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-f9931fb539f91959d118b722f4263459cb4034662261dce8f48b35a403d63667 | A tramp has a coat of area $1$ with five patches, each having an area of at least $1/2$. Let $A_i$ denote the area of the $i$-th patch for $i = 1, 2, 3, 4, 5$. Define $S$ as the sum of the areas of the intersections between all pairs of patches. Calculate the minimum possible value of $S$. | [
"2"
] | [
"A tramp has a coat of area $1$ with five patches, each having an area of at least $1/2$. Let $A_i$ denote the area of the $i$-th patch for $i = 1, 2, 3, 4, 5$. Define $S$ as the sum of the areas of the intersections between all pairs of patches. Calculate the minimum possible value of $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-581d7b22cf0c20d20b9b21c0390ee354429dc17753e92201ee3b30911d68214b | A tramp has a coat of area 1 covered by five patches, each having an area of at least $\frac{1}{2}$. Let $A_i$ be the area of the $i$-th patch for $i = 1, 2, 3, 4, 5$. Calculate the minimum possible value of the sum of the areas of the intersections between all pairs of patches, $\sum_{1 \leq i < j \leq 5} (A_i \cap A_... | [
"2"
] | [
"A tramp has a coat of area 1 covered by five patches, each having an area of at least $\\frac{1}{2}$. Let $A_i$ be the area of the $i$-th patch for $i = 1, 2, 3, 4, 5$. Calculate the minimum possible value of the sum of the areas of the intersections between all pairs of patches, $\\sum_{1 \\leq i < j \\leq 5} (A_... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-6e490731a9e562153d9a1971de1fff5cec26e284883be8d1928877e7cc533a43 | A tramp has a coat of area 1 with five patches, each having an area of at least 1/2. Let \( A \) be the total area of the intersections between all pairs of patches. Find the minimum possible value of \( A \). | [
"2"
] | [
"A tramp has a coat of area 1 with five patches, each having an area of at least 1/2. Let \\( A \\) be the total area of the intersections between all pairs of patches. Find the minimum possible value of \\( A \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-df3824e41f125e8b9a2f53952f22231f7db7f196599f62e7b229056163fa5395 | A tramp has a coat with five patches, each having an area of at least $\frac{1}{2}$. Let $A$ be the total area of the coat. If the maximum area of overlap between any two patches is denoted by $M$, find the minimum possible value of $M$ given that $A = 3$. | [
"0"
] | [
"A tramp has a coat with five patches, each having an area of at least $\\frac{1}{2}$. Let $A$ be the total area of the coat. If the maximum area of overlap between any two patches is denoted by $M$, find the minimum possible value of $M$ given that $A = 3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-e8bc71417df6cc20d9750b6d9e9117f11d4b65124d9c2de2adb1b2920e5759bc | A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\sqrt{x},\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find the grea... | [
"314"
] | [
"A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\\sqrt{x},\\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 8,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-c8ec53ae12da75047776bb12e2534e3b8c795b20814356237213787544d55f18 | A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\sqrt{x},\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k_{}$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find the g... | [
"314"
] | [
"A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\\sqrt{x},\\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k_{}$ be the area of the region enclosed by the image of quadrilateral $ABCD.... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 7
} | 20 | 0 | 0 |
omr-ws-ab33c747f9cb0e58efcf150774932d9785de6b60a02c775eb02e58d52bc5d2d6 | A transformation of the first quadrant of the coordinate plane maps each point \((x, y)\) to the point \((\sqrt{x}, \sqrt{y})\). The vertices of quadrilateral \(ABCD\) are \(A = (900, 300)\), \(B = (1800, 600)\), \(C = (600, 1800)\), and \(D = (300, 900)\). Let \(k\) be the area of the region enclosed by the image of q... | [
"300",
"314"
] | [
"A transformation of the first quadrant of the coordinate plane maps each point \\((x, y)\\) to the point \\((\\sqrt{x}, \\sqrt{y})\\). The vertices of quadrilateral \\(ABCD\\) are \\(A = (900, 300)\\), \\(B = (1800, 600)\\), \\(C = (600, 1800)\\), and \\(D = (300, 900)\\). Let \\(k\\) be the area of the region enc... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 17,
"genselect": 7,
"tir": 3
} | 27 | 0 | 0 |
omr-ws-de5952b0621772acf34d429b533b7459ee4f10f64948c9f877b3d314d6560a47 | A translation in the xy-coordinate plane involves a positive horizontal move of 4 units and a positive vertical move of 2 units. Starting at the point (5,7), the translation ends at the point (x,31). What is the value of x? | [
"53"
] | [
"A translation in the xy-coordinate plane involves a positive horizontal move of 4 units and a positive vertical move of 2 units. Starting at the point (5,7), the translation ends at the point (x,31). What is the value of x?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-5e9d9ae9aad20d09c4cd71df48bda6fb1783c546c3ab8052297b514ff0472731 | A translation maps (1,4) onto (7,-3). Write the image of (5,10) under the same translation. | [
"(11, 3)"
] | [
"A translation maps (1,4) onto (7,-3). Write the image of (5,10) under the same translation."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff2217d22aacf0cf1d9c34cac876381d30d62a1671d3982cd711918ade1c7c87 | A translation of the plane takes $-3 + 2i$ to $-7 - i.$ Find the complex number that the translation takes $-4 + 5i$ to. | [
"-8 + 2i"
] | [
"A translation of the plane takes $-3 + 2i$ to $-7 - i.$ Find the complex number that the translation takes $-4 + 5i$ to."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c6f09907238275c9280094bda8f3b7ce9288ca4e9d5df2e5dc805edd397a8893 | A transmitter is sending a message consisting of $4$ binary digits (either ones or zeros) to a receiver. The probability that the transmitter sends the correct digit to the receiver is $80\%$. The receiver only accepts a message if the sum of the first three digits equals the last digit modulo $2$. If the receiver acce... | [
"\\dfrac{256}{353}"
] | [
"A transmitter is sending a message consisting of $4$ binary digits (either ones or zeros) to a receiver. The probability that the transmitter sends the correct digit to the receiver is $80\\%$. The receiver only accepts a message if the sum of the first three digits equals the last digit modulo $2$. If the receive... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-8cab5a7c28847cf7a93cd9e5a9c96c9cbe2b5da93686b731e7fa28030f44a29a | A transversal cuts the sides \(OL\), \(OM\), and the diagonal \(ON\) of a parallelogram \(OLNM\) at points \(A\), \(B\), and \(C\) respectively. If \(OL = 12\), \(OM = 16\), and \(ON = 20\), calculate the value of \(\frac{OA}{OL} + \frac{OB}{OM} + \frac{OC}{ON}\). | [
"1"
] | [
"A transversal cuts the sides \\(OL\\), \\(OM\\), and the diagonal \\(ON\\) of a parallelogram \\(OLNM\\) at points \\(A\\), \\(B\\), and \\(C\\) respectively. If \\(OL = 12\\), \\(OM = 16\\), and \\(ON = 20\\), calculate the value of \\(\\frac{OA}{OL} + \\frac{OB}{OM} + \\frac{OC}{ON}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-07b044f74b4e4c89ae75b25d7ac241eb0141eae413e04fff1fbfcb62bb98b323 | A trapezium \( ABCD \) with \( AB \parallel CD \) is inscribed in a circle with center \( O \). Suppose the diagonals \( AC \) and \( BD \) of the trapezium intersect at \( M \), and \( OM = 2 \). (a) If \( \angle AMB = 60^\circ \), find the difference between the lengths of the parallel sides. (b) If \( \angle AMD = 6... | [
"2\\sqrt{3}"
] | [
"A trapezium \\( ABCD \\) with \\( AB \\parallel CD \\) is inscribed in a circle with center \\( O \\). Suppose the diagonals \\( AC \\) and \\( BD \\) of the trapezium intersect at \\( M \\), and \\( OM = 2 \\).\n(a) If \\( \\angle AMB = 60^\\circ \\), find the difference between the lengths of the parallel sides.... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-c6bc1d6deb1eafb9d40e06f74c52aa174830d6c0af59b443d9ea1514469b0d30 | A trapezium has height \( h \), parallel sides \( a \) and \( b \), and angles at the base of \( 90^\circ \) and \( 30^\circ \). The area of the trapezium is a multiple of its perimeter. Find all integer values of \( (a, b, h) \). | [
