7fa79528b84202131c3b97f1ba92638c9da082ea2e4 | Two points $A$ and $B$ are given on the plane. An arbitrary circle passes through $B$ and intersects the straight line $AB$ for a second time at a point $K$, different from $A$. A circle passing through $A$, $K$, and the center of the first circle intersects the first circle for a second time at point $M$. Find the loc... | [
"\\text{The circle with center } A \\text{ and radius } AB"
] | [
"Two points $A$ and $B$ are given on the plane. An arbitrary circle passes through $B$ and intersects the straight line $AB$ for a second time at a point $K$, different from $A$. A circle passing through $A$, $K$, and the center of the first circle intersects the first circle for a second time at point $M$. Find th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-77dd8ef13ce45b129e86b4620e02af8b7b44e1e6f1279480aeefb5de1ed00c01 | Two points $A$ and $B$ are inside a circle $(O)$ with radius $r=4$, where $OA=OB=\sqrt{13}$ and $AB=4$. Find the minimum value of $PA + PB$ as point $P$ moves on the circle, without using calculus or a coordinate system. | [
"4"
] | [
"Two points $A$ and $B$ are inside a circle $(O)$ with radius $r=4$, where $OA=OB=\\sqrt{13}$ and $AB=4$. Find the minimum value of $PA + PB$ as point $P$ moves on the circle, without using calculus or a coordinate system."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-32411c1f5f65b1a8bd19544195410c63f6f9093d8a2885aebfb7f4e4ce58e38a | Two points $A$ and $B$ are randomly selected on Circle $O$. Point $C$ is labeled as the midpoint of Chord $AB$. Then, a point $D$ is randomly chosen on the border of Circle $O$. What is the probability that the Circle centered at $C$ passing through $D$ has a greater area than that of Circle $O$? | [
"\\dfrac{2}{3} - \\dfrac{(\\ln 2)^2}{\\pi^2}"
] | [
"Two points $A$ and $B$ are randomly selected on Circle $O$. Point $C$ is labeled as the midpoint of Chord $AB$. Then, a point $D$ is randomly chosen on the border of Circle $O$. What is the probability that the Circle centered at $C$ passing through $D$ has a greater area than that of Circle $O$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-a0a0c0fcaf1e159babdc7039b001b88ae53a279f10ed0eefd099fa3187e907a2 | Two points $A$ and $B$ are randomly selected on a line segment $OC$. What is the probability $p$ that the line segments $OA$, $AB$, and $BC$ can form the sides of a triangle? Find $\left\lfloor{\frac{1}{p}}\right\rfloor$. | [
"4"
] | [
"Two points $A$ and $B$ are randomly selected on a line segment $OC$. What is the probability $p$ that the line segments $OA$, $AB$, and $BC$ can form the sides of a triangle? Find $\\left\\lfloor{\\frac{1}{p}}\\right\\rfloor$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-b0a01c2c2c9ace11dfd78d2ecb5fbc715f263a591cbc0b9535f2a5456c2c1427 | Two points $A$ and $B$ are randomly selected on the surface of Sphere $O$. Point $C$ is labeled as the midpoint of the line segment $AB$ inside the sphere. Then, a point $D$ is randomly chosen on the surface of Sphere $O$. What is the probability that the Sphere centered at $C$ passing through $D$ has a greater volume ... | [
"\\dfrac{2}{3}"
] | [
"Two points $A$ and $B$ are randomly selected on the surface of Sphere $O$. Point $C$ is labeled as the midpoint of the line segment $AB$ inside the sphere. Then, a point $D$ is randomly chosen on the surface of Sphere $O$. What is the probability that the Sphere centered at $C$ passing through $D$ has a greater vo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-5970bebd286e7c2c1ac6fbd6448c1e51f9f60a201f3ac3342635d56dc62b641a | Two points $A$ and $B$ lie on a circle $k$ with center $M$. The point $C$ is located such that the line $AC$ is a tangent to the circle $k$. Point $D$ is the image of point $C$ as it rotates around point $M$, which maps point $A$ to point $B$. If the radius of the circle $k$ is $r$ and the length of the segment $CD$ is... | [
"r\\sqrt{3}"
] | [
"Two points $A$ and $B$ lie on a circle $k$ with center $M$. The point $C$ is located such that the line $AC$ is a tangent to the circle $k$. Point $D$ is the image of point $C$ as it rotates around point $M$, which maps point $A$ to point $B$. If the radius of the circle $k$ is $r$ and the length of the segment $C... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-3818f2a6739036ad6a55ef9669c85a13981b779aeecc8f17e5ead72dc13c8d55 | Two points $A$ and $B$ lie on a regular hexagon, such that $\overline{AB}$ divides the hexagon into two pentagons. Both pentagons have an area of $36 \sqrt{3}$ and a perimeter of $19 \sqrt{3}$. What is $AB$? | [
"\\( 7 \\sqrt{3} \\)"
] | [
"Two points $A$ and $B$ lie on a regular hexagon, such that $\\overline{AB}$ divides the hexagon into two pentagons. Both pentagons have an area of $36 \\sqrt{3}$ and a perimeter of $19 \\sqrt{3}$. What is $AB$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 7
} | 29 | 0 | 0 |
omr-ws-3e48a31e1079be8b9ec762c17bc34e2f8e3043ba638ca30093b9ed707e9772e8 | Two points $A$ and $B$ move around two different circles in the plane with the same angular velocity. Given that there exists a point $C$ such that it is equidistant from $A$ and $B$ at every instant, find the number of distinct positions where $A$ and $B$ can coincide within one complete revolution of the circles. | [
"2"
] | [
"Two points $A$ and $B$ move around two different circles in the plane with the same angular velocity. Given that there exists a point $C$ such that it is equidistant from $A$ and $B$ at every instant, find the number of distinct positions where $A$ and $B$ can coincide within one complete revolution of the circles... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dd8999a0fc721ddc18dc3f4f19081ceae555cfcceed5549254054b3536abd30f | Two points $A$ and $B$ move around two different circles in the plane with the same angular velocity. Suppose that there is a point $C$ which is equidistant from $A$ and $B$ at every moment. If the radii of the circles are $r_A$ and $r_B$, and the distance between the centers of the circles is $d$, find the ratio $\fra... | [
"no_valid_answer_found"
] | [
