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-f36ce61e577ed9200d13e44711ab536093403462f7a56160f82ed33786223c13 | A circle with center \( O \) is inscribed in triangle \( ABC \). Point \( L \) lies on the extension of side \( AB \) beyond \( A \). The tangent from \( L \) intersects side \( AC \) at point \( K \). Find \( \angle KOL \) if \( \angle BAC = 50^\circ \). | [
"\\(65^\\circ\\)"
] | [
"A circle with center \\( O \\) is inscribed in triangle \\( ABC \\). Point \\( L \\) lies on the extension of side \\( AB \\) beyond \\( A \\). The tangent from \\( L \\) intersects side \\( AC \\) at point \\( K \\). Find \\( \\angle KOL \\) if \\( \\angle BAC = 50^\\circ \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 9
} | 23 | 0 | 0 |
omr-ws-04504a60a2b41583f38e9a70a00b8db39c76c3f81c5dd760dc02ad234dca271b | A circle with center \( O \) is internally tangent to two circles at points \( S \) and \( T \). Suppose the two circles intersect at points \( M \) and \( N \) with \( N \) closer to \( ST \). If \( S, N, T \) are collinear, find the value of \( \cos \angle OMN \). | [
"no_valid_answer_found"
] | [
"A circle with center \\( O \\) is internally tangent to two circles at points \\( S \\) and \\( T \\). Suppose the two circles intersect at points \\( M \\) and \\( N \\) with \\( N \\) closer to \\( ST \\). If \\( S, N, T \\) are collinear, find the value of \\( \\cos \\angle OMN \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af05340a7f2162fc592437e0e262639836dd860848f7d80913f39fe6c7c6f0de | A circle with center \( O \) is internally tangent to two circles inside it at points \( S \) and \( T \). Suppose the two circles inside intersect at \( M \) and \( N \) with \( N \) closer to \( ST \). If \( S, N, T \) are collinear, calculate the value of \( \cos(\angle OMN) \). | [
"\\dfrac{1}{2}"
] | [
"A circle with center \\( O \\) is internally tangent to two circles inside it at points \\( S \\) and \\( T \\). Suppose the two circles inside intersect at \\( M \\) and \\( N \\) with \\( N \\) closer to \\( ST \\). If \\( S, N, T \\) are collinear, calculate the value of \\( \\cos(\\angle OMN) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-f843a3f4ac918890c7e8fb39401d0d52539366796943a3b87d80bb81c1c82dd4 | A circle with center \( O \) is internally tangent to two circles inside it at points \( S \) and \( T \). Suppose the two circles inside intersect at \( M \) and \( N \) with \( N \) closer to \( ST \). If \( S, N, T \) are collinear, find the value of \( \frac{OM^2}{ON^2} \) given that \( OS = OT = 1 \) and \( SN = N... | [
"2"
] | [
"A circle with center \\( O \\) is internally tangent to two circles inside it at points \\( S \\) and \\( T \\). Suppose the two circles inside intersect at \\( M \\) and \\( N \\) with \\( N \\) closer to \\( ST \\). If \\( S, N, T \\) are collinear, find the value of \\( \\frac{OM^2}{ON^2} \\) given that \\( OS ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-6df5e8bd89dd42cb844207de3d109cea981e909b72510434f46e6172453b4c9d | A circle with center \( O \) is tangent to the coordinate axes and to the hypotenuse of a \( 30^\circ - 60^\circ - 90^\circ \) triangle \( ABC \), where \( AB = 1 \), \( \angle C = 30^\circ \), and \( BC \) is the hypotenuse. What is the radius of the circle? | [
"\\(\\frac{3 + \\sqrt{3}}{2}\\)"
] | [
"A circle with center \\( O \\) is tangent to the coordinate axes and to the hypotenuse of a \\( 30^\\circ - 60^\\circ - 90^\\circ \\) triangle \\( ABC \\), where \\( AB = 1 \\), \\( \\angle C = 30^\\circ \\), and \\( BC \\) is the hypotenuse. What is the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-214272ac4e35a15dc1fcd0932eda80f0bf67fe35e8b25531c063cf0210cf992e | A circle with center \( O \) is tangent to the coordinate axes and to the hypotenuse of a \( 30^\circ \)-\( 60^\circ \)-\( 90^\circ \) triangle \( ABC \), where \( AB = 1 \). To the nearest hundredth, what is the radius of the circle? | [
"2.37"
] | [
"A circle with center \\( O \\) is tangent to the coordinate axes and to the hypotenuse of a \\( 30^\\circ \\)-\\( 60^\\circ \\)-\\( 90^\\circ \\) triangle \\( ABC \\), where \\( AB = 1 \\). To the nearest hundredth, what is the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d1e254e98c2500cbc6e3cbedc0fc2ce9001d1952f088a18168d2094d6978b018 | A circle with center \( O \) is tangent to the sides \( AB \), \( BC \), and \( AC \) of a triangle \( ABC \) at points \( E \), \( F \), and \( D \) respectively. The lines \( AO \) and \( CO \) meet \( EF \) at points \( N \) and \( M \). Let \( P \) be the intersection of the circumcircle of triangle \( OMN \) and l... | [
"30"
] | [