"\\text{No solution}"
] | [
"A trapezium has height \\( h \\), parallel sides \\( a \\) and \\( b \\), and angles at the base of \\( 90^\\circ \\) and \\( 30^\\circ \\). The area of the trapezium is a multiple of its perimeter. Find all integer values of \\( (a, b, h) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-f8896d8df5048fec15af86a7613cb54d0cf888be4e130b57a1e18b90d5086e85 | A trapezium has parallel bases of lengths $1$ and $4$. It is split into two trapeziums by a cut parallel to the bases of length $3$. The two new trapeziums are to be divided into $m$ and $n$ trapeziums, respectively, such that all $m + n$ trapezoids have the same area. Determine the minimum possible value for $m + n$ a... | [
"15"
] | [
"A trapezium has parallel bases of lengths $1$ and $4$. It is split into two trapeziums by a cut parallel to the bases of length $3$. The two new trapeziums are to be divided into $m$ and $n$ trapeziums, respectively, such that all $m + n$ trapezoids have the same area. Determine the minimum possible value for $m +... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 5
} | 18 | 0 | 0 |
omr-ws-ba9802ed4afc42d7454bee223f01aa271f3ca48c194090dbeb77a840e091bcb9 | A trapezium has parallel sides of length $a$ and $b$ ($a < b$), and the distance between the parallel sides is the altitude $h$. The extensions of the non-parallel sides intersect at a point, forming two triangles with the parallel sides of the trapezium. Express the areas of these triangles as functions of $a$, $b$, a... | [
"$\\text{Smaller triangle: } \\frac{a^2h}{2(b-a)}, \\text{ Larger triangle: } \\frac{hb^2}{2(b-a)}$"
] | [
"A trapezium has parallel sides of length $a$ and $b$ ($a < b$), and the distance between the parallel sides is the altitude $h$. The extensions of the non-parallel sides intersect at a point, forming two triangles with the parallel sides of the trapezium. Express the areas of these triangles as functions of $a$, $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 17
} | 18 | 0 | 0 |
omr-ws-8f29dab202826a8d08bc50f4149898549a3a92a61392ccd7aeac07c482ccebe5 | A trapezium has perpendicular diagonals and an altitude of 4. If one of the diagonals has a length of 5, find the area of the trapezium. | [
"\\(\\frac{50}{3}\\)"
] | [
"A trapezium has perpendicular diagonals and an altitude of 4. If one of the diagonals has a length of 5, find the area of the trapezium."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-4d5f2b3f2ffd25596a7c5fb9f852b3c6c421258faca3f3d285cec21e8e17fd1e | A trapezium is divided into four triangles by its two diagonals. If X and Y are the areas of the triangles adjacent to the parallel sides, find the area of the trapezium in terms of X and Y. | [
"\\( X + Y + 2\\sqrt{XY} \\)",
"\\((\\sqrt{X} + \\sqrt{Y})^2\\)"
] | [
"A trapezium is divided into four triangles by its two diagonals. If X and Y are the areas of the triangles adjacent to the parallel sides, find the area of the trapezium in terms of X and Y."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 18
} | 50 | 0 | 0 |
omr-ws-07dc3bad72c6fde5c011863630d1b8c1e46827f187322f679d1a1b3977b9ed6c | A trapezium is formed from three equilateral triangles, two adjacent and pointing upwards, and one between them pointing downwards. Three copies of this trapezium are placed together without gaps or overlaps, and only complete edges coincide, to form a polygon with $N$ sides. How many different values of $N$ are possib... | [
"3"
] | [
"A trapezium is formed from three equilateral triangles, two adjacent and pointing upwards, and one between them pointing downwards. Three copies of this trapezium are placed together without gaps or overlaps, and only complete edges coincide, to form a polygon with $N$ sides. How many different values of $N$ are p... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-280b592e2b01e12b20444684dbb259ae64bf60b0b93c8167e4781bd467668990 | A trapezoid $ABCD$ (with $AD$ and $BC$ as bases) is inscribed in a circle. Its diagonals intersect at point $M$. A line perpendicular to the bases of $ABCD$ intersects $BC$ at $K$ and the circle at $I$ (where $M$ lies on line segment $KL$). Given $MK = a$ and $LM = b$, express the radius of the circle tangent to segmen... | [
"\\(\\frac{a + b}{2}\\)"
] | [
"A trapezoid $ABCD$ (with $AD$ and $BC$ as bases) is inscribed in a circle. Its diagonals intersect at point $M$. A line perpendicular to the bases of $ABCD$ intersects $BC$ at $K$ and the circle at $I$ (where $M$ lies on line segment $KL$). Given $MK = a$ and $LM = b$, express the radius of the circle tangent to s... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1eb7c3145f2c233127446aeec7686384b55513b78ca1357aa0dfb12393f4c23 | A trapezoid $ABCD$ has $AB \parallel DC$ and $AD = BC = 10$. Given that the bisector of angle $D$ intersects $B$ and that $DB = 10\sqrt{3}$, find the area of trapezoid $ABCD$. | [
"\\(75\\sqrt{3}\\)"
] | [
"A trapezoid $ABCD$ has $AB \\parallel DC$ and $AD = BC = 10$. Given that the bisector of angle $D$ intersects $B$ and that $DB = 10\\sqrt{3}$, find the area of trapezoid $ABCD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 22
} | 43 | 0 | 0 |
omr-ws-4294930e8a72316ab0bb62eae334a4e90be8db1f4eac7263645d820e4f2b1962 | A trapezoid $ABCD$ is inscribed in a circle $(\omega)$ such that $AD=AB=BC$. Let $E$ be a point on ray $BA$ such that $BE=\frac{AC}{2}$. Let $F$ be the point on $CD$ such that $EF \perp CD$. If the radius of circle $(\omega)$ is $r$, find the length of $EF$ in terms of $r$. | [
"r"
] | [
"A trapezoid $ABCD$ is inscribed in a circle $(\\omega)$ such that $AD=AB=BC$. Let $E$ be a point on ray $BA$ such that $BE=\\frac{AC}{2}$. Let $F$ be the point on $CD$ such that $EF \\perp CD$. If the radius of circle $(\\omega)$ is $r$, find the length of $EF$ in terms of $r$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-957869d4f70067a62750e2edacdd0b362e83717f201ba2cc61079aae624f1b7a | A trapezoid $ABCD$ is inscribed into a semi-circle of radius $l$ so that the base $AD$ of the trapezoid is a diameter and the vertices $B$ and $C$ lie on the circumference. Find the base angle $\varphi $ of the trapezoid $ABCD$ which has the greatest perimeter. | [
"$\\varphi =\\pi /3$"
] | [
"A trapezoid $ABCD$ is inscribed into a semi-circle of radius $l$ so that the base $AD$ of the trapezoid is a diameter and the vertices $B$ and $C$ lie on the circumference. Find the base angle $\\varphi $ of the trapezoid $ABCD$ which has the greatest perimeter."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad5e0221204fa5685609d7a9c758d4b2ea62a422204b694abd32f5da02aae060 | A trapezoid $ABCD$ with $AB \parallel CD$ is inscribed in a circle $k$. Points $P$ and $Q$ are chosen on the arc $ADCB$ in the order $A-P-Q-B$. Lines $CP$ and $AQ$ meet at $X$, and lines $BP$ and $DQ$ meet at $Y$. If the circle passing through points $P, Q, X, Y$ has radius $r$ and the distance between the centers of c... | [
"100"
] | [
"A trapezoid $ABCD$ with $AB \\parallel CD$ is inscribed in a circle $k$. Points $P$ and $Q$ are chosen on the arc $ADCB$ in the order $A-P-Q-B$. Lines $CP$ and $AQ$ meet at $X$, and lines $BP$ and $DQ$ meet at $Y$. If the circle passing through points $P, Q, X, Y$ has radius $r$ and the distance between the center... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b204031d9747f433e1200981736467675d35e75af27fa85755a27a0f14d87821 | A trapezoid $ABCD$ with $AB \parallel CD$ is inscribed in a circle $k$. Points $P$ and $Q$ are chosen on the arc $ADCB$ in the order $A-P-Q-B$. Lines $CP$ and $AQ$ meet at $X$, and lines $BP$ and $DQ$ meet at $Y$. If the radius of circle $k$ is $r$, and the radius of the circle passing through points $P, Q, X, Y$ is $r... | [
"1"
] | [