"Two points $A$ and $B$ move around two different circles in the plane with the same angular velocity. Suppose that there is a point $C$ which is equidistant from $A$ and $B$ at every moment. If the radii of the circles are $r_A$ and $r_B$, and the distance between the centers of the circles is $d$, find the ratio ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7bd531afb31bfbbc13c50ff5643284108058551b4aef86eca9399c2fa5460ac4 | Two points $A$ and $C$ are chosen on a circle $\Gamma$. Let $O$ be an arbitrary point on the perpendicular bisector of $\overline{AC}$. A circle centered at $O$ intersects $\overline{OA}$ and $\overline{OC}$ at points $A'$ and $C'$, respectively. The tangents to this circle at $A'$ and $C'$ meet $\Gamma$ at points $B$ ... | [
"6"
] | [
"Two points $A$ and $C$ are chosen on a circle $\\Gamma$. Let $O$ be an arbitrary point on the perpendicular bisector of $\\overline{AC}$. A circle centered at $O$ intersects $\\overline{OA}$ and $\\overline{OC}$ at points $A'$ and $C'$, respectively. The tangents to this circle at $A'$ and $C'$ meet $\\Gamma$ at p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-c7bd5368cae6852d5862334599a60d5b75fa6b2547fa0047f3a8034d2c0c69bd | Two points $A(1,1)$ and $B(a,a^{2})$ are on the parabola $y=x^{2}$. Two normal lines at $A$ and $B$ meet each other at point $P$. Find the value of the $x$-coordinate of $P$ as $B$ approaches $A$, using Euclidean geometric methods. | [
"-4"
] | [
"Two points $A(1,1)$ and $B(a,a^{2})$ are on the parabola $y=x^{2}$. Two normal lines at $A$ and $B$ meet each other at point $P$. Find the value of the $x$-coordinate of $P$ as $B$ approaches $A$, using Euclidean geometric methods."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d5c6c7e1667b60f64e182d90531158d365c9c10bf4ee2acc3ea419cc9050481e | Two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are chosen on the graph of $f(x) = \ln x$, with $0 < x_1 < x_2$. The points $C$ and $D$ trisect $\overline{AB}$, with $AC < CB$. Through $C$ a horizontal line is drawn to cut the curve at $E(x_3, y_3)$. Find $x_3$ if $x_1 = 1$ and $x_2 = 1000$. | [
"10"
] | [
"Two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are chosen on the graph of $f(x) = \\ln x$, with $0 < x_1 < x_2$. The points $C$ and $D$ trisect $\\overline{AB}$, with $AC < CB$. Through $C$ a horizontal line is drawn to cut the curve at $E(x_3, y_3)$. Find $x_3$ if $x_1 = 1$ and $x_2 = 1000$.\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-34a14d91ce306b5efb740d8954bb2da61efdd5e79a6dfd4cc187f14ca760b688 | Two points $C$ and $D$ lie in a semicircle with diameter $AB$. Given that $AC=2\sqrt{5}$, $AD=8$, and $\tan \angle CAD=\frac{1}{2}$, find the length of $AB$. | [
"10"
] | [
"Two points $C$ and $D$ lie in a semicircle with diameter $AB$. Given that $AC=2\\sqrt{5}$, $AD=8$, and $\\tan \\angle CAD=\\frac{1}{2}$, find the length of $AB$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
"genselect": 8,
"tir": 4
} | 55 | 0 | 0 |
omr-ws-405f77d3d276f314fabd53ce946252e56f6f27f3b25824894b14f9070b984e10 | Two points $C$ and $D$ with a constant distance move on a semicircle with diameter $AB$. Lines $AC$ and $BD$ intersect at $E$, and lines $AD$ and $BC$ intersect at $F$. If the radius of the semicircle is $5$ and the distance between $C$ and $D$ is $6$, calculate the area of quadrilateral $AEBF$. | [
"\\dfrac{75}{2}"
] | [
"Two points $C$ and $D$ with a constant distance move on a semicircle with diameter $AB$. Lines $AC$ and $BD$ intersect at $E$, and lines $AD$ and $BC$ intersect at $F$. If the radius of the semicircle is $5$ and the distance between $C$ and $D$ is $6$, calculate the area of quadrilateral $AEBF$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ee23ccc48e13ec35dd9032293dbfb5ed36104b07b21923b41fbad9d68c115bd0 | Two points $M$ and $N$ have been selected from the edge $BC$ of a tetrahedron $ABCD$ so that the point $M$ lies between $B$ and $N$, the point $N$ lies between $M$ and $C$, and the angle between the planes $AND$ and $ACD$ is equal to the angle between the planes $AMD$ and $ABD$. Calculate the minimum value of \[ \frac{... | [
"0"
] | [
"Two points $M$ and $N$ have been selected from the edge $BC$ of a tetrahedron $ABCD$ so that the point $M$ lies between $B$ and $N$, the point $N$ lies between $M$ and $C$, and the angle between the planes $AND$ and $ACD$ is equal to the angle between the planes $AMD$ and $ABD$. Calculate the minimum value of\n\\[... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-5e3cc8afb21bfd94d4ee9c1c4092309b0930e696f204f94bbf0f6ce71be09c2d | Two points $P$ and $Q$ are on sides $AB$ and $AC$ of triangle $ABC$. $CP$ and $BQ$ intersect at $O$. $(I_1)$ and $(I_2)$ are the incircles of triangles $POB$ and $QOC$, respectively. $W$ is the exsimilicenter of $(I_1)$ and $(I_2)$. An arbitrary line passing through $W$ intersects $AB$, $AC$, and $BC$ at points $X$, $Y... | [
"4"
] | [
"Two points $P$ and $Q$ are on sides $AB$ and $AC$ of triangle $ABC$. $CP$ and $BQ$ intersect at $O$. $(I_1)$ and $(I_2)$ are the incircles of triangles $POB$ and $QOC$, respectively. $W$ is the exsimilicenter of $(I_1)$ and $(I_2)$. An arbitrary line passing through $W$ intersects $AB$, $AC$, and $BC$ at points $X... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-a6bb571ce83876e5328afd27f262afa68702b2e936dd756bfb902c80f34ef783 | Two points $P$ and $Q$ lie on side $BC$ of triangle $ABC$ such that their distances from the midpoint of $BC$ are equal. The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$, respectively. Let $M$ be the point of intersection of $PF$ and $EQ$. If $H_1$ and $H_2$ are the orthocenters of tri... | [
"90"
] | [
"Two points $P$ and $Q$ lie on side $BC$ of triangle $ABC$ such that their distances from the midpoint of $BC$ are equal. The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$, respectively. Let $M$ be the point of intersection of $PF$ and $EQ$. If $H_1$ and $H_2$ are the orthocenters o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f683d24b12b8d3bee102a45303be8a0fa7b994d4bbeb940b4a71002ba42d499 | Two points $P$ and $Q$ lie on the side $BC$ of $\triangle ABC$ and have the same distance to the midpoint of $BC$. The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$, respectively. Let $M$ be the intersection point of $PF$ and $EQ$. If $H_{1}$ and $H_{2}$ denote the orthocenter of $\tria... | [
"90"
] | [