"A circle with center \\( O \\) is tangent to the sides \\( AB \\), \\( BC \\), and \\( AC \\) of a triangle \\( ABC \\) at points \\( E \\), \\( F \\), and \\( D \\) respectively. The lines \\( AO \\) and \\( CO \\) meet \\( EF \\) at points \\( N \\) and \\( M \\). Let \\( P \\) be the intersection of the circumc... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-61bb93a5ddad8fcfd2597f8b77caa854567f6038a32c7fbe1603240c37cbe2fe | A circle with center \( O \) is tangent to the sides of an angle with vertex \( A \) at points \( B \) and \( C \). Let \( M \) be a point on the larger arc \( BC \) of this circle (different from \( B \) and \( C \)) such that \( M \) does not lie on the line \( AO \). Lines \( BM \) and \( CM \) intersect the line \(... | [
"\\dfrac{3\\sqrt{3}}{4} r"
] | [
"A circle with center \\( O \\) is tangent to the sides of an angle with vertex \\( A \\) at points \\( B \\) and \\( C \\). Let \\( M \\) be a point on the larger arc \\( BC \\) of this circle (different from \\( B \\) and \\( C \\)) such that \\( M \\) does not lie on the line \\( AO \\). Lines \\( BM \\) and \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17246c760c094eb409421a26c5e589b68392ae4d8b616c587f2f395d149616ce | A circle with center \( O \) touches the sides of an angle with vertex \( A \) at points \( B \) and \( C \). On the larger arc of this circle with ends \( B \) and \( C \), a point \( M \) (different from points \( B \) and \( C \)) is chosen, which does not belong to the line \( AO \). The lines \( BM \) and \( CM \)... | [
""
] | [
"A circle with center \\( O \\) touches the sides of an angle with vertex \\( A \\) at points \\( B \\) and \\( C \\). On the larger arc of this circle with ends \\( B \\) and \\( C \\), a point \\( M \\) (different from points \\( B \\) and \\( C \\)) is chosen, which does not belong to the line \\( AO \\). The li... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-c6bd564167ef5280709602ab3685c52385755cccabf0d1ecb19c31900a654d22 | A circle with center \( P \) is tangent to the negative \( x \)-axis and \( y \)-axis and is externally tangent to a circle with center \((-6,0)\) and radius \(2\). What is the sum of all possible radii of the circle with center \( P \)? | [
"16"
] | [
"A circle with center \\( P \\) is tangent to the negative \\( x \\)-axis and \\( y \\)-axis and is externally tangent to a circle with center \\((-6,0)\\) and radius \\(2\\). What is the sum of all possible radii of the circle with center \\( P \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dc69d9226d27b2424af1bd5aaf55f48543ab8500fcd27efde4daf2e8f0fe7b6d | A circle with center \((3\alpha, 3\beta)\) and variable radius intersects the rectangular hyperbola \(x^2 - y^2 = 9a^2\) at points \(P\), \(Q\), \(R\), and \(S\). Find the locus of the centroid of triangle \(PQR\). | [
"\\(\\left( \\frac{3\\alpha}{2}, \\frac{3\\beta}{2} \\right)\\)"
] | [
"A circle with center \\((3\\alpha, 3\\beta)\\) and variable radius intersects the rectangular hyperbola \\(x^2 - y^2 = 9a^2\\) at points \\(P\\), \\(Q\\), \\(R\\), and \\(S\\). Find the locus of the centroid of triangle \\(PQR\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aa5fa087bd16036b797d46d3d5eef5bbc0826f987edad1deec64cdc9cd91ee58 | A circle with center \(I\) is entirely inside a circle with center \(O\). Consider all possible chords \(AB\) of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle \(AIB\). | [
"\\dfrac{R^2 - |OI|^2}{2r}"
] | [
"A circle with center \\(I\\) is entirely inside a circle with center \\(O\\). Consider all possible chords \\(AB\\) of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle \\(AIB\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-251bbf9a207f17292217be5d27e69531bb4d294ccfe84f6de7daa1bcbe87111f | A circle with center \(O\) has an area of \(156\pi\). Triangle \(ABC\) is equilateral, \(BC\) is a chord of the circle, \(OA = 4\sqrt{3}\), and point \(O\) is outside the triangle \(ABC\). What is the side length of triangle \(ABC\)? | [
"6"
] | [
"A circle with center \\(O\\) has an area of \\(156\\pi\\). Triangle \\(ABC\\) is equilateral, \\(BC\\) is a chord of the circle, \\(OA = 4\\sqrt{3}\\), and point \\(O\\) is outside the triangle \\(ABC\\). What is the side length of triangle \\(ABC\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-bc83e51856de0f6c6aac3b9da9e9f6054db6d1659ce05c6de96ea5b0275da9cc | A circle with center \(O\) has an area of \(28\pi\). Points \(A\) and \(B\) lie within the circle such that \(OA = 4\), \(OB = 1\), and \(\angle AOB = 120^\circ\). Let \(P\) be the point on the circle for which \(\angle APB\) is maximized. Find \(AP + BP\). | [
"5\\sqrt{3}"
] | [