"A trapezoid $ABCD$ with $AB \\parallel CD$ is inscribed in a circle $k$. Points $P$ and $Q$ are chosen on the arc $ADCB$ in the order $A-P-Q-B$. Lines $CP$ and $AQ$ meet at $X$, and lines $BP$ and $DQ$ meet at $Y$. If the radius of circle $k$ is $r$, and the radius of the circle passing through points $P, Q, X, Y$... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-42c285f4d192bac6b6419a69b4b65a0c28b3cb8f874bd9cd7561870f4312f464 | A trapezoid $ABCD$ with a longer base $AB$ is inscribed in a circle $k$. Let $A_0$ and $B_0$ be the midpoints of segments $BC$ and $CA$, respectively. Let $N$ be the foot of the perpendicular from vertex $C$ to $AB$, and $G$ be the centroid of triangle $ABC$. Circle $k_1$ passes through points $A_0$ and $B_0$ and is ta... | [
"r"
] | [
"A trapezoid $ABCD$ with a longer base $AB$ is inscribed in a circle $k$. Let $A_0$ and $B_0$ be the midpoints of segments $BC$ and $CA$, respectively. Let $N$ be the foot of the perpendicular from vertex $C$ to $AB$, and $G$ be the centroid of triangle $ABC$. Circle $k_1$ passes through points $A_0$ and $B_0$ and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4fa346407a3b106bc3a3fae31f7cfdaa5c24c932bef2946da5bd8fe887b5c59 | A trapezoid $ABCD$ with bases $AD$ and $BC$ is circumscribed about a circle, and $E$ is the intersection point of the diagonals. If $\angle AED = \theta$, find the minimum value of $\cos(\theta)$. | [
"0"
] | [
"A trapezoid $ABCD$ with bases $AD$ and $BC$ is circumscribed about a circle, and $E$ is the intersection point of the diagonals. If $\\angle AED = \\theta$, find the minimum value of $\\cos(\\theta)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-20c562f94a6bcc66452442d758865d5c0969312c68f262e6386553fda709bd0e | A trapezoid $ABCD$ with bases $BC$ and $AD$ satisfies that $\angle DAB = \angle ABC = 90^\circ$. The circle of diameter $AB$ is tangent to side $CD$. If $BC = 8$ and $AD = 18$, find the area of the trapezoid $ABCD$. | [
"312"
] | [
"A trapezoid $ABCD$ with bases $BC$ and $AD$ satisfies that $\\angle DAB = \\angle ABC = 90^\\circ$. The circle of diameter $AB$ is tangent to side $CD$. If $BC = 8$ and $AD = 18$, find the area of the trapezoid $ABCD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 5
} | 53 | 0 | 0 |
omr-ws-1ad54d138f139a3dac4a9e083e9f545ca4a43198c24829ffcd9aa8f708dfe619 | A trapezoid $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle $k$. Let $k_a$ and $k_c$ be the circles with diameters $AB$ and $CD$, respectively. Compute the area of the region which is inside the circle $k$, but outside the circles $k_a$ and $k_c$. | [
"2ab"
] | [
"A trapezoid $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle $k$. Let $k_a$ and $k_c$ be the circles with diameters $AB$ and $CD$, respectively. Compute the area of the region which is inside the circle $k$, but outside the circles $k_a$ and $k_c$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-c7f363b97fdff189bf66958d85e2c130344ae9de914e1231d1060b021662b132 | A trapezoid $ABCD$, with parallel sides $AB$ and $CD$, is inscribed in a circle of radius $25$. It is known that $CD$ is a diameter and the height of this trapezoid is $24$. Let $E$ be a point on the smallest arc determined by $A$ and $B$. Let $F$ and $G$ be the points of intersection of $ED$ and $EC$ with $AB$, respec... | [
"32"
] | [
"A trapezoid $ABCD$, with parallel sides $AB$ and $CD$, is inscribed in a circle of radius $25$. It is known that $CD$ is a diameter and the height of this trapezoid is $24$. Let $E$ be a point on the smallest arc determined by $A$ and $B$. Let $F$ and $G$ be the points of intersection of $ED$ and $EC$ with $AB$, r... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-c28f0d360c3c09c3a50db274aa7e44983a39d937a014a4696b868507590a1baf | A trapezoid $ABRS$ is inscribed in an ellipse with a large axis $a$ such that $AB \parallel RS$. The reflections of points $A$ and $B$ in the axis $a$ are points $P$ and $Q$, respectively. Given that the coordinates of $A, B, R, S$ are $(x_1, y_1), (x_2, y_1), (x_3, y_3), (x_4, y_4)$ respectively, and the ellipse equat... | [
" \\sqrt{ \\left(x_3 - \\frac{x_1 + x_2}{2}\\right)^2 + \\left( \\frac{y_3(a^2 + b^2) - y_1(a^2 - b^2)}{2b^2} \\right)^2 } "
] | [
"A trapezoid $ABRS$ is inscribed in an ellipse with a large axis $a$ such that $AB \\parallel RS$. The reflections of points $A$ and $B$ in the axis $a$ are points $P$ and $Q$, respectively. Given that the coordinates of $A, B, R, S$ are $(x_1, y_1), (x_2, y_1), (x_3, y_3), (x_4, y_4)$ respectively, and the ellipse... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-79e58116ce5da4215c72f3e3ca915475f4590e79e54427fa73e98ba8775f40c8 | A trapezoid ABCD has AB parallel to DC, and DC = AB + 100. The line joining the midpoints of AD and BC divides the trapezoid into two regions with areas in the ratio 2 : 3. Find the length of the segment parallel to DC that joins AD and BC and divides the trapezoid into two regions of equal area. | [
"\\( 25\\sqrt{29} \\)"
] | [
"A trapezoid ABCD has AB parallel to DC, and DC = AB + 100. The line joining the midpoints of AD and BC divides the trapezoid into two regions with areas in the ratio 2 : 3. Find the length of the segment parallel to DC that joins AD and BC and divides the trapezoid into two regions of equal area."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 28
} | 76 | 0 | 0 |
omr-ws-bf39f42fa9acd49c40dd953874163fd6f7160909dbce74a340e65ce4063bd801 | A trapezoid ABCD with AB parallel to CD is inscribed in a circle. Points P and Q are chosen on the arc ADCB in order A-P-Q-B. Lines CP and AQ meet at X, and lines BP and DQ meet at Y. If the measure of angle APB is $75^\circ$, find the measure of angle XYQ in degrees. | [
"75"
] | [
"A trapezoid ABCD with AB parallel to CD is inscribed in a circle. Points P and Q are chosen on the arc ADCB in order A-P-Q-B. Lines CP and AQ meet at X, and lines BP and DQ meet at Y. If the measure of angle APB is $75^\\circ$, find the measure of angle XYQ in degrees."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c909199bb77545401f3183db352aa29c3f0ebdfda74ae32fedc143fa904c050a | A trapezoid \( ABCD \) has \( AB \) and \( CD \) both perpendicular to \( AD \) and \( BC = AB + AD \). If \( AB = 26 \), what is \( \frac{CD^2}{AD + CD} \)? | [
"52"
] | [
"A trapezoid \\( ABCD \\) has \\( AB \\) and \\( CD \\) both perpendicular to \\( AD \\) and \\( BC = AB + AD \\). If \\( AB = 26 \\), what is \\( \\frac{CD^2}{AD + CD} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c5bec9601a42bc91efc9a0bd351ccb3247c940cc72de23464d97a2964381d635 | A trapezoid \(ABCD\) with \(AB \parallel CD\) and \(AD < CD\) is inscribed in a circle \(c\). Let \(DP\) be a chord parallel to \(AC\). The tangent to \(c\) at \(D\) meets the line \(AB\) at \(E\), and the lines \(PB\) and \(DC\) meet at \(Q\). If \(AC = 10\), find the length of \(EQ\). | [
"10"
] | [
"A trapezoid \\(ABCD\\) with \\(AB \\parallel CD\\) and \\(AD < CD\\) is inscribed in a circle \\(c\\). Let \\(DP\\) be a chord parallel to \\(AC\\). The tangent to \\(c\\) at \\(D\\) meets the line \\(AB\\) at \\(E\\), and the lines \\(PB\\) and \\(DC\\) meet at \\(Q\\). If \\(AC = 10\\), find the length of \\(EQ\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-d4ffee38499cce800ae8116c922ee2a9a35232caec11e17d938f440a17cebac9 | A trapezoid \(ABCD\) with bases \(AB\) and \(CD\) is given, in which \(AB + CD = AD\). The diagonals \(AC\) and \(BD\) intersect at point \(E\). A line passing through point \(E\) and parallel to the bases of the trapezoid intersects \(AD\) at point \(F\). What can be the angle \(\angle BFC\)? | [
"90^\\circ"
] | [
"A trapezoid \\(ABCD\\) with bases \\(AB\\) and \\(CD\\) is given, in which \\(AB + CD = AD\\). The diagonals \\(AC\\) and \\(BD\\) intersect at point \\(E\\). A line passing through point \\(E\\) and parallel to the bases of the trapezoid intersects \\(AD\\) at point \\(F\\). What can be the angle \\(\\angle BFC\\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fc75cfd35ee97ce940aa62079e724150b82017812c4a3b292336ba47e5cc35d5 | A trapezoid can be decomposed into a unit square and two similar but incongruent triangles, with one side of the triangles having a length of 1/2. What is the trapezoid's perimeter? | [
"\\( \\frac{9}{2} + \\frac{3\\sqrt{5}}{2} \\)"
] | [