"Two points $P$ and $Q$ lie on the side $BC$ of $\\triangle ABC$ and have the same distance to the midpoint of $BC$. The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$, respectively. Let $M$ be the intersection point of $PF$ and $EQ$. If $H_{1}$ and $H_{2}$ denote the orthocenter of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e0495957de87606e51103882a3bd0eda5e4f6359dbe76633da86db0f36761c2 | Two points $P$ and $Q$ lie on the side $BC$ of triangle $ABC$ and have the same distance to the midpoint of $BC$. The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$ respectively. Let $M$ be the intersection point of $PF$ and $EQ$. If $H_1$ and $H_2$ denote the orthocenters of triangles $... | [
"90^\\circ"
] | [
"Two points $P$ and $Q$ lie on the side $BC$ of triangle $ABC$ and have the same distance to the midpoint of $BC$. The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$ respectively. Let $M$ be the intersection point of $PF$ and $EQ$. If $H_1$ and $H_2$ denote the orthocenters of triang... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-39782a502b82a379dbb2fd55a5b4a632946f782b4ec405c62e804fe8f98ffa9d | Two points $P$ and $Q$ lie on the side $BC$ of triangle $ABC$ and have the same distance to the midpoint of $BC$. The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$ respectively. Let $M$ be the intersection point of $PF$ and $EQ$. If $H_{1}$ and $H_{2}$ denote the orthocenter of $\triang... | [
"90^\\circ"
] | [
"Two points $P$ and $Q$ lie on the side $BC$ of triangle $ABC$ and have the same distance to the midpoint of $BC$. The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$ respectively. Let $M$ be the intersection point of $PF$ and $EQ$. If $H_{1}$ and $H_{2}$ denote the orthocenter of $\\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-91e1b267e802adf0053420592e1616d1b7c343ee1d64090a4e76978fe12e66be | Two points $X$ and $Y$ lie on the arc $BC$ of the circumcircle of $\triangle ABC$ (this arc does not contain $A$) such that $\angle BAX = \angle CAY$. Let $M$ denote the midpoint of the chord $AX$. If $BM + CM = k \cdot AY$, find the minimum possible value of $k$. | [
"1"
] | [
"Two points $X$ and $Y$ lie on the arc $BC$ of the circumcircle of $\\triangle ABC$ (this arc does not contain $A$) such that $\\angle BAX = \\angle CAY$. Let $M$ denote the midpoint of the chord $AX$. If $BM + CM = k \\cdot AY$, find the minimum possible value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-41020904c5fc739874181e249524fad05251bc23aab35ff18940d7ee6ab96725 | Two points A and B are 1 unit apart. Let region S be the set of all disks with radius 1 that completely cover AB. What is the area of region S? | [
"\\( 3\\pi - \\frac{\\sqrt{3}}{2} \\)"
] | [
"Two points A and B are 1 unit apart. Let region S be the set of all disks with radius 1 that completely cover AB. What is the area of region S?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-d7eea33fc2a0b67759b124b656a7e9b6496e2c88d3873067dac4a39ce05da13f | Two points A and B are chosen on the graph $y=x^2$. If they form an equilateral triangle with the origin, what is the area of the triangle? | [
"3\\sqrt{3}"
] | [
"Two points A and B are chosen on the graph $y=x^2$. If they form an equilateral triangle with the origin, what is the area of the triangle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1d1487f5c6c32b036f1b33a47748731ba76b7334ce9937a93fc587fe5d2c5206 | Two points A and B are randomly selected on Circle O. Point C is labeled as the midpoint of Chord AB. Then, a point D is randomly chosen on the border of Circle O. What is the probability that the circle centered at C passing through D has a greater area than Circle O? | [
"$\\dfrac{1}{3}+\\dfrac{\\sqrt{3}}{2\\pi}$"
] | [
"Two points A and B are randomly selected on Circle O. Point C is labeled as the midpoint of Chord AB. Then, a point D is randomly chosen on the border of Circle O. What is the probability that the circle centered at C passing through D has a greater area than Circle O?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cc4cb60b58cf781402d4585d692d846b9abe2d1e16d26cb65a2c086dc2dfcce2 | Two points P and Q lie in the interior of a regular tetrahedron ABCD. If the length of each edge of the tetrahedron is 1, find the maximum possible value of the cosine of angle PAQ. | [
"1"
] | [
"Two points P and Q lie in the interior of a regular tetrahedron ABCD. If the length of each edge of the tetrahedron is 1, find the maximum possible value of the cosine of angle PAQ."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f90e1d8cb8f86b6a8253527d9c9d7cdc54f1fa02f6c19dd0682513a8e92346f2 | Two points \( A = (x_1, y_1) \) and \( B = (x_2, y_2) \) lie on the graph \( y^2 = 6x \). The perpendicular bisector of \( AB \) meets the \( x \)-axis at point \( C \). Given that \( x_1 \neq x_2 \) and \( x_1 + x_2 = 4 \), what is the maximum area of \( \triangle ABC \)? | [
"\\(\\frac{14\\sqrt{7}}{3}\\)"
] | [
"Two points \\( A = (x_1, y_1) \\) and \\( B = (x_2, y_2) \\) lie on the graph \\( y^2 = 6x \\). The perpendicular bisector of \\( AB \\) meets the \\( x \\)-axis at point \\( C \\). Given that \\( x_1 \\neq x_2 \\) and \\( x_1 + x_2 = 4 \\), what is the maximum area of \\( \\triangle ABC \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 8,
"tir": 12
} | 49 | 0 | 0 |
omr-ws-ff70ad38978f0637034e672aeba583c966aba6fe0e85fb4d1595c15ae59e6b43 | Two points \( A \) and \( B \) are fixed, and \( M \) is a variable point on the circle \( C(O, R) \). Find the locus of the foot of the perpendicular from the midpoint of \( AM \) to the line \( BM \). | [
"\\text{A circle}"
] | [
"Two points \\( A \\) and \\( B \\) are fixed, and \\( M \\) is a variable point on the circle \\( C(O, R) \\). Find the locus of the foot of the perpendicular from the midpoint of \\( AM \\) to the line \\( BM \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-caa88d2d2b97a0829848a685ca24b99cfd1684afb455ef4b0664a883cf416338 | Two points \( A \) and \( B \) are given in the plane. A point \( X \) is called their [i]preposterous midpoint[/i] if there is a Cartesian coordinate system in the plane such that the coordinates of \( A \) and \( B \) in this system are non-negative, the abscissa of \( X \) is the geometric mean of the abscissae of \... | [
"Disk with diameter \\( AB \\), excluding the center."