"A circle with center \\(O\\) has an area of \\(28\\pi\\). Points \\(A\\) and \\(B\\) lie within the circle such that \\(OA = 4\\), \\(OB = 1\\), and \\(\\angle AOB = 120^\\circ\\). Let \\(P\\) be the point on the circle for which \\(\\angle APB\\) is maximized. Find \\(AP + BP\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-9f930986f050c29090da577d6689d1894146da3a76ea19669657799e5c1c2f68 | A circle with center at $ I $ tangent to the sides $ BC $, $ AC $, and $ AB $ at points $ D $, $ E $, and $ F $, respectively, is inscribed in the acute triangle $ ABC $. Let the circumscribed circle of $ \Delta ABC $ intersect the line $ EF $ at the points $ P $ and $ Q $. Denote by $ O_1 $ and $ O_2 $ the centers of ... | [
"5"
] | [
"A circle with center at $ I $ tangent to the sides $ BC $, $ AC $, and $ AB $ at points $ D $, $ E $, and $ F $, respectively, is inscribed in the acute triangle $ ABC $. Let the circumscribed circle of $ \\Delta ABC $ intersect the line $ EF $ at the points $ P $ and $ Q $. Denote by $ O_1 $ and $ O_2 $ the cente... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-b1ba5b74edfe2bc8435b1cac66990e141828b3d1c23b09398290d4e14ecf8c43 | A circle with center at $O$ is circumscribed about the triangle $ABC$ $(\angle S>\pi /2)$. The point $F$ is the midpoint of the larger of the arcs subtended by the chord $BC$. Let us designate the point of intersection of the side $BC$ with the extension of the radius $AO$ as $E$ and with the chord $AF$ as $P$. Assume ... | [
"22"
] | [
"A circle with center at $O$ is circumscribed about the triangle $ABC$ $(\\angle S>\\pi /2)$. The point $F$ is the midpoint of the larger of the arcs subtended by the chord $BC$. Let us designate the point of intersection of the side $BC$ with the extension of the radius $AO$ as $E$ and with the chord $AF$ as $P$. ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7974f7d3f43166a31bdef1727caebeb36b0473c2d7a247b0dbe0c27e10daffaa | A circle with center at $O$ is inscribed in a trapezoid $A BCD$ with bases $BC$ and $AD$ and with the nonparallel sides $AB$ and $CD$. Find the area of the trapezoid if $\angle DAB$ is a right angle and $| OC | =2$, $| OD | = 4$. | [
"14.4"
] | [
"A circle with center at $O$ is inscribed in a trapezoid $A BCD$ with bases $BC$ and $AD$ and with the nonparallel sides $AB$ and $CD$. Find the area of the trapezoid if $\\angle DAB$ is a right angle and $| OC | =2$, $| OD | = 4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f12334aab44940087b8f3a707cd9313e83daff199a383e9aebcb86a4f0dc9d11 | A circle with center at \( O \) is inscribed in quadrilateral \( ABCD \) with points of tangency at \( E \) (on \( AB \)), \( F \) (on \( BC \)), \( G \) (on \( CD \)), and \( H \) (on \( AD \)). Given that \( AB \parallel DC \), \( AD = 337 \), \( BE = 62 \), \( FC = 225 \), and \( DG = AE + 175 \), find the ratio \( ... | [
"\\(\\frac{16}{9}\\)"
] | [
"A circle with center at \\( O \\) is inscribed in quadrilateral \\( ABCD \\) with points of tangency at \\( E \\) (on \\( AB \\)), \\( F \\) (on \\( BC \\)), \\( G \\) (on \\( CD \\)), and \\( H \\) (on \\( AD \\)). Given that \\( AB \\parallel DC \\), \\( AD = 337 \\), \\( BE = 62 \\), \\( FC = 225 \\), and \\( D... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-4d9f8f61c91cba94ffe5f92a96da1d4171ee91ce619a168ce5db110a56284812 | A circle with center at \( O \) is inscribed in quadrilateral \( ABCD \) with points of tangency at \( E \) (on \( AB \)), \( F \) (on \( BC \)), \( G \) (on \( CD \)), and \( H \) (on \( AD \)). \( AB \) is parallel to \( DC \), but \( AD \) is not parallel to \( BC \). Given \( AD = 337 \), \( BE = 62 \), \( FC = 225... | [
"25"
] | [
"A circle with center at \\( O \\) is inscribed in quadrilateral \\( ABCD \\) with points of tangency at \\( E \\) (on \\( AB \\)), \\( F \\) (on \\( BC \\)), \\( G \\) (on \\( CD \\)), and \\( H \\) (on \\( AD \\)). \\( AB \\) is parallel to \\( DC \\), but \\( AD \\) is not parallel to \\( BC \\). Given \\( AD = ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-85a154ea57425650fa2515c31ab3c2e083630a26621e34dd433c26426302b06b | A circle with center at point $O$ is inscribed in the quadrilateral $ABCD$. Straight lines $\ell_A, \ell_B, \ell_C, \ell_D$ are drawn through points $A, B, C, D$ perpendicular to $OA$, $OB$, $OC$, $OD$, respectively, and $\ell_A \cap \ell_B = K$, $\ell_B \cap \ell_C = L$, $\ell_C \cap \ell_D = M$, $\ell_D \cap \ell_A =... | [
"0"
] | [
"A circle with center at point $O$ is inscribed in the quadrilateral $ABCD$. Straight lines $\\ell_A, \\ell_B, \\ell_C, \\ell_D$ are drawn through points $A, B, C, D$ perpendicular to $OA$, $OB$, $OC$, $OD$, respectively, and $\\ell_A \\cap \\ell_B = K$, $\\ell_B \\cap \\ell_C = L$, $\\ell_C \\cap \\ell_D = M$, $\\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-b33019f3ad59373f6f2dfcb561c7e25882e61c19c06a5c774236f5eb044409a7 | A circle with center at the origin and radius $5$ intersects the x-axis at points $A$ and $B$. Let $P$ be a point on the line $x = 11$, and let $Q$ be the intersection point of line $AP$ with the circle such that $Q$ is the midpoint of $AP$. Find the coordinates of point $P$. | [