"A trapezoid can be decomposed into a unit square and two similar but incongruent triangles, with one side of the triangles having a length of 1/2. What is the trapezoid's perimeter?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-15bd98737d7a486768c96462c581c4d66a62a697591edab09363d0b5737a849d | A trapezoid has \( BC \parallel AD \) and \( AB = CD \). Given \( BC = 8 \), \( AD = 16 \), and the area of the trapezoid is 48, what is the perimeter of the trapezoid? | [
"\\( 24 + 8\\sqrt{2} \\)"
] | [
"A trapezoid has \\( BC \\parallel AD \\) and \\( AB = CD \\). Given \\( BC = 8 \\), \\( AD = 16 \\), and the area of the trapezoid is 48, what is the perimeter of the trapezoid?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-cc2952d8f29e658233a8656a99844b0a59e49e4af34a925d8a344d982bce1099 | A trapezoid has a height of $10$, its legs are integers, and the sum of the sines of the acute base angles is $\frac{1}{2}$. Compute the largest sum of the lengths of the two legs. | [
"441"
] | [
"A trapezoid has a height of $10$, its legs are integers, and the sum of the sines of the acute base angles is $\\frac{1}{2}$. Compute the largest sum of the lengths of the two legs."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-0341a5006f581e9edd6e82084c14ad1032a0358eed588ae6926abd331363209d | A trapezoid has an area of $2\, m^2$ and the sum of its diagonals is $4\,m$. Determine the height of this trapezoid. | [
"\\(\\frac{\\sqrt{2}}{2} \\, m\\)"
] | [
"A trapezoid has an area of $2\\, m^2$ and the sum of its diagonals is $4\\,m$. Determine the height of this trapezoid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f1ad6ddb3952f2deb50ccc77e2576c9df5be4615ddffb6d87c165be10432d65 | A trapezoid has area $\mathcal A$. If one of its base lengths were halved, its area would be $\frac{2}{3}\mathcal A$. If instead the other base length were halved, what would the trapezoid's area be? | [
"\\dfrac{5}{6}\\mathcal{A}"
] | [
"A trapezoid has area $\\mathcal A$. If one of its base lengths were halved, its area would be $\\frac{2}{3}\\mathcal A$. If instead the other base length were halved, what would the trapezoid's area be?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3dc6afc444a8fe9b1f52719f2e37030ef2d2d2b69993a93bef9f265714665ab3 | A trapezoid has bases of length $x$ and 4. A line segment PH, parallel to the bases, divides the trapezoid into two quadrilaterals of the same area. Given that the length of PH is 6, find the length of $x$. | [
"\\(2\\sqrt{14}\\)"
] | [
"A trapezoid has bases of length $x$ and 4. A line segment PH, parallel to the bases, divides the trapezoid into two quadrilaterals of the same area. Given that the length of PH is 6, find the length of $x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 27
} | 75 | 0 | 0 |
omr-ws-1d8aed1bd5e02d6c88bfa0262d5b78d98e3409b1fe04c181155f307c2347e729 | A trapezoid has diagonals of lengths 17 and 15. The length of the segment connecting the midpoints of the two bases is 4. The height of this trapezoid can be expressed as $\dfrac{a}{\sqrt{b}}$. Find $a+b$. | [
"301"
] | [
"A trapezoid has diagonals of lengths 17 and 15. The length of the segment connecting the midpoints of the two bases is 4. The height of this trapezoid can be expressed as $\\dfrac{a}{\\sqrt{b}}$. Find $a+b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 11
} | 34 | 0 | 0 |
omr-ws-fdcb7be747269205088246f321f972e42a4bb0f6538d375f167ca05f361ba7e3 | A trapezoid has legs of lengths 4 and 5, and bases of lengths 10 and \( x \). Given that \( x < 19 \), determine the values that \( x \) cannot take. | [
"\\( x \\notin (0, 1] \\cup (7, 13) \\cup [19, \\infty) \\)"
] | [
"A trapezoid has legs of lengths 4 and 5, and bases of lengths 10 and \\( x \\). Given that \\( x < 19 \\), determine the values that \\( x \\) cannot take."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 4
} | 10 | 0 | 0 |
omr-ws-598452e775ebb998f88eae8554245753a6c864f3151c66e665150eeaaaf99d58 | A trapezoid has one base equal to twice its height, $x$, and the other base is three times as long as the height. Write the expression for the area of the trapezoid as a common fraction in terms of the height $x$. | [
"\\dfrac{5x^2}{2}"
] | [
"A trapezoid has one base equal to twice its height, $x$, and the other base is three times as long as the height. Write the expression for the area of the trapezoid as a common fraction in terms of the height $x$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b8a99b7f6854e7fb0d7e0d6e5bbf5895289d1fc3af3fb57dead4eccd41e76ed9 | A trapezoid has perpendicular diagonals, a height of 4, and one diagonal of length 5. Find the area of the trapezoid. | [
"\\(\\frac{50}{3}\\)"
] | [
"A trapezoid has perpendicular diagonals, a height of 4, and one diagonal of length 5. Find the area of the trapezoid."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 24
} | 72 | 0 | 0 |
omr-ws-cad1226c12b3712e8022dc51395e41c9dca52d18d3908786bb9b34ca70610f51 | A trapezoid has perpendicular diagonals. If one of these diagonals has a length of 30, and the altitude of the trapezoid has a length of 24, find the area of the trapezoid. | [
"600"
] | [
"A trapezoid has perpendicular diagonals. If one of these diagonals has a length of 30, and the altitude of the trapezoid has a length of 24, find the area of the trapezoid."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-537b51362bef922ba54af4baeb02b94fec28883f32a8006d48bba0e2a9db45cd | A trapezoid has side lengths $3, 5, 7,$ and $11$. The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \sqrt{n_1} + r_2 \sqrt{n_2} + r_3$, where $r_1, r_2,$ and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of a prime. What is the great... | [
"63"
] | [
"A trapezoid has side lengths $3, 5, 7,$ and $11$. The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \\sqrt{n_1} + r_2 \\sqrt{n_2} + r_3$, where $r_1, r_2,$ and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of a prime. What is th... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 34,
"genselect": 8,
"tir": 2
} | 44 | 0 | 0 |
omr-ws-89383a5a032eeba8219d28923dbc20cac5905f3758a06669dd69ea4032ffd906 | A trapezoid has side lengths 1, 2, 3, and 4, but not necessarily in that order. The area of this trapezoid can be written as \(\frac{a \sqrt{b}}{c}\) in simplest form. Find \(a + b + c\). | [
"15"
] | [
"A trapezoid has side lengths 1, 2, 3, and 4, but not necessarily in that order. The area of this trapezoid can be written as \\(\\frac{a \\sqrt{b}}{c}\\) in simplest form. Find \\(a + b + c\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 5
} | 27 | 0 | 0 |
omr-ws-61750e765031dd37eae11c31b810ae45f93d61c930af9d1e9c8fe37c916a0351 | A trapezoid has the property that the sum of each pair of opposite sides is 23. The lengths of the legs differ by 7, and the lengths of the bases differ by 17. The area of the trapezoid can be written as a reduced fraction \( \frac{x}{y} \) where \( x > y \). Evaluate \( x - y \). | [
"1363"
] | [
"A trapezoid has the property that the sum of each pair of opposite sides is 23. The lengths of the legs differ by 7, and the lengths of the bases differ by 17. The area of the trapezoid can be written as a reduced fraction \\( \\frac{x}{y} \\) where \\( x > y \\). Evaluate \\( x - y \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-c385184390f08470c1a3d180e6ce30c9cf3cca677802b66ee95626ba46cd6750 | A trapezoid has the ratio of its bases as 2:1, the ratio of its legs (nonparallel sides) as 2:1, and the ratio of its diagonals as 2:1. What is the ratio of its shorter base to its shorter leg? | [
"\\dfrac{\\sqrt{10}}{2}"
] | [
"A trapezoid has the ratio of its bases as 2:1, the ratio of its legs (nonparallel sides) as 2:1, and the ratio of its diagonals as 2:1. What is the ratio of its shorter base to its shorter leg?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-7bb809eb903134643b4829984e932c9beaa42c7ceab4888a9fb50c2ad382329d | A trapezoid inscribed in a circle of radius $r$ has three sides of length $s < r$ and one side of length $r+s$. Find the angles of the trapezoid. | [
"The angles of the trapezoid are \\( 36^\\circ, 36^\\circ, 144^\\circ, 144^\\circ \\)."