] | [
"Two points \\( A \\) and \\( B \\) are given in the plane. A point \\( X \\) is called their [i]preposterous midpoint[/i] if there is a Cartesian coordinate system in the plane such that the coordinates of \\( A \\) and \\( B \\) in this system are non-negative, the abscissa of \\( X \\) is the geometric mean of t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-01190078dc490e70b849ea35009d8e4ec78c419f1191adef3f019608292f180a | Two points \( A \) and \( B \) are marked on circle \( O \). Points \( M \) and \( N \) are then marked on circle \( O \), on opposite sides of the line \( AB \). The bisectors of angles \( AMB \) and \( BNA \) are constructed, intersecting circle \( O \) at points \( X \) and \( Y \). If the radius of circle \( O \) i... | [
"\\sqrt{4 + 2\\sqrt{2}}"
] | [
"Two points \\( A \\) and \\( B \\) are marked on circle \\( O \\). Points \\( M \\) and \\( N \\) are then marked on circle \\( O \\), on opposite sides of the line \\( AB \\). The bisectors of angles \\( AMB \\) and \\( BNA \\) are constructed, intersecting circle \\( O \\) at points \\( X \\) and \\( Y \\). If t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-5d27ea402d93813890fb94c396e5a1a2cf9c787d4bba6ead4ca8d09114cc5c95 | Two points \( A \) and \( B \) lie on a sphere of radius 12. The length of the straight line segment joining \( A \) and \( B \) is \( 12 \sqrt{3} \). Find the shortest path from \( A \) to \( B \) if every point of the path is on the sphere of radius 12. | [
"8\\pi"
] | [
"Two points \\( A \\) and \\( B \\) lie on a sphere of radius 12. The length of the straight line segment joining \\( A \\) and \\( B \\) is \\( 12 \\sqrt{3} \\). Find the shortest path from \\( A \\) to \\( B \\) if every point of the path is on the sphere of radius 12."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f3e601eaa70804c94dff79b952044a1a17dd22eb20449dbfa7103b7e96b9ce32 | Two points \( A \) and \( B \) lie on the hyperbola given by the equation \( y = \frac{1}{x} \). Let \( A_x \) and \( A_y \) be the projections of \( A \) onto the coordinate axes, and similarly, let \( B_x \) and \( B_y \) be the projections of \( B \) onto the coordinate axes. Calculate the sum of the areas of triang... | [
"1"
] | [
"Two points \\( A \\) and \\( B \\) lie on the hyperbola given by the equation \\( y = \\frac{1}{x} \\). Let \\( A_x \\) and \\( A_y \\) be the projections of \\( A \\) onto the coordinate axes, and similarly, let \\( B_x \\) and \\( B_y \\) be the projections of \\( B \\) onto the coordinate axes. Calculate the su... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-8b7a6a3f0eae8a4581c63083ae8622a6718ddd334486e7af296401f619cae760 | Two points \( A \) and \( C \) are chosen on a circle \( \Gamma \). Let \( O \) be an arbitrary point on the perpendicular bisector of \( AC \). The perpendicular bisectors of \( OA \) and \( OC \) meet \( \Gamma \) at points \( B \) and \( D \) respectively such that \( A, B, C, D \) lie on \( \Gamma \) in this order.... | [
"5\\sqrt{3}"
] | [
"Two points \\( A \\) and \\( C \\) are chosen on a circle \\( \\Gamma \\). Let \\( O \\) be an arbitrary point on the perpendicular bisector of \\( AC \\). The perpendicular bisectors of \\( OA \\) and \\( OC \\) meet \\( \\Gamma \\) at points \\( B \\) and \\( D \\) respectively such that \\( A, B, C, D \\) lie o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-173135820cfed7fb239a9928a8df2919746eec5d03e98fc21ca9d26e8752dff0 | Two points \( A(x_A, y_B) \) and \( B(x_A + 5, y_B + 12) \) are on the parabola \( x^2 - px - 5y + q = 0 \) such that \( x_A + y_B = 5 \). How many possible positive integer pairs \((p, q)\) are there where \( q \leq 2017 \)? | [
"42"
] | [
"Two points \\( A(x_A, y_B) \\) and \\( B(x_A + 5, y_B + 12) \\) are on the parabola \\( x^2 - px - 5y + q = 0 \\) such that \\( x_A + y_B = 5 \\). How many possible positive integer pairs \\((p, q)\\) are there where \\( q \\leq 2017 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9a4fdaae99642ca51a0260c4b6f879e2ed10040889a2dff2ed2d1de21b2179e6 | Two points \( A(x_A, y_B) \) and \( B(x_A+5, y_B+12) \) are on the parabola \( 5x^2 - px - 5y + q = 0 \) such that \( x_A + y_B = 5 \). How many possible positive integer pairs \((p, q)\) are there where \( q \leq 2050 \)? | [
"41"
] | [
"Two points \\( A(x_A, y_B) \\) and \\( B(x_A+5, y_B+12) \\) are on the parabola \\( 5x^2 - px - 5y + q = 0 \\) such that \\( x_A + y_B = 5 \\). How many possible positive integer pairs \\((p, q)\\) are there where \\( q \\leq 2050 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3451985b016b4c15af7d3a47fbc51d604332d350537e5778a8c0f9f75de03662 | Two points \( A(x_a, y_b) \) and \( B(x_a + 5, y_b + 12) \) are on the parabola \( 5x^2 - px - 5y + q = 0 \) such that \( x_a + y_b = 5 \). How many possible positive integer pairs \((p, q)\) are there where \( q \leq 2050 \)? | [
"41"
] | [
"Two points \\( A(x_a, y_b) \\) and \\( B(x_a + 5, y_b + 12) \\) are on the parabola \\( 5x^2 - px - 5y + q = 0 \\) such that \\( x_a + y_b = 5 \\). How many possible positive integer pairs \\((p, q)\\) are there where \\( q \\leq 2050 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 41,
"genselect": 6,
"tir": 9
} | 56 | 0 | 0 |
omr-ws-9f0105fc2842876b2b08a050792bef08995caad489c0bb9351fe91271f9ca21a | Two points \( M \) and \( N \) are chosen inside a non-equilateral triangle \( ABC \) such that \( \angle BAM = \angle CAN \), \( \angle ABM = \angle CBN \), and \[ AM \cdot AN \cdot BC = BM \cdot BN \cdot CA = CM \cdot CN \cdot AB = k \] for some real \( k \). Given that \( AB = 13 \), \( BC = 14 \), and \( CA = 15 \)... | [
"910"
] | [
"Two points \\( M \\) and \\( N \\) are chosen inside a non-equilateral triangle \\( ABC \\) such that \\( \\angle BAM = \\angle CAN \\), \\( \\angle ABM = \\angle CBN \\), and \n\\[ AM \\cdot AN \\cdot BC = BM \\cdot BN \\cdot CA = CM \\cdot CN \\cdot AB = k \\]\nfor some real \\( k \\). Given that \\( AB = 13 \\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-b6e1e670cd36c8f3aac01a528fffc1831f732dc1d6d92ff879942061a6cd5aab | Two points \( P \) and \( Q \) lie in the interior of a regular tetrahedron \( ABCD \). Calculate the maximum possible value of \( \cos(\angle PAQ) \). | [
"1"
] | [
"Two points \\( P \\) and \\( Q \\) lie in the interior of a regular tetrahedron \\( ABCD \\). Calculate the maximum possible value of \\( \\cos(\\angle PAQ) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c6deeabbe68b2d51df5853dafa099d94c0fc04e418a2e81c53b76b0d56494db | Two points \( P \) and \( Q \) lie within, or on the boundary of, a square of side 1 cm, one corner of which is the point \( O \). Calculate the maximum possible value of the minimum length among the lines \( OP \), \( PQ \), and \( QO \). | [
"1"
] | [
"Two points \\( P \\) and \\( Q \\) lie within, or on the boundary of, a square of side 1 cm, one corner of which is the point \\( O \\). Calculate the maximum possible value of the minimum length among the lines \\( OP \\), \\( PQ \\), and \\( QO \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-87a548d1fa8d9e07e280a443d3339288740d272b1b7c94819c5b289ec48da2ce | Two points \( P \) and \( Q \) move on the curves \( y = x + 2 \) and \( y = \ln x \) respectively. Find the minimum value of the distance \( \overline{PQ} \). | [
"\\(\\frac{3\\sqrt{2}}{2}\\)"
] | [
"Two points \\( P \\) and \\( Q \\) move on the curves \\( y = x + 2 \\) and \\( y = \\ln x \\) respectively. Find the minimum value of the distance \\( \\overline{PQ} \\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-95d4f86b1ee35aac5395bf482b84e9ec2ff768f3ff2288ba4564a627fd62c3a2 | Two points \( P \) and \( Q \) move on the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (where \( a > b > 0 \)) such that \(\angle POQ = \frac{\pi}{2}\). Find the maximum and minimum values of \(\frac{1}{OP} + \frac{1}{OQ}\). | [
"\\(\\min = \\frac{a+b}{ab}\\) and \\(\\max = \\frac{\\sqrt{2(a^2+b^2)}}{ab}\\)"
] | [