"\\( (11, 8) \\) or \\( (11, -8) \\)"
] | [
"A circle with center at the origin and radius $5$ intersects the x-axis at points $A$ and $B$. Let $P$ be a point on the line $x = 11$, and let $Q$ be the intersection point of line $AP$ with the circle such that $Q$ is the midpoint of $AP$. Find the coordinates of point $P$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 10
} | 11 | 0 | 0 |
omr-ws-3cbd94745224a30db112fa5e46a80eac0888fe9a890e6cd8b6d5e103402b211f | A circle with circumference \(6n\) units is given, and \(3n\) points divide the circumference into \(n\) intervals of 1 unit, \(n\) intervals of 2 units, and \(n\) intervals of 3 units. Let \(P\) be the number of pairs of points that are diametrically opposite to each other. Calculate the minimum possible value of \(P\... | [
"n"
] | [
"A circle with circumference \\(6n\\) units is given, and \\(3n\\) points divide the circumference into \\(n\\) intervals of 1 unit, \\(n\\) intervals of 2 units, and \\(n\\) intervals of 3 units. Let \\(P\\) be the number of pairs of points that are diametrically opposite to each other. Calculate the minimum possi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-192c5b1296275c816690722c2b4ad22d0091a843163d7fa62bda3db19c0cd09e | A circle with diameter $\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. A square $ABCD$ is constructed such that $A$ and $B$ are on the larger circle, $\overline{CD}$ is tangent at $Q$ to the smaller circle, and the smaller circle is outside $ABCD$. The length of $\overline{AB}$ can b... | [
"312"
] | [
"A circle with diameter $\\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. A square $ABCD$ is constructed such that $A$ and $B$ are on the larger circle, $\\overline{CD}$ is tangent at $Q$ to the smaller circle, and the smaller circle is outside $ABCD$. The length of $\\overline{AB... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-df758fffc0896feff4c492213e4623e249cbf6fe4798455503ce206a36d3f390 | A circle with diameter $\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$. The length of $\overline{AB}$ can be written in the ... | [
"312"
] | [
"A circle with diameter $\\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$. The length of $\\overline{AB}$ can be written... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 8
} | 56 | 0 | 0 |
omr-ws-00035fe27b98952d823bb7e9464e7287ee2dbca9f79601b8486d9e3ef9211283 | A circle with diameter 2 is tangent to the diagonals of a square with side length 2. The circle intersects the square at points P and Q. Find the length of segment PQ. | [
"\\( 2\\sqrt{2\\sqrt{2}-2} \\)"
] | [
"A circle with diameter 2 is tangent to the diagonals of a square with side length 2. The circle intersects the square at points P and Q. Find the length of segment PQ."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-e85e42363175649ed156232247baf1ddf513558a879bb27e55b01aaa2c756f40 | A circle with diameter 2 is tangent to the diagonals of a square with side length 2. The circle intersects the square at points \( P \) and \( Q \). Find the length of segment \( PQ \). | [
"\\( 2\\sqrt{2\\sqrt{2} - 2} \\)"
] | [
"A circle with diameter 2 is tangent to the diagonals of a square with side length 2. The circle intersects the square at points \\( P \\) and \\( Q \\). Find the length of segment \\( PQ \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-7a58f25811955f863e83a40d06228ac895466423cbae6f4d2a5bea5a91885f50 | A circle with diameter 3 is internally tangent to two externally tangent circles with diameters 1 and 2. Find the radius of the circle that is externally tangent to all three of them. | [
"\\dfrac{3}{7}"
] | [
"A circle with diameter 3 is internally tangent to two externally tangent circles with diameters 1 and 2. Find the radius of the circle that is externally tangent to all three of them."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ad533b56b06b91eecb68b500d4230080e8414d48027c72d0b3205441a6a6f695 | A circle with diameter \(AB\) intersects a circle with center \(A\) at points \(C\) and \(D\). The point \(E\) is the intersection of \(AB\) and \(CD\). Given that \(P\) is a point on the circle such that \(PC = 16\) cm, \(PD = 28\) cm, and \(PE = 14\) cm, find the length of \(PB\) in cm. | [
"49"
] | [
"A circle with diameter \\(AB\\) intersects a circle with center \\(A\\) at points \\(C\\) and \\(D\\). The point \\(E\\) is the intersection of \\(AB\\) and \\(CD\\). Given that \\(P\\) is a point on the circle such that \\(PC = 16\\) cm, \\(PD = 28\\) cm, and \\(PE = 14\\) cm, find the length of \\(PB\\) in cm."
... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-362ef566a8f82953ce3405f8a2aa5ab9fb3f9b497ae7535314250c1f94afd284 | A circle with diameter \(AB\) is drawn, and point \(P\) is chosen on segment \(AB\) such that \(\frac{AP}{AB} = \frac{1}{42}\). Two new circles \(a\) and \(b\) are drawn with diameters \(AP\) and \(PB\) respectively. The perpendicular line to \(AB\) passing through \(P\) intersects the circle twice at points \(S\) and ... | [
"\\(\\frac{205}{1681}\\)"
] | [
"A circle with diameter \\(AB\\) is drawn, and point \\(P\\) is chosen on segment \\(AB\\) such that \\(\\frac{AP}{AB} = \\frac{1}{42}\\). Two new circles \\(a\\) and \\(b\\) are drawn with diameters \\(AP\\) and \\(PB\\) respectively. The perpendicular line to \\(AB\\) passing through \\(P\\) intersects the circle... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b79168589fc4aa5ece06c29cf09a5e243cbeae635146651170a060d5c35ec1d | A circle with diameter \(AB\) is given. Let \(C\) be an arbitrary point on the diameter \(AB\). Let \(D\) be the center of the circle tangent to the given circle and tangent to \(AB\) at \(C\). The circle with diameter \(BC\) meets \(BD\) at \(P\). Let \(O\) be the circumcenter of \(\triangle APC\). Find the locus of \... | [
"The locus of \\(O\\) is a line."
] | [
"A circle with diameter \\(AB\\) is given. Let \\(C\\) be an arbitrary point on the diameter \\(AB\\). Let \\(D\\) be the center of the circle tangent to the given circle and tangent to \\(AB\\) at \\(C\\). The circle with diameter \\(BC\\) meets \\(BD\\) at \\(P\\). Let \\(O\\) be the circumcenter of \\(\\triangle... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-e8bf1712bd3bce864c5908d65fd502e45d267d5373fd2dd46a6cfb9f5d2ca0c3 | A circle with diameter \(\overline{AC}\) is intersected by a secant at points \( B \) and \( D \). The secant and the diameter intersect at point \( P \) outside the circle. Perpendiculars \(\overline{AE}\) and \(\overline{CF}\) are drawn from the extremities of the diameter to the secant. If \( EB = 2 \) and \( BD = 6... | [
"2"
] | [
"A circle with diameter \\(\\overline{AC}\\) is intersected by a secant at points \\( B \\) and \\( D \\). The secant and the diameter intersect at point \\( P \\) outside the circle. Perpendiculars \\(\\overline{AE}\\) and \\(\\overline{CF}\\) are drawn from the extremities of the diameter to the secant. If \\( EB... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-32f9833e1a836a8bfd31151bb68c9b7c83479444c3eabf1888f8f0b8e65150b9 | A circle with equation (x-3)² + (y-3)² = 16 is rotated around the line y=x. What is the volume of the solid formed? | [
"\\(\\frac{256\\pi}{3}\\)"
] | [
"A circle with equation (x-3)² + (y-3)² = 16 is rotated around the line y=x. What is the volume of the solid formed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-1cc0b66b4fdbd9538dae4c3993c415bd7deff1e05e463d6fb2156d86946ba230 | A circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$. What is the ratio $\frac{c_{14}}{c... | [
"17"
] | [
"A circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \\le i \\le 14$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers and $c_1 \\le c_2 \\le \\cdots \\le c_{14}$. What is the ratio $\\fr... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 7,
"tir": 1
} | 38 | 0 | 0 |
omr-ws-f5cf93cb7d4d0b0e39bba49140a039dbd74f8855432440002686224e69f768ab | A circle with its center on the line \(x + 2y = 4\) passes through the points \(A(6,9)\) and \(B(12,-9)\). Write the equation of the circle. | [
"\\((x-6)^{2}+(y+1)^{2}=100\\)"
] | [
"A circle with its center on the line \\(x + 2y = 4\\) passes through the points \\(A(6,9)\\) and \\(B(12,-9)\\). Write the equation of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-08ee591dfbe05e224a335db4f69317fcbbdebf687c2173c1220c2b15b33126ec | A circle with radius $1$ is inscribed in a convex quadrilateral $ABCD$, tangent to the sides $AB, BC, CD, DA$ at points $K, L, M, N$ respectively. Given that $\angle KLM = 4 \angle AKN$ and $\angle KNM = 4 \angle BKL$, calculate the length of the segment $LN$. | [
"\\sqrt{2}"
] | [
"A circle with radius $1$ is inscribed in a convex quadrilateral $ABCD$, tangent to the sides $AB, BC, CD, DA$ at points $K, L, M, N$ respectively. Given that $\\angle KLM = 4 \\angle AKN$ and $\\angle KNM = 4 \\angle BKL$, calculate the length of the segment $LN$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-b4decaecca834e5d0193718580f2919b56ef18de5e1bebfafa7a77d044ce349e | A circle with radius $1$ is inscribed in a square and circumscribed about another square. What is the ratio of the circle's shaded area to the area between the two squares? | [