] | [
"A trapezoid inscribed in a circle of radius $r$ has three sides of length $s < r$ and one side of length $r+s$. Find the angles of the trapezoid."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-844d8e8cfaef7f73487eb0865f152d40aa6369d1cc437014f7be020c54e6be0b | A trapezoid inscribed in a circle of radius $r$ has three sides of length $s$ and a fourth side of length $r + s$, with $s < r$. Find the measures of the angles of the trapezoid. | [
"\\(\\frac{\\pi}{5}\\) and \\(\\frac{4\\pi}{5}\\)"
] | [
"A trapezoid inscribed in a circle of radius $r$ has three sides of length $s$ and a fourth side of length $r + s$, with $s < r$. Find the measures of the angles of the trapezoid."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-d204d8aaa0edc9db0e0ccab4bfbd1e783d93fd9ad23b1acafd936497110315b8 | A trapezoid into which a circle can be inscribed has its larger base on the plane \(a\), and the intersection point of the diagonals is 9 cm from this plane. If the sides of the trapezoid form angles of \(60^\circ\) with its larger base, find the distance from the smaller base to the plane \(a\). | [
"12 cm"
] | [
"A trapezoid into which a circle can be inscribed has its larger base on the plane \\(a\\), and the intersection point of the diagonals is 9 cm from this plane. If the sides of the trapezoid form angles of \\(60^\\circ\\) with its larger base, find the distance from the smaller base to the plane \\(a\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 39,
"genselect": 6,
"tir": 1
} | 46 | 0 | 0 |
omr-ws-ca190bfde2dffa60eb88aed8187dcc64c829c59c99d3ba739450acfe2daabf24 | A trapezoid is constructed with two parallel horizontal lines of different lengths joined at each end. Another line, parallel to these two lines, bisects the area of the trapezoid (not the height). Find the length of this line that bisects the area. | [
"\\(\\sqrt{\\frac{x^2 + y^2}{2}}\\)"
] | [
"A trapezoid is constructed with two parallel horizontal lines of different lengths joined at each end. Another line, parallel to these two lines, bisects the area of the trapezoid (not the height). Find the length of this line that bisects the area."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-e32b82a95ecf4356368ab17cbe36580a2d6fbf0ea478c62adecb3ff3ea541b67 | A trapezoid is divided into four triangles by its diagonals. Find the area of the trapezoid if the areas of the triangles adjacent to the bases of the trapezoid are equal to $S_1$ and $S_2$. | [
"\\(\\left(\\sqrt{S_1} + \\sqrt{S_2}\\right)^2\\)"
] | [
"A trapezoid is divided into four triangles by its diagonals. Find the area of the trapezoid if the areas of the triangles adjacent to the bases of the trapezoid are equal to $S_1$ and $S_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ea0054721f88478dc9d1235bd276157ff7c536a35a5a7b4cfbcaa3a613b7967e | A trapezoid is inscribed and circumscribed about two circles. The lengths of the bases of the trapezoid are 4 and 8. What is the area of the trapezoid? | [
"\\( 24\\sqrt{2} \\)"
] | [
"A trapezoid is inscribed and circumscribed about two circles. The lengths of the bases of the trapezoid are 4 and 8. What is the area of the trapezoid?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-1815dd7017655a206d2020fe3343cce8caf219e1196c9dd5cc5d289bfa01effe | A trapezoid is inscribed in a circle with base lengths of $10$ and $20$. One of the legs has a length of $\sqrt{89}$. Find the length of one of the diagonals. | [
"17"
] | [
"A trapezoid is inscribed in a circle with base lengths of $10$ and $20$. One of the legs has a length of $\\sqrt{89}$. Find the length of one of the diagonals."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a583a86ac127acc733b369827f9e0244748cc7a85261cd5bc2c2fe68e5078099 | A trapezoid is inscribed in a semicircle of radius \( R \). One of the bases of the trapezoid is the diameter of the semicircle, and one of the non-parallel sides makes an angle \( \alpha \) with the diameter. Compute the area of the trapezoid. | [
"\\(4R^2\\sin^3\\alpha\\cos\\alpha\\)"
] | [
"A trapezoid is inscribed in a semicircle of radius \\( R \\). One of the bases of the trapezoid is the diameter of the semicircle, and one of the non-parallel sides makes an angle \\( \\alpha \\) with the diameter. Compute the area of the trapezoid."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4960fac9ec8823c5a347c43e952c717eaaa3cda80fbcaccf0b1d0bc8d1e3e912 | A trapezoid whose legs are both integers has acute base angles such that the sum of their sines is equal to \(\frac{1}{3}\). Given that the trapezoid has a height of 20, calculate the minimum value of the sum of the legs. | [
"240"
] | [
"A trapezoid whose legs are both integers has acute base angles such that the sum of their sines is equal to \\(\\frac{1}{3}\\). Given that the trapezoid has a height of 20, calculate the minimum value of the sum of the legs."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0bf0be95b7601dcf6ff4314d0fc4b8421d21c39e44fb28e3da48c48526c56d5d | A trapezoid with bases \(AB\) and \(CD\) is inscribed in a circle centered at \(O\). Let \(AP\) and \(AQ\) be the tangents from \(A\) to the circumcircle of triangle \(CDO\). Let \(M\) be the midpoint of \(AB\). If the circumcircle of triangle \(APQ\) intersects \(AB\) at \(M\) and another point \(N\), find the ratio \... | [
"no_valid_answer_found"
] | [
"A trapezoid with bases \\(AB\\) and \\(CD\\) is inscribed in a circle centered at \\(O\\). Let \\(AP\\) and \\(AQ\\) be the tangents from \\(A\\) to the circumcircle of triangle \\(CDO\\). Let \\(M\\) be the midpoint of \\(AB\\). If the circumcircle of triangle \\(APQ\\) intersects \\(AB\\) at \\(M\\) and another ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-85cead40ee3ef89c7b3064e710cdac8871883338b1e597dcd048eb932f3619a9 | A trapezoid with height 12 has legs of length 20 and 15 and a larger base of length 42. What are the possible lengths of the other base? | [
"\\( 17, 35 \\)"
] | [
"A trapezoid with height 12 has legs of length 20 and 15 and a larger base of length 42. What are the possible lengths of the other base?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 8,