"Two points \\( P \\) and \\( Q \\) move on the ellipse \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\) (where \\( a > b > 0 \\)) such that \\(\\angle POQ = \\frac{\\pi}{2}\\). Find the maximum and minimum values of \\(\\frac{1}{OP} + \\frac{1}{OQ}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-75a619a7a8a5a1bb721275a4c28db18db94a9036694fab2d912b7ff3580afc10 | Two points \(S\) and \(F\) are marked inside a round billiard table, and the straight line \(SF\) does not pass through the center of the circle; there is a ball at point \(S\). It can be hit so that it reflects once relative to the table boundary (according to the law “the angle of incidence is equal to the angle of r... | [
"2"
] | [
"Two points \\(S\\) and \\(F\\) are marked inside a round billiard table, and the straight line \\(SF\\) does not pass through the center of the circle; there is a ball at point \\(S\\). It can be hit so that it reflects once relative to the table boundary (according to the law “the angle of incidence is equal to t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-19b05dd72f3296d44e1c8ef4e51f719bf4576a655981914c6c4d7177c4c2fff3 | Two points \(a\) and \(b\) (\(a < b\)) are chosen at random in the interval \([0, 1]\). What is the probability that \(a\), \(b-a\), and \(1-b\) are the lengths of the sides of a nondegenerate triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"Two points \\(a\\) and \\(b\\) (\\(a < b\\)) are chosen at random in the interval \\([0, 1]\\). What is the probability that \\(a\\), \\(b-a\\), and \\(1-b\\) are the lengths of the sides of a nondegenerate triangle?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-02355fe91fb73148cf536ababf70d24769e7c6b960931561d0f69aa11c1f6eb3 | Two points are chosen at random on the perimeter of a circle with radius $1$. What is the expected distance between the two points? | [
"\\dfrac{4}{\\pi}"
] | [
"Two points are chosen at random on the perimeter of a circle with radius $1$. What is the expected distance between the two points?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c076973bd26785961d564142056f3cc3472a320500156d657c18621cd255cc53 | Two points are chosen randomly inside a circle of radius 1. What is the expected value of the distance between these two points? (Or show that there is no such value) | [
"\\(\\frac{128}{45\\pi}\\)"
] | [
"Two points are chosen randomly inside a circle of radius 1. What is the expected value of the distance between these two points? (Or show that there is no such value)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-ee6507f0b315c83f1746f022ec834c02252ce08395131072b181a254968b7791 | Two points are chosen randomly on the unit circle and a line segment is drawn between the two points to form a chord A. This process is repeated 3 more times to obtain chords B, C, and D. What is the probability that no two chords intersect each other? | [
"\\dfrac{2}{15}"
] | [
"Two points are chosen randomly on the unit circle and a line segment is drawn between the two points to form a chord A. This process is repeated 3 more times to obtain chords B, C, and D. What is the probability that no two chords intersect each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-16d8a2c0fb20cd7d88b6bd180fa5336128a478392b1c03830262ea42e79ffb61 | Two points are chosen uniformly at random inside an equilateral triangle of side length $1$. Find the expected distance between the two points. | [
"\\dfrac{4 + 3\\sqrt{3}}{45}"
] | [
"Two points are chosen uniformly at random inside an equilateral triangle of side length $1$. Find the expected distance between the two points."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-551ce72ef63d862ec005ac5912daaf29d654139d496f6b8b32886e00f2cc8449 | Two points are chosen uniformly at random on the sides of a square of length $1$. If $p$ is the probability that the distance between them is greater than $1$, what is $\lfloor 100p \rfloor$? | [
"35"
] | [
"Two points are chosen uniformly at random on the sides of a square of length $1$. If $p$ is the probability that the distance between them is greater than $1$, what is $\\lfloor 100p \\rfloor$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-242300516294f3ca047efcf5b74da24a2fb143664bcc11833aa84fe9a6543389 | Two points are fixed on a circle, and the third point moves along the circle. Find the locus of the intersection points of the medians of the triangle formed by these three points. | [
"A circle with radius \\( \\frac{R}{3} \\) and center on the line segment \\( OM \\) such that \\( MK:KO = 1:2 \\)."
] | [
"Two points are fixed on a circle, and the third point moves along the circle. Find the locus of the intersection points of the medians of the triangle formed by these three points."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 19
} | 31 | 0 | 0 |
omr-ws-947795b2581c3478444bfd4fffa28b33be39704a878a5f441b262933aa35f16f | Two points are given on one of the sides of the angle $\alpha $, the distances between the points and the other side being equal to $b$ and $c$. Find the radius of the circle which passes through those two points and touches the other side of the angle. | [
"$\\frac{b+c-2\\sqrt {bc}\\cos \\alpha }{2\\sin ^2 \\alpha }$"
] | [
"Two points are given on one of the sides of the angle $\\alpha $, the distances between the points and the other side being equal to $b$ and $c$. Find the radius of the circle which passes through those two points and touches the other side of the angle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e8227b4f2c53adf8036dcf4a07403ebd65534aecc2ab26cfef32950b1d3082c | Two points are located 10 units apart, and a circle with radius \( r \) is centered at one of the points. A tangent line to the circle is drawn from the other point. What value of \( r \) maximizes the area of the triangle formed by the two points and the point of tangency? | [
"\\( \\sqrt{50} \\)"
] | [
"Two points are located 10 units apart, and a circle with radius \\( r \\) is centered at one of the points. A tangent line to the circle is drawn from the other point. What value of \\( r \\) maximizes the area of the triangle formed by the two points and the point of tangency?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-34915c6a3b95e255bd887f1e353b6edd7a60648a49b01e13b20dddc2dad38e96 | Two points are picked in the unit circle. Find the probability that they are more than $1$ unit apart. | [
"\\dfrac{3\\sqrt{3}}{4\\pi}"
] | [
"Two points are picked in the unit circle. Find the probability that they are more than $1$ unit apart."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-d010d5e511ee54562a75075d067f50885debb13279546cc9b15c2458b829af1e | Two points are placed on two randomly selected faces of a unit cube such that one point lies on one face and the other lies on another face. Find the expected length of the segment connecting the two points. | [
"\\dfrac{4 + 3\\sqrt{2} + 5 \\ln(1 + \\sqrt{2})}{15}"
] | [
"Two points are placed on two randomly selected faces of a unit cube such that one point lies on one face and the other lies on another face. Find the expected length of the segment connecting the two points."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-a924d506cadc90057370f0de15f35c64d94eac457bd3abb1e88faa29cef1e99a | Two points are randomly and independently chosen on the interval $[0,1]$ according to a uniform distribution. These points divide the interval into three parts. What is the probability that these three parts can form a triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"Two points are randomly and independently chosen on the interval $[0,1]$ according to a uniform distribution. These points divide the interval into three parts. What is the probability that these three parts can form a triangle?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-5081828267cb269911a23b30bb209c498c8b7cf9afe47f03ff32f03c95ba1afb | Two points are randomly and simultaneously chosen from a 5 x 4 grid of 20 lattice points. The probability that the two points have a rational distance is \(\frac{a}{b}\) where \((a,b) = 1\). If \(\frac{a+b}{b-a} = \frac{p}{q}\) in lowest terms, find \( p + q \). | [
"190"
] | [
"Two points are randomly and simultaneously chosen from a 5 x 4 grid of 20 lattice points. The probability that the two points have a rational distance is \\(\\frac{a}{b}\\) where \\((a,b) = 1\\). If \\(\\frac{a+b}{b-a} = \\frac{p}{q}\\) in lowest terms, find \\( p + q \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-d66d6451706d06c3fe5ced98da58bad191aa2cb0261cf3487c05ade65c602903 | Two points are randomly chosen from the interior of a circle with radius 1. Compute the probability that the sum of their distances from the center of the circle is less than or equal to 1. | [