"\\dfrac{\\pi - 2}{2}"
] | [
"A circle with radius $1$ is inscribed in a square and circumscribed about another square. What is the ratio of the circle's shaded area to the area between the two squares?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-1938ad82b3bfcf4b5665e2a473862e8cb3df8373afdd26ed37e3bd5f76c7b6c2 | A circle with radius $1$ is internally inscribed in a circle of radius $3$. Find the area of the largest triangle whose vertices are the tangent point, a point on the smaller circle, and a point on the larger circle. | [
"2"
] | [
"A circle with radius $1$ is internally inscribed in a circle of radius $3$. Find the area of the largest triangle whose vertices are the tangent point, a point on the smaller circle, and a point on the larger circle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b5f7b8d4f0a5b639a6e8b17f8835aa0343a5561f2f846d16ac305c3814c93a4 | A circle with radius $1$ is tangent to a line $l$ at point $a$. The circle is rolled on line $l$ at an angle $\theta$ until point $a$ makes a full rotation around the circle. The path that $a$ traveled from its initial position to its current position has length $m \pi$. What is $m$? | [
"$\\frac{8}{\\pi}$"
] | [
"A circle with radius $1$ is tangent to a line $l$ at point $a$. The circle is rolled on line $l$ at an angle $\\theta$ until point $a$ makes a full rotation around the circle. The path that $a$ traveled from its initial position to its current position has length $m \\pi$. What is $m$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-bc65cce12ad2195682ab7df9a71614cd31800c2985aace87a68456a5f7f4218e | A circle with radius $1$ rolls around the inside of a right triangle with side lengths $6$, $8$, and $10$, always being tangent to one of the sides. If $P$ is a point on the circle, how far does $P$ travel when the circle makes a complete trip around the triangle? | [
"12 + 2\\pi"
] | [
"A circle with radius $1$ rolls around the inside of a right triangle with side lengths $6$, $8$, and $10$, always being tangent to one of the sides. If $P$ is a point on the circle, how far does $P$ travel when the circle makes a complete trip around the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-9eb62b96b4ea123889c30509d74317af45a8bbbf1cf078ee0287f8d2c1d447b2 | A circle with radius $2$ and an equilateral triangle with side length $s$ are concentric, with the triangle entirely inside the circle. The three circles whose diameters are the sides of the triangle intersect the larger circle at $6$ points equally spaced around its circumference. Find $s$. | [
"\\(\\sqrt{13} - 1\\)"
] | [
"A circle with radius $2$ and an equilateral triangle with side length $s$ are concentric, with the triangle entirely inside the circle. The three circles whose diameters are the sides of the triangle intersect the larger circle at $6$ points equally spaced around its circumference. Find $s$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b8b00d9b1bb65bcce2c2986d07872cf4be0eb091783f4fcc4e34c365d9b6a71 | A circle with radius $3c/2$ passes through the origin $O(0,0)$ and intersects the coordinate axes at points $A$ and $B$. Find the locus of the centroid of triangle $OAB$. | [
"\\(x^2 + y^2 = c^2\\)"
] | [
"A circle with radius $3c/2$ passes through the origin $O(0,0)$ and intersects the coordinate axes at points $A$ and $B$. Find the locus of the centroid of triangle $OAB$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 13
} | 30 | 0 | 0 |
omr-ws-e0916bbda2046c7720a30b7bd652c85dc5c4541f91f84cbd00ff6324a30394a0 | A circle with radius $4$ is centered at the origin. For any right triangle with legs perpendicular to the coordinate axes inscribed within this circle, a segment is constructed from its incenter to the origin, and the midpoint of this segment is labeled $M$. The locus of all possible points $M$ forms the boundary of a ... | [
"4"
] | [
"A circle with radius $4$ is centered at the origin. For any right triangle with legs perpendicular to the coordinate axes inscribed within this circle, a segment is constructed from its incenter to the origin, and the midpoint of this segment is labeled $M$. The locus of all possible points $M$ forms the boundary ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-2015e8ad0ad6ceeadeccf74deedfa7585d839a3cd570222241443e683d91dda2 | A circle with radius $5$ and center in the first quadrant is tangent to the $y$-axis. The line passing through the origin that is tangent to the circle has a slope of $\frac{1}{2}$. The $y$-coordinate of the center of the circle can be written in the form $\frac{m+\sqrt{n}}{p}$ where $m$, $n$, and $p$ are positive inte... | [