"tir": 10
} | 20 | 0 | 0 |
omr-ws-2c77d2df72578627fc9cc8c8e9d2a0cb16e5cd44deb53512868c2138ea9e0a5c | A trapezoid with perimeter $58 + 14 \sqrt{11}$ is inscribed in a circle with diameter $17 + 7 \sqrt{11}$. Find the dimensions of its sides. | [
"12 + 7\\sqrt{11}"
] | [
"A trapezoid with perimeter $58 + 14 \\sqrt{11}$ is inscribed in a circle with diameter $17 + 7 \\sqrt{11}$. Find the dimensions of its sides."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-087bbc234743fcd7344e47f5e4dba1c1271e9728d8d96074b98264d6c157300e | A trapezoid with the base angles equal to $\alpha $ and $\beta $ is circumscribed about a circle. Find the ratio of the area of the trapezoid to that of the circle. | [
"$2\\pi ^{-1}({\\rm cosec}\\, \\alpha +{\\rm cosec}\\, \\beta )$"
] | [
"A trapezoid with the base angles equal to $\\alpha $ and $\\beta $ is circumscribed about a circle. Find the ratio of the area of the trapezoid to that of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cfcf415f24262e467fb1ef4e93447d3bd82efff6a875f2b28d86a143a30ac4bc | A trapezoidal prism fish tank has a bottom base of 80 cm, a top base of 60 cm, a height of 30 cm, and a depth of 40 cm. If 63 L (or 63,000 cm³) of water is poured into the tank, what is the height of the water level in the tank? | [
"22.5 cm"
] | [
"A trapezoidal prism fish tank has a bottom base of 80 cm, a top base of 60 cm, a height of 30 cm, and a depth of 40 cm. If 63 L (or 63,000 cm³) of water is poured into the tank, what is the height of the water level in the tank?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d3a594a9c8b5df33169feb63dda6927e09ba31d6454625ad42f716685851fde | A travel agency offers an all-inclusive tour for Y8000 per person if not more than 100 persons join the tour. However, the cost per person will be reduced by Y50 for each person in excess of 100. Find the largest possible gross sales revenue for the travel agency. How many persons should join the tour and what is the c... | [
"Y845000, 130 persons, Y6500 per person"
] | [
"A travel agency offers an all-inclusive tour for Y8000 per person if not more than 100 persons join the tour. However, the cost per person will be reduced by Y50 for each person in excess of 100. Find the largest possible gross sales revenue for the travel agency. How many persons should join the tour and what is ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 27
} | 37 | 0 | 0 |
omr-ws-ec7ce1db9e137c1a12c33c608ff42349cc2b9942ae229dbc82a88ded03f8da46 | A traveler has a chain made of gold rings and can break \( n \) rings. The traveler needs to pay the innkeeper one ring per day. Determine the maximum duration of the stay in terms of \( n \). | [
"\\( 2^n - 1 \\)"
] | [
"A traveler has a chain made of gold rings and can break \\( n \\) rings. The traveler needs to pay the innkeeper one ring per day. Determine the maximum duration of the stay in terms of \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ccc2228569b0c9632e8afedcd9759562f20298f80a89b38a9db569865d3bc274 | A traveler has a straight gold chain with 7 links. She plans to stay at an inn that accepts one link as payment for one night's lodging. She wants to pay on a daily basis for up to a week. What is the smallest number of links she must cut in order to be able to pay for every night, possibly getting change in the form o... | [
"1 (cut the third link)"
] | [
"A traveler has a straight gold chain with 7 links. She plans to stay at an inn that accepts one link as payment for one night's lodging. She wants to pay on a daily basis for up to a week. What is the smallest number of links she must cut in order to be able to pay for every night, possibly getting change in the f... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-390f7feb35fa07b1c3c57a2ffec6b87e9f902fbd922c8d2895b24f6b7ab8f28a | A traveler has to cross a river. At what angle $\alpha $ should he sail to attain the least drift if the speed of the boat is $V_b$ and that of the river is $V_r$? | [
"$\\alpha =\\arccos \\frac{V_r}{V_b}$"
] | [
"A traveler has to cross a river. At what angle $\\alpha $ should he sail to attain the least drift if the speed of the boat is $V_b$ and that of the river is $V_r$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19d1cc5ae3c129092b7153484af857fe441012814f632f43693d74da5be05d54 | A traveler walked in a sequence of steps: 1 meter East, 2 meters North, 3 meters West, 4 meters South, and so on, up to 2008 meters South. Calculate the nearest integer value of the distance between the traveler's start and end positions. | [
"2846831"
] | [
"A traveler walked in a sequence of steps: 1 meter East, 2 meters North, 3 meters West, 4 meters South, and so on, up to 2008 meters South. Calculate the nearest integer value of the distance between the traveler's start and end positions."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31af74938804ef8c4086c4fc6d069d3b6be856a1630545d62bf1ed6018259ccd | A traveling party is 10 miles North of their destination when they lost their map and compass. Each day, the party travels 10 miles in a direction chosen at random from North, South, East, or West. If the party arrives at their destination, they stop traveling. What is the chance that the party reaches their destinatio... | [
"\\dfrac{93}{256}"
] | [
"A traveling party is 10 miles North of their destination when they lost their map and compass. Each day, the party travels 10 miles in a direction chosen at random from North, South, East, or West. If the party arrives at their destination, they stop traveling. What is the chance that the party reaches their desti... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-fa02f867dc03b070a4a3784a4b5c15b19a4c676225e1d4974d4ee846267c3ac8 | A traveling salesman has to visit six of the ten cities in his territory, including cities A and B. How many ways can he choose the six cities? In how many of these ways will city A come before city B? | [
"50,400 ways to choose the six cities, and 25,200 ways in which A comes before B."