"\\(\\frac{1}{6}\\)"
] | [
"Two points are randomly chosen from the interior of a circle with radius 1. Compute the probability that the sum of their distances from the center of the circle is less than or equal to 1."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-da1a690f5c9b7878eba63508b60f0d18abcce812c117c8e18080de299b7d1606 | Two points are randomly chosen in the interval $(0,2)$. Find the probability that the distance between them is less than a fixed number $c$, where $0 < c < 1$. | [
"\\(\\frac{4-(2-c)^2}{4}\\)"
] | [
"Two points are randomly chosen in the interval $(0,2)$. Find the probability that the distance between them is less than a fixed number $c$, where $0 < c < 1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-f993ac5196773ea662563d807cc85bcd6cd4228a0f49d179c4f27ebc02c50121 | Two points are randomly chosen on the circumference of a circle of radius $r$. Find the probability that the distance between them is at least $r$. | [
"\\(\\frac{2}{3}\\)"
] | [
"Two points are randomly chosen on the circumference of a circle of radius $r$. Find the probability that the distance between them is at least $r$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 30
} | 78 | 0 | 0 |
omr-ws-96978b3e4430ac67432c0b114d7398c2fd5891adfb5266be75e7f4149319d3a7 | Two points are randomly selected from inside the unit circle. Let \( X \) be a random variable representing the distance between them. What is the expected value of \( X \)? | [
"\\(\\frac{128}{45\\pi}\\)",
"\\dfrac{128}{45\\pi}"
] | [
"Two points are randomly selected from inside the unit circle. Let \\( X \\) be a random variable representing the distance between them. What is the expected value of \\( X \\)?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 28,
"tir": 29
} | 57 | 0 | 0 |
omr-ws-b628e2bf2c926864db34d60188b969e28147878eb3b899dd86dd91aaf6f1d1c8 | Two points are randomly selected from inside the unit circle. Let \( X \) be a random variable representing the distance between them. What is the information entropy of \( X \) in nats? | [
"2 - \\ln 2"
] | [
"Two points are randomly selected from inside the unit circle. Let \\( X \\) be a random variable representing the distance between them. What is the information entropy of \\( X \\) in nats?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-8a44fe562c40be3bc47725bb98bc649033e40be4e7f6e3c4167bc5d89b4b8583 | Two points are randomly selected on a line segment of length one, and the line segment is split at these two points into three shorter line segments. What is the expected value of the length of the longest of the three resultant line segments? | [
"\\dfrac{11}{18}"
] | [
"Two points are randomly selected on a line segment of length one, and the line segment is split at these two points into three shorter line segments. What is the expected value of the length of the longest of the three resultant line segments?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-4073a8aeeab764cb16045b4482910a6f52ee9e07b061b29714fe8c056e1b6e78 | Two points are randomly selected on the circumference of a unit circle. What is the expected straight-line distance between them? | [
"\\(\\frac{4}{\\pi}\\)"
] | [
"Two points are randomly selected on the circumference of a unit circle. What is the expected straight-line distance between them?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-db28ea7f3916d08f9fccca2da899d49ee20251dbd65f5ad9ef7d3a4be130016e | Two points are selected independently and at random from a segment of length \( \beta \). What is the probability that they are at least a distance \( \alpha \) (\( \alpha < \beta \)) apart? | [
"\\(\\frac{(\\beta - \\alpha)^2}{\\beta^2}\\)"
] | [
"Two points are selected independently and at random from a segment of length \\( \\beta \\). What is the probability that they are at least a distance \\( \\alpha \\) (\\( \\alpha < \\beta \\)) apart?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-dea2df11610a4530409229c5e9fd619f358a4c399ab14278d85d0224017504da | Two points are selected with equal probability on a line segment of length 1. What is the probability that the three resulting line segments can form a triangle? | [
"$\\frac{1}{4}$"
] | [
"Two points are selected with equal probability on a line segment of length 1. What is the probability that the three resulting line segments can form a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-084476d05ec7c94dd366f37c568a82f1ab97fa32860536638173e93f27e7750e | Two points are taken at random on the circumference of a circle of radius \( r \). What is the probability that the distance between them will not exceed \( \frac{\pi r}{4} \)? | [
"\\(\\frac{1}{4}\\)"
] | [
"Two points are taken at random on the circumference of a circle of radius \\( r \\). What is the probability that the distance between them will not exceed \\( \\frac{\\pi r}{4} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 1,
"tir": 7
} | 18 | 0 | 0 |
omr-ws-e868099401f8428663f4d96c0d3592afe529d384c1573878d6fb3a503b74ea88 | Two points are uniformly and independently marked inside a square. What is the probability that the center of the square is inside the circle formed by the two points as the diameter? | [
"\\dfrac{1}{2}"
] | [
"Two points are uniformly and independently marked inside a square. What is the probability that the center of the square is inside the circle formed by the two points as the diameter?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-32e41a798f987ff9c0ee955a79a37f8fc69995aa4d895ef56f05dfe85eaf5259 | Two points are uniformly moving in the same direction along the circumference of a circle of length $c$ and come together every $t$ seconds. Find the speed of each point, knowing that one of them makes the whole circle $n$ seconds faster than the other. | [
"The higher speed is numerically equal to $\\frac{c\\left(\\sqrt {n^2+4nt}+n\\right)}{2nt}$, the lower, $\\frac{c\\left(\\sqrt {n^2+4nt}-n\\right)}{2nt}$"
] | [
"Two points are uniformly moving in the same direction along the circumference of a circle of length $c$ and come together every $t$ seconds. Find the speed of each point, knowing that one of them makes the whole circle $n$ seconds faster than the other."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d0811838c036b25816351a375366e9bbade8ad4607ae9ed580f1f383e7cdc064 | Two points in the Cartesian plane, \((x_1, y_1)\) and \((x_2, y_2)\), are a distance 2019 apart. Let \(|x_1 - x_2| = 673\). What is the slope of the line passing through these two points? | [
"2.83"
] | [
"Two points in the Cartesian plane, \\((x_1, y_1)\\) and \\((x_2, y_2)\\), are a distance 2019 apart. Let \\(|x_1 - x_2| = 673\\). What is the slope of the line passing through these two points?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-345e4beffca6c81a2f77a1d19e0edb9878f9a8ff073e3b7323c69334750c0fc2 | Two points lie on the curve \( f(x) = x^2 - 2x - 3 \). The midpoint of these two points is \( (0, 0) \). Find the coordinates of the two points. | [
"\\((\\sqrt{3}, -2\\sqrt{3})\\) and \\((- \\sqrt{3}, 2\\sqrt{3})\\)"
] | [
"Two points lie on the curve \\( f(x) = x^2 - 2x - 3 \\). The midpoint of these two points is \\( (0, 0) \\). Find the coordinates of the two points."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 28
} | 30 | 0 | 0 |
omr-ws-f59caf8fd357ed8a60fd9951e4653b23d71e715a46ddd83b55261626baf2d012 | Two points move along a circle of radius $R$ uniformly in the same direction. One of them makes a full rotation $t$ s quicker than the other, and the periods between successive meetings are equal to $T$. Find the speeds of the points. | [
"$\\frac{\\pi R}{T}\\left(\\sqrt {1+\\frac{4T}{t}}\\pm 1\\right)$"
] | [