"132"
] | [
"A circle with radius $5$ and center in the first quadrant is tangent to the $y$-axis. The line passing through the origin that is tangent to the circle has a slope of $\\frac{1}{2}$. The $y$-coordinate of the center of the circle can be written in the form $\\frac{m+\\sqrt{n}}{p}$ where $m$, $n$, and $p$ are posit... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d3191f0dbc8a58911ae190d30a51bf9cefde3b6dced99c57516fca4c05194672 | A circle with radius $5$ is inscribed in a right triangle with hypotenuse $34$. What is the area of the triangle? | [
"195"
] | [
"A circle with radius $5$ is inscribed in a right triangle with hypotenuse $34$. What is the area of the triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 17
} | 40 | 0 | 0 |
omr-ws-3f2e57075a10b2f426e4c4990120fd25de2d9a7f5140da97d2f6183819c1915c | A circle with radius $6$ is externally tangent to a circle with radius $24$. Find the area of the triangular region bounded by the three common tangent lines of these two circles. | [
"192"
] | [
"A circle with radius $6$ is externally tangent to a circle with radius $24$. Find the area of the triangular region bounded by the three common tangent lines of these two circles."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-602d595ab8c9fbcea5b79c2a5003939509c99ec55338b2abf022490731fd19ac | A circle with radius $8$ lies on a coordinate plane, tangent to the $x$-axis. The point of tangency is marked, and the circle is rolled such that its center travels $20$ units. Find the length of the chord connecting the new location of the mark and the original point of tangency. | [
"16"
] | [
"A circle with radius $8$ lies on a coordinate plane, tangent to the $x$-axis. The point of tangency is marked, and the circle is rolled such that its center travels $20$ units. Find the length of the chord connecting the new location of the mark and the original point of tangency."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-d93747e11202c1e503af99b865c5bff17de25d4d97a82bf724c937c771a9459f | A circle with radius $R$ is divided into twelve equal parts. The twelve dividing points are connected with the centre of the circle, producing twelve rays. Starting from one of the dividing points, a segment is drawn perpendicular to the next ray in the clockwise sense; from the foot of this perpendicular, another perp... | [
"R(2 + \\sqrt{3})"
] | [
"A circle with radius $R$ is divided into twelve equal parts. The twelve dividing points are connected with the centre of the circle, producing twelve rays. Starting from one of the dividing points, a segment is drawn perpendicular to the next ray in the clockwise sense; from the foot of this perpendicular, another... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5be12244b94bed52f7fd33891de625425131aaf93b24b62190ce6b0b4249a4ca | A circle with radius $\frac{1}{2}$ is inscribed in a square. In the regions outside the circle and inside the square, four smaller circles are drawn. What is the diameter of one of the smaller circles? | [
"\\(3 - 2\\sqrt{2}\\)"
] | [
"A circle with radius $\\frac{1}{2}$ is inscribed in a square. In the regions outside the circle and inside the square, four smaller circles are drawn. What is the diameter of one of the smaller circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-328e25a1af30a8376697e336c89412fd1de44c756adfddd6eebda25f48698952 | A circle with radius $\sqrt{2}$ and center $P$ rolls on the curve $\alpha: y^2 = 4x$. Find the maximum acute angle between the line segment $AP$ and the $x$-axis, where $A = (-3, 0)$. | [
"\\dfrac{\\pi}{4}"
] | [
"A circle with radius $\\sqrt{2}$ and center $P$ rolls on the curve $\\alpha: y^2 = 4x$. Find the maximum acute angle between the line segment $AP$ and the $x$-axis, where $A = (-3, 0)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-029ba9cc32c76b4fa822fe4dd4aadd2c1020beffba25f4695ef136942bc62313 | A circle with radius $r$ rolls along a side of an equilateral triangle with side length $s$. Given that $r$ is equal to the height of the triangle, calculate the measure of the arc intercepted by the sides of the triangle on this circle as the circle rolls along the side. Express your answer in degrees. | [
"120"
] | [