] | [
"A traveling salesman has to visit six of the ten cities in his territory, including cities A and B. How many ways can he choose the six cities? In how many of these ways will city A come before city B?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-d79d76ef57fae3d3794601521278ef11198542caa182a80277a24a4858ef1d48 | A traveller is riding a car on a straight highway $AB$ from town $A$ to a point $X \in (AB)$ and then leaves on foot in a straight line to point $C \not\in AB$. The speed of the car is $v_1$ and the speed on foot is $v_2$, with $v_2 < v_1$. Determine the position of point $X \in (AB)$ such that the total time to travel... | [
"c - \\dfrac{v_2 d}{\\sqrt{v_1^2 - v_2^2}}"
] | [
"A traveller is riding a car on a straight highway $AB$ from town $A$ to a point $X \\in (AB)$ and then leaves on foot in a straight line to point $C \\not\\in AB$. The speed of the car is $v_1$ and the speed on foot is $v_2$, with $v_2 < v_1$. Determine the position of point $X \\in (AB)$ such that the total time ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-d481dedb515de7dc7bb0a809dcab5a8f5fe6e9ab722d933d8daf9c9a1e186515 | A traveller starts from a certain place and travels 1 km on the first day, increasing the distance by 2 km each subsequent day. A second traveller starts from the same place 3 days later and travels 12 km on the first day, increasing the distance by 1 km each subsequent day. On how many days would the second traveller ... | [
"6"
] | [
"A traveller starts from a certain place and travels 1 km on the first day, increasing the distance by 2 km each subsequent day. A second traveller starts from the same place 3 days later and travels 12 km on the first day, increasing the distance by 1 km each subsequent day. On how many days would the second trave... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-19d8c7a1c5046049cc2cae77fee320e2efb4b9c156fbcab2aa80f9e16a2ee7da | A tray contains 6 cups of coffee, 4 of which have a dose of $x$ grams each of a deadly powder, and the other 2 contain $2x$ grams each. A dose of $4x$ grams or more is instantly fatal. Fred Innocent drinks two cups of the coffee at random and survives. What is the probability that he will survive if he takes a third cu... | [
"\\(\\frac{3}{14}\\)"
] | [
"A tray contains 6 cups of coffee, 4 of which have a dose of $x$ grams each of a deadly powder, and the other 2 contain $2x$ grams each. A dose of $4x$ grams or more is instantly fatal. Fred Innocent drinks two cups of the coffee at random and survives. What is the probability that he will survive if he takes a thi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-2d991e46eab7a2bb68d03befed28541bb056aeb41d39475feee7abbd07b1f2c3 | A tree 123 feet high breaks off a certain distance up, and the moment the top strikes a stump 15 feet high, the broken part points to a spot 108 feet from the base of the tree. Find the length of the part broken off. | [
"78"
] | [
"A tree 123 feet high breaks off a certain distance up, and the moment the top strikes a stump 15 feet high, the broken part points to a spot 108 feet from the base of the tree. Find the length of the part broken off."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-7470e908f8370f85f7d08959b477a74796b458ed6f437a17066cda43f384ac15 | A tree doubled its height every year until it reached a height of 32 feet at the end of 6 years. What was the height of the tree, in feet, at the end of 3 years? | [
"4 \\text{ feet}"
] | [
"A tree doubled its height every year until it reached a height of 32 feet at the end of 6 years. What was the height of the tree, in feet, at the end of 3 years?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2b9b61414f71475dbd45909b024a112aa28b5ba6e73282f3f4f45bf174dabd17 | A tree grows in the following manner. On the first day, one branch grows out of the ground. On the second day, a leaf grows on the branch and the branch tip splits up into two new branches. On each subsequent day, a new leaf grows on every existing branch, and each branch tip splits up into two new branches. How many l... | [
"1013"
] | [
"A tree grows in the following manner. On the first day, one branch grows out of the ground. On the second day, a leaf grows on the branch and the branch tip splits up into two new branches. On each subsequent day, a new leaf grows on every existing branch, and each branch tip splits up into two new branches. How m... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b25dde33dd82a395b5b692d13d216744d768dffa4bca88b7909b8e032e30695 | A tree grows vertically on a slope with an incline of 25 degrees. When the sun is at an elevation of 70 degrees, the tree casts a shadow of 20 feet directly down the slope. What is the height of the tree? | [
"20 feet"
] | [
"A tree grows vertically on a slope with an incline of 25 degrees. When the sun is at an elevation of 70 degrees, the tree casts a shadow of 20 feet directly down the slope. What is the height of the tree?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-15bf040f53ef768a8cccbe93046483abb8c835b2e92020b48d52c624fcbdf7c4 | A tree has 10 pounds of apples at dawn. Every afternoon, a bird eats \( x \) pounds of apples. Overnight, the amount of food on the tree increases by 10%. What is the maximum value of \( x \) such that the bird can sustain itself indefinitely on the tree without the tree running out of food? | [
"\\(\\frac{10}{11}\\)"
] | [
"A tree has 10 pounds of apples at dawn. Every afternoon, a bird eats \\( x \\) pounds of apples. Overnight, the amount of food on the tree increases by 10%. What is the maximum value of \\( x \\) such that the bird can sustain itself indefinitely on the tree without the tree running out of food?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-1f89325bead09c3ed06e52d52f2c22d1466b2e4e0a0ad16869240390735bba9a | A tree increases in height by 25 percent each year. If it was 2.5 feet tall when planted in January 2000, approximately how tall will the tree be in December 2008? | [
"14.9 feet"
] | [
"A tree increases in height by 25 percent each year. If it was 2.5 feet tall when planted in January 2000, approximately how tall will the tree be in December 2008?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 3
} | 8 | 0 | 0 |
omr-ws-c1a6b01d42c132d46490918f1568638ffa4a5057fcd527098dc6b4d4f40879c1 | A tree is located 50 feet east of a fence that runs north to south. The dog buried Billy's hat a distance of 25 feet from the fence and also 20 feet from the tree. How many possible locations are there for the hat? | [
"0"
] | [
"A tree is located 50 feet east of a fence that runs north to south. The dog buried Billy's hat a distance of 25 feet from the fence and also 20 feet from the tree. How many possible locations are there for the hat?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-28561c6e415da8b2038b7b7f059cd8bc1fde42c11a49950b9155e7c416f105df | A tree shop starts with 95 saplings. Each year, 50% of the saplings are sold, and 95 saplings are then planted. a) What happens to the number of saplings after a long time? b) What would happen if they cut the amount of saplings they planted each year in half? | [
"a) 190, b) 95"
] | [
"A tree shop starts with 95 saplings. Each year, 50% of the saplings are sold, and 95 saplings are then planted.\na) What happens to the number of saplings after a long time?\nb) What would happen if they cut the amount of saplings they planted each year in half?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-2db1f4abf3705638e1d12f7ce117275876d0359def5be170a1f327ff1e29e7aa | A tree stands at a corner of a square playing field with each side measuring 100 meters. At the center of the field, the tree subtends an angle of 20 degrees. What angle does the tree subtend at each of the other three corners of the field? | [
"\\(\\tan^{-1}\\left( \\frac{\\tan 20^{\\circ}}{\\sqrt2}\\right) \\approx 14.4^{\\circ}\\)"
] | [
"A tree stands at a corner of a square playing field with each side measuring 100 meters. At the center of the field, the tree subtends an angle of 20 degrees. What angle does the tree subtend at each of the other three corners of the field?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-e816fec5dbaea6e54f1f09525ad08f47dbfc969da00b4cc5688056af6cb80911 | A triangle $ABC$ has $\angle ACB = 45^\circ.$ $G$ is the centroid and $O$ is the circumcenter. If $OG \parallel BC$ and $OG = 1,$ find $BC.$ | [
"12"
] | [
"A triangle $ABC$ has $\\angle ACB = 45^\\circ.$ $G$ is the centroid and $O$ is the circumcenter. If $OG \\parallel BC$ and $OG = 1,$ find $BC.$"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f710e5abed5063101d7ca5c27d1c86923e8b25d1880434e9e8c2ab645c736e65 | A triangle $ABC$ has angles whose measures satisfy the condition $\angle A \le \angle B \le \angle C$. If the maximum possible value of $\cos A + \cos C$ is $\lambda$, calculate the value of $(24\lambda)^2$. | [
"729"
] | [