"Two points move along a circle of radius $R$ uniformly in the same direction. One of them makes a full rotation $t$ s quicker than the other, and the periods between successive meetings are equal to $T$. Find the speeds of the points."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75dbd27ced1a3fa13ea5dd80500a4042f166ad3118b3fbcb0e7d849da07e12f1 | Two points move in a circle whose circumference is $l$ meters at the speeds $v$ and $w< v$. At what moments of time reckoned from the start of the first point will successive meetings of the points occur if they move in the same direction, and the first point starts $t$ seconds before the second and is $a$ meters behin... | [
"t_n=\\frac{a-wt+l(n-1)}{v-w}"
] | [
"Two points move in a circle whose circumference is $l$ meters at the speeds $v$ and $w< v$. At what moments of time reckoned from the start of the first point will successive meetings of the points occur if they move in the same direction, and the first point starts $t$ seconds before the second and is $a$ meters ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-acb6880f273ae9987b2acf4e052f2cf97070e133f9d20201708f6c16d5b6be05 | Two points move in a circle whose circumference is equal to 360 m, the first point completing the circle 1 second earlier than the second point. Find the velocity of either point if it is known that during one second the first point covers 4 m more than the second point. | [
"40 m/s and 36 m/s"
] | [
"Two points move in a circle whose circumference is equal to 360 m, the first point completing the circle 1 second earlier than the second point. Find the velocity of either point if it is known that during one second the first point covers 4 m more than the second point."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d86e8e04a0bb6c943438d7fff4391f41114b623a4dd2b40b91e5bfad5637fdc | Two points on the circumference of a circle of radius \( r \) are selected independently and at random. From each point, a chord of length \( r \) is drawn in a clockwise direction. What is the probability that the two chords intersect? | [
"\\(\\frac{1}{3}\\)",
"\\dfrac{1}{3}"
] | [
"Two points on the circumference of a circle of radius \\( r \\) are selected independently and at random. From each point, a chord of length \\( r \\) is drawn in a clockwise direction. What is the probability that the two chords intersect?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 126,
"genselect": 22,
"tir": 216
} | 364 | 0 | 0 |
omr-ws-adc09cf480e655db02230844475ea5588afb4ea1adaabb92eee61a49a7b12626 | Two points rotate uniformly along two concentric circles. One of them makes a full circle 5 s quicker than the other and, therefore, makes two rotations more per minute. How many rotations a minute does each point make? | [
"4 and 6"
] | [
"Two points rotate uniformly along two concentric circles. One of them makes a full circle 5 s quicker than the other and, therefore, makes two rotations more per minute. How many rotations a minute does each point make?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3fe44c31bb39ae733dfd06c711e5a359ba2ded7635c85a9256db9ac73967c06b | Two points were chosen on each side of a square. Each of these points is connected to both ends of the opposite side of the square. If any three (or four) of these segments intersect at one point, then such an intersection point is taken into account only once. Find the smallest number of intersection points inside the... | [
"7"
] | [
"Two points were chosen on each side of a square. Each of these points is connected to both ends of the opposite side of the square. If any three (or four) of these segments intersect at one point, then such an intersection point is taken into account only once. Find the smallest number of intersection points insid... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-17f3d95c60bbc6fba54c2edb4f857518e2f75eb586538febd4a62caa3d592bdd | Two points, $B$ and $C$, are thrown at random onto a unit segment of the abscissa. Find the probability that the length of the segment $BC$ will be smaller than the distance between the origin and the nearest point. | [
"$\\frac{1}{2}$"
] | [
"Two points, $B$ and $C$, are thrown at random onto a unit segment of the abscissa. Find the probability that the length of the segment $BC$ will be smaller than the distance between the origin and the nearest point."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1c724c0153b4f8a5dbd065e9e02395b19433966337726d5a4ab12012679f478 | Two points, P and Q, are on a circle and start from the same point. They move at constant speeds in the same direction around the circle, with P moving twice as fast as Q. What path is followed by the midpoint of the chord PQ? | [
"\\( 4(x^2 + y^2)^2 - 3r^2(x^2 + y^2) - r^3x = 0 \\)"
] | [
"Two points, P and Q, are on a circle and start from the same point. They move at constant speeds in the same direction around the circle, with P moving twice as fast as Q. What path is followed by the midpoint of the chord PQ?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-8b973cffb4592be9d9ece096975c967a18e52ddd4198c12acba43c9794708cc7 | Two points, \( P \) and \( Q \), lie in the interior of a regular tetrahedron \( ABCD \). Let \( \theta \) be the measure of the angle \( \angle PAQ \). Find the maximum possible value of \( \cos(\theta) \). | [
"1"
] | [
"Two points, \\( P \\) and \\( Q \\), lie in the interior of a regular tetrahedron \\( ABCD \\). Let \\( \\theta \\) be the measure of the angle \\( \\angle PAQ \\). Find the maximum possible value of \\( \\cos(\\theta) \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-5d664d292b3efa56a945ea54ebdefbcca459f6ed656d14949f2b60d47c6c04af | Two poles XY and ZW of heights \(a\) and \(b\) respectively above the ground are vertical. A wire joins \(X\), the base of XY, to \(W\), the top of ZW, and another wire joins \(Z\), the base of ZW to \(Y\), the top of XY. The wires cross at \(P\). How high is \(P\) above the ground? | [
"\\(\\frac{ab}{a + b}\\)"
] | [
"Two poles XY and ZW of heights \\(a\\) and \\(b\\) respectively above the ground are vertical. A wire joins \\(X\\), the base of XY, to \\(W\\), the top of ZW, and another wire joins \\(Z\\), the base of ZW to \\(Y\\), the top of XY. The wires cross at \\(P\\). How high is \\(P\\) above the ground?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-e2a00e73dbf64cd8a4d8149f8167dbc6b294e04aee9d9f3ec191c569b1a1ee84 | Two poles are 50 feet apart on level ground. One pole is 10 feet taller than the other. A cable hangs from the top of the poles in the shape of a parabola. Its lowest point is 10 feet above a point on the ground 20 feet from the shorter pole. Find the height of the taller pole. | [
"28"
] | [
"Two poles are 50 feet apart on level ground. One pole is 10 feet taller than the other. A cable hangs from the top of the poles in the shape of a parabola. Its lowest point is 10 feet above a point on the ground 20 feet from the shorter pole. Find the height of the taller pole."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8ed9d5bf691e9dccd0ac7ef8655420f82409050eb3c201f3cf6ebf8aa239833e | Two poles are 50 feet apart on level ground. One pole is 10 feet taller than the other. A cable hangs from the top of the poles in the shape of a parabola. The lowest point of the cable is 10 feet above a point on the ground 20 feet from the shorter pole. Find the height of the taller pole. | [
"28"
] | [
"Two poles are 50 feet apart on level ground. One pole is 10 feet taller than the other. A cable hangs from the top of the poles in the shape of a parabola. The lowest point of the cable is 10 feet above a point on the ground 20 feet from the shorter pole. Find the height of the taller pole."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-d4b988f568cd403a225f532c4f67c21c7e71c6cf5d58292e8062cc6de5592956 | Two poles of heights 11 feet and 33 feet are placed 240 feet apart. A wire needs to be connected between the tops of the two poles and also touch the ground at a point between them. What is the length of the shortest wire that can be used, and how far away from the pole of 11 feet will the wire touch the ground? | [
"244 feet, 60 feet"
] | [
"Two poles of heights 11 feet and 33 feet are placed 240 feet apart. A wire needs to be connected between the tops of the two poles and also touch the ground at a point between them. What is the length of the shortest wire that can be used, and how far away from the pole of 11 feet will the wire touch the ground?"
... | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-5b15840a02bfa3c3fa8b83be5d7814016a7ccfbf4ee7e01c97f9d1328f6b7906 | Two poles with diameters 10 and 30 are bound together by a metal band. How could I find the length of the metal band? | [
"\\( 20\\sqrt{3} + \\frac{125\\pi}{3} \\)"
] | [
"Two poles with diameters 10 and 30 are bound together by a metal band. How could I find the length of the metal band?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b8c19ed3719beccd6a18c598fcf9f84d2bb5ef107741b8ee6e6fecd4d5d9b81a | Two polygons together have 89 diagonals. How many sides do each of the polygons have? | [
"22"
] | [
"Two polygons together have 89 diagonals. How many sides do each of the polygons have?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88989e30c9b897bac8a4c352c18cf9b72d10d56dcddba063c5737abfccb7ee8e | Two polynomials of the same degree \(A(x) = a_nx^n + \cdots + a_1x + a_0\) and \(B(x) = b_nx^n + \cdots + b_1x + b_0\) are called friends if the coefficients \(b_0, b_1, \ldots, b_n\) are a permutation of the coefficients \(a_0, a_1, \ldots, a_n\). Given that \(P(x)\) and \(Q(x)\) are two friendly polynomials with inte... | [
"1"
] | [
"Two polynomials of the same degree \\(A(x) = a_nx^n + \\cdots + a_1x + a_0\\) and \\(B(x) = b_nx^n + \\cdots + b_1x + b_0\\) are called friends if the coefficients \\(b_0, b_1, \\ldots, b_n\\) are a permutation of the coefficients \\(a_0, a_1, \\ldots, a_n\\). Given that \\(P(x)\\) and \\(Q(x)\\) are two friendly ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-4081c8f5d5bd8a45dab9130ef2bdbb34ea0536430ebad15d854ad81f1cf2d627 | Two polynomials with real coefficients have leading coefficients equal to 1 and each has an odd degree that is equal to the number of its distinct real roots. The product of the values of the first polynomial at the roots of the second polynomial is equal to 2024. Find the product of the values of the second polynomial... | [
"-2024"
] | [
"Two polynomials with real coefficients have leading coefficients equal to 1 and each has an odd degree that is equal to the number of its distinct real roots. The product of the values of the first polynomial at the roots of the second polynomial is equal to 2024. Find the product of the values of the second polyn... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 47
} | 47 | 0 | 0 |
omr-ws-2935469fc50c357d3e988037d8a46c9736e7205bed7abc42a5645e0477409e93 | Two positive fractions in their lowest form are written on a board. Every minute, one of the fractions is replaced with a new fraction formed by the sum of the numerators divided by the sum of the denominators. After 10 minutes, let $p/q$ be the fraction that was last replaced, where $p$ and $q$ are coprime. Find the s... | [
"377"
] | [
"Two positive fractions in their lowest form are written on a board. Every minute, one of the fractions is replaced with a new fraction formed by the sum of the numerators divided by the sum of the denominators. After 10 minutes, let $p/q$ be the fraction that was last replaced, where $p$ and $q$ are coprime. Find ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-180964ec04f5b4148e55dc8c834f5e3ed6ee0a8095b85ff626bd118d4b5e4bd9 | Two positive integers $m$ and $n$ are both less than $500$ and $\text{lcm}(m,n) = (m-n)^2$. What is the maximum possible value of $m+n$? | [
"840"
] | [
"Two positive integers $m$ and $n$ are both less than $500$ and $\\text{lcm}(m,n) = (m-n)^2$. What is the maximum possible value of $m+n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-5e7519d34776cfcdbb25a9904ded8a1c1cb60b84552881f9d666ebeb9262379f | Two positive integers $m$ and $n$ are chosen such that $m$ is the smallest positive integer with only two positive divisors and $n$ is the largest integer less than $100$ with exactly three positive divisors. What is $m+n$? | [
"51"
] | [
"Two positive integers $m$ and $n$ are chosen such that $m$ is the smallest positive integer with only two positive divisors and $n$ is the largest integer less than $100$ with exactly three positive divisors. What is $m+n$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-72b7cc3f05502e2f6924f1497d000685296d1ce9c4e2acab0e0c5c8e4ba42c90 | Two positive integers \( m \) and \( n \) (where \( m, n \geq 2 \)) are called allied if the sum of the prime factors of \( m \) (including multiplicities) is equal to the sum of the prime factors of \( n \) (including multiplicities). Find the largest number that is allied to 2020. | [
"\\( 2^{55} \\)"
] | [
"Two positive integers \\( m \\) and \\( n \\) (where \\( m, n \\geq 2 \\)) are called allied if the sum of the prime factors of \\( m \\) (including multiplicities) is equal to the sum of the prime factors of \\( n \\) (including multiplicities). Find the largest number that is allied to 2020."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8fcce0e624523795da7e7581ca21015381403a85f6a5b38a5744a693ef7efd1f | Two positive integers \( m \) and \( n \) satisfy the equations \[ \max(m, n) = (m - n)^2 \] \[ \gcd(m, n) = \frac{\min(m, n)}{6} \] Find \( \text{lcm}(m, n) \). | [
"294"
] | [
"Two positive integers \\( m \\) and \\( n \\) satisfy the equations\n\\[ \\max(m, n) = (m - n)^2 \\]\n\\[ \\gcd(m, n) = \\frac{\\min(m, n)}{6} \\]\nFind \\( \\text{lcm}(m, n) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-0fd9016dd349b1664fb826139f96fe0ae9d5abb8d6d305354d32dc199ddc1f35 | Two positive integers \( m \) and \( n \) satisfy the equations: \[ \max(m, n) = (m - n)^2 \] \[ \gcd(m, n) = \frac{\min(m, n)}{6} \] Find \( \text{lcm}(m, n) \). | [
"294"
] | [
"Two positive integers \\( m \\) and \\( n \\) satisfy the equations:\n\\[ \\max(m, n) = (m - n)^2 \\]\n\\[ \\gcd(m, n) = \\frac{\\min(m, n)}{6} \\]\nFind \\( \\text{lcm}(m, n) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 30,
"tir": 4
} | 34 | 0 | 0 |
omr-ws-5a0e8903d43c24e68fe4294f3d8260947941c9f2d105838562877afb97debe64 | Two positive integers \( m \) and \( n \) where \( m > n \) can be expressed as \( a_{0} + 7a_{1} + 7^{2}a_{2} + 7^{3}a_{3} \) where \( a_{i} \in \{0, 1, 2, 3, 4, 5, 6\} \) for \( i = 0, 1, 2, 3 \) and \( a_{3} \neq 0 \). Given that \( m + n = 2010 \), how many different values of \( m \) satisfying the condition are t... | [
"662"
] | [
"Two positive integers \\( m \\) and \\( n \\) where \\( m > n \\) can be expressed as \\( a_{0} + 7a_{1} + 7^{2}a_{2} + 7^{3}a_{3} \\) where \\( a_{i} \\in \\{0, 1, 2, 3, 4, 5, 6\\} \\) for \\( i = 0, 1, 2, 3 \\) and \\( a_{3} \\neq 0 \\). Given that \\( m + n = 2010 \\), how many different values of \\( m \\) sat... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 7
} | 29 | 0 | 0 |
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