"A triangle $ABC$ has angles whose measures satisfy the condition $\\angle A \\le \\angle B \\le \\angle C$. If the maximum possible value of $\\cos A + \\cos C$ is $\\lambda$, calculate the value of $(24\\lambda)^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8ca92b535c49a2d54dcb74169906e56e681b70d940bb24332c72ac906b8c2164 | A triangle $ABC$ is divided into four regions by three lines parallel to $BC$. The lines divide $AB$ into four equal segments. If the second largest region has an area of $225$, what is the area of triangle $ABC$? | [
"\\(\\frac{3600}{7}\\)"
] | [
"A triangle $ABC$ is divided into four regions by three lines parallel to $BC$. The lines divide $AB$ into four equal segments. If the second largest region has an area of $225$, what is the area of triangle $ABC$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b5f56f19a863593ebbbd3a46f71ffcdcd6a409401572ff4b9937173061ae358 | A triangle $ABC$ is divided into three equal figures by straight lines which arc parallel to the side $AC$. Compute the parts into which the side $AB$, equal to $a$, is divided by the parallel lines. | [
"The side $AB$ is divided into the following parts: $\\frac{a}{\\sqrt 3}$, $\\frac{a}{\\sqrt 3}\\left(\\sqrt 2-1\\right)$ and $\\frac{a}{\\sqrt 3}\\left(\\sqrt 3-\\sqrt 2\\right)$"
] | [
"A triangle $ABC$ is divided into three equal figures by straight lines which arc parallel to the side $AC$. Compute the parts into which the side $AB$, equal to $a$, is divided by the parallel lines."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab7c13b1b2a0db912b7373a505e0a2d99c7616ecc13bbbb822a96510e5e54e5b | A triangle $ABC$ is inscribed in a circle of radius 1, with $\angle BAC=60^{\circ}$. Altitudes $AD$ and $BE$ intersect at $H$. Find the smallest possible length of the segment $AH$. | [
"1"
] | [
"A triangle $ABC$ is inscribed in a circle of radius 1, with $\\angle BAC=60^{\\circ}$. Altitudes $AD$ and $BE$ intersect at $H$. Find the smallest possible length of the segment $AH$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ee293dab3f4bded2b97e8702b4d0e08656d18124dd8ffd2efad3a22321e3f30 | A triangle $ABC$ is inscribed in a circle; through the vertex $A$ a tangent line is drawn to intersect the extension of the side $BC$ at the point $D$. From the vertices $B$ and $C$ perpendiculars are dropped to the tangent line, the shorter of these perpendiculars being equal to 6 cm. Determine the area of the trapezo... | [
"$S=30\\mbox{ cm}^2$"
] | [
"A triangle $ABC$ is inscribed in a circle; through the vertex $A$ a tangent line is drawn to intersect the extension of the side $BC$ at the point $D$. From the vertices $B$ and $C$ perpendiculars are dropped to the tangent line, the shorter of these perpendiculars being equal to 6 cm. Determine the area of the tr... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-287a6d8b6bd4d6cd190c9ea2517ba04d1516f7e116cf103290bc02cbe8f7a8e1 | A triangle $ABC$ is right-angled at $A$ and moves such that its sides touch the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Determine the locus of the point $A$. | [
"\\( x^2 + y^2 = a^2 + b^2 \\)"
] | [
"A triangle $ABC$ is right-angled at $A$ and moves such that its sides touch the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$. Determine the locus of the point $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-504a98b589746b572dcfb63fd57dd2290204cfde8d97839449de0631e53fdb29 | A triangle $ABC$ satisfies that the lengths of its sides are three consecutive integers (in some order). Let $M$ be the midpoint of side $BC$. If the bisector of the angle $\angle ABC$ is perpendicular to the segment $AM$, calculate the perimeter of the triangle $ABC$. | [
"9"
] | [
"A triangle $ABC$ satisfies that the lengths of its sides are three consecutive integers (in some order). Let $M$ be the midpoint of side $BC$. If the bisector of the angle $\\angle ABC$ is perpendicular to the segment $AM$, calculate the perimeter of the triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-e94f4123d3f769621d52c1a78d707c2146342941c454878f7508e25251245351 | A triangle $ABC$ with $\angle ABC = 135^\circ$ is inscribed in a circle $\omega$. Lines tangent to $\omega$ at points $A$ and $C$ meet at point $D$. If $AB$ divides the segment $CD$ in half, find $\angle ABD$. | [
"22.5"
] | [
"A triangle $ABC$ with $\\angle ABC = 135^\\circ$ is inscribed in a circle $\\omega$. Lines tangent to $\\omega$ at points $A$ and $C$ meet at point $D$. If $AB$ divides the segment $CD$ in half, find $\\angle ABD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-968755ec07c76f05622c5dd2c7f37d069f16347d5d05d1883c0295d4a9293ac9 | A triangle $ABC$ with area $1$ is given. Anja chooses a point $X$ on side $BC$, Bernd chooses a point $Y$ on side $CA$, and Anja then chooses a point $Z$ on side $AB$. The points $X$, $Y$, and $Z$ cannot be vertices of triangle $ABC$. Anja aims to maximize the area of triangle $XYZ$, while Bernd aims to minimize it. Wh... | [
"\\(\\frac{1}{4}\\)"
] | [
"A triangle $ABC$ with area $1$ is given. Anja chooses a point $X$ on side $BC$, Bernd chooses a point $Y$ on side $CA$, and Anja then chooses a point $Z$ on side $AB$. The points $X$, $Y$, and $Z$ cannot be vertices of triangle $ABC$. Anja aims to maximize the area of triangle $XYZ$, while Bernd aims to minimize i... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-423e36315bff38d3c736745764e9afa1fc0bb820115c1aefc9a0d9291836e88a | A triangle $ABC$ with circumcenter $O$ has side lengths $2,3$ and $4$. Find the sum of the areas of triangles $AOB,BOC$ and $COA$. | [
"\\dfrac{77\\sqrt{15}}{60}"
] | [
"A triangle $ABC$ with circumcenter $O$ has side lengths $2,3$ and $4$. Find the sum of the areas of triangles $AOB,BOC$ and $COA$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-cfe986601eb12efa40c01f3eaa834d0bc75b487715e06331c2be06969ab08287 | A triangle $ABC$ with orthocenter $H$ is inscribed in a circle with center $K$ and radius $1$, where the angles at $B$ and $C$ are non-obtuse. If the lines $HK$ and $BC$ meet at point $S$ such that $SK(SK - SH) = 1$, compute the area of the concave quadrilateral $ABHC$. | [
"\\(\\frac{\\sqrt{3}}{2}\\)"
] | [
"A triangle $ABC$ with orthocenter $H$ is inscribed in a circle with center $K$ and radius $1$, where the angles at $B$ and $C$ are non-obtuse. If the lines $HK$ and $BC$ meet at point $S$ such that $SK(SK - SH) = 1$, compute the area of the concave quadrilateral $ABHC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-824ddfc5f4360ed0f34571f96d86d8bf1e448065d117dac2cf03c356c2a05ba1 | A triangle $ABC$ with positive integer side lengths has a perimeter of $35$. The centroid is $G$ and the incenter is $I$. If $\angle{GIC} = 90^{\circ}$, what is the length of $\overline{AB}$? | [
"11"
] | [
"A triangle $ABC$ with positive integer side lengths has a perimeter of $35$. The centroid is $G$ and the incenter is $I$. If $\\angle{GIC} = 90^{\\circ}$, what is the length of $\\overline{AB}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 4
} | 27 | 0 | 0 |
omr-ws-350d05cee5b455e232062e5f06f9b2d7df6f742cfedbdda5c8dfc02607f9fc05 | A triangle $\triangle ABC$ with sides $AB=3$, $AC=4$, and $BC=5$ is placed on the coordinate plane with $BC$ on the $x$-axis and the midpoint of $BC$ at the origin. Point $A$ is reflected across the $y$-axis to get $A'$. Find the distance $AA'$. | [
"\\(\\frac{7}{5}\\)"
] | [
"A triangle $\\triangle ABC$ with sides $AB=3$, $AC=4$, and $BC=5$ is placed on the coordinate plane with $BC$ on the $x$-axis and the midpoint of $BC$ at the origin. Point $A$ is reflected across the $y$-axis to get $A'$. Find the distance $AA'$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-da58872d52f8612b4af8ba7dead3bf4294d2d12d158708b754965a2a2799170d | A triangle ABC has an area of 20 square centimeters. We extend side AC to obtain a point M such that A is between M and C and AM = (1/2) AC. We extend side AB to obtain a point N such that B is between N and A and BN = (1/2) AB. We extend side BC to obtain a point P such that C is between P and B and CP = (1/2) BC. Wha... | [
"65"
] | [
"A triangle ABC has an area of 20 square centimeters. We extend side AC to obtain a point M such that A is between M and C and AM = (1/2) AC. We extend side AB to obtain a point N such that B is between N and A and BN = (1/2) AB. We extend side BC to obtain a point P such that C is between P and B and CP = (1/2) BC... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 8,
"tir": 3
} | 56 | 0 | 0 |
omr-ws-d38b3ebcb0713ea8fc39074468578c5a69899b9d198c8f67e53afc540dccb9f7 | A triangle ABC has sides AB=8, BC=6, and AC=10. Three circles are constructed such that each side of the triangle is the diameter of one of the circles. What is the radius of the fourth circle that is externally tangent to all three circles? | [
"\\(\\frac{144}{23}\\)"
] | [
"A triangle ABC has sides AB=8, BC=6, and AC=10. Three circles are constructed such that each side of the triangle is the diameter of one of the circles. What is the radius of the fourth circle that is externally tangent to all three circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 8
} | 24 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.