"A circle with radius $r$ rolls along a side of an equilateral triangle with side length $s$. Given that $r$ is equal to the height of the triangle, calculate the measure of the arc intercepted by the sides of the triangle on this circle as the circle rolls along the side. Express your answer in degrees."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-79d4d0fce78fb305ae5258a064ead8c91d59212cd227296d5c86b5b1dee6695a | A circle with radius $r$ rolls around the inside of a circle with radius $3r$. A point on the circumference of the smaller circle traces out a curvilinear triangle. Find the area inside this figure. | [
"\\(\\frac{2}{3} \\pi r^2\\)"
] | [
"A circle with radius $r$ rolls around the inside of a circle with radius $3r$. A point on the circumference of the smaller circle traces out a curvilinear triangle. Find the area inside this figure."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-1e8fa01b936bf75c00b92892aaf2f77bf665d3d9685a1b810640aa24742e16a5 | A circle with radius $r=2$ is inscribed in a right-angle triangle. Given that one of the angles is $\alpha = 67.38^\circ$, calculate the lengths of the sides $a$, $b$, and $c$ of the triangle. | [
"\\(a = 12\\), \\(b = 5\\), \\(c = 13\\)"
] | [
"A circle with radius $r=2$ is inscribed in a right-angle triangle. Given that one of the angles is $\\alpha = 67.38^\\circ$, calculate the lengths of the sides $a$, $b$, and $c$ of the triangle."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 30
} | 30 | 0 | 0 |
omr-ws-13ff1b5f893035fe520b3eaedce68808790894c621d4398b6ac65b2eecd557e5 | A circle with radius $r=6$ is inscribed in a rhombus. If the sum of the lengths of the diagonals of the rhombus is twice the perimeter of the rhombus, what is the side length $a$ of the rhombus? | [
"\\dfrac{4}{5}"
] | [
"A circle with radius $r=6$ is inscribed in a rhombus. If the sum of the lengths of the diagonals of the rhombus is twice the perimeter of the rhombus, what is the side length $a$ of the rhombus?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-552250cd2892f889d49d708c54f87b134a407bb183c86b5e0140eb9e13059d74 | A circle with radius 1 and a square with side length 1 share the same center. Find the area of the shaded region formed by the intersection of the circle and the square. | [
"\\(\\frac{\\pi}{3} + 1 - \\sqrt{3}\\)"
] | [
"A circle with radius 1 and a square with side length 1 share the same center. Find the area of the shaded region formed by the intersection of the circle and the square."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-baf4ab7825e2d34004ee83dd2eacc64d81300a5f52b8df5d919899d5ed4aef55 | A circle with radius 1 circumscribes a pentagon. Find the maximum value of the sum of the squares of the lengths of the pentagon's edges and diagonals. | [
"25"
] | [
"A circle with radius 1 circumscribes a pentagon. Find the maximum value of the sum of the squares of the lengths of the pentagon's edges and diagonals."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 15
} | 37 | 0 | 0 |
omr-ws-91aa29aedfd5c86320e991a10af6de8ad2e3246799be4619bd3129f4bae2835c | A circle with radius 1 is escribed on a right triangle ABC. Points D and E are marked on the hypotenuse AB such that AD = AC and BE = BC. Calculate the length of DE. | [
"2"
] | [
"A circle with radius 1 is escribed on a right triangle ABC. Points D and E are marked on the hypotenuse AB such that AD = AC and BE = BC. Calculate the length of DE."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-c1ef18df3c84c05e08b22870a68fd14eb9e71aa4e3d64fd8910616aabaa52742 | A circle with radius 1 is inscribed in an equilateral triangle. Progressively smaller circles are inscribed at each of the vertices, with each smaller circle being successively congruent and tangent to two sides of the triangle. What is the sum of the circumferences of all the circles? | [
"5\\pi"
] | [
"A circle with radius 1 is inscribed in an equilateral triangle. Progressively smaller circles are inscribed at each of the vertices, with each smaller circle being successively congruent and tangent to two sides of the triangle. What is the sum of the circumferences of all the circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-74bf2871a6ba8fea2334ea69dad4f8cc8a45b2c9e81ecc995d9fdb72719dfefb | A circle with radius 1 is inscribed inside an isosceles triangle \(ABC\) where \(AB = AC\) and \(BC = 2\sqrt{3}\). Find the area of the shaded region, which is the area inside the triangle but outside the circle. | [
"\\(3\\sqrt{3} - \\pi\\)"
] | [
"A circle with radius 1 is inscribed inside an isosceles triangle \\(ABC\\) where \\(AB = AC\\) and \\(BC = 2\\sqrt{3}\\). Find the area of the shaded region, which is the area inside the triangle but outside the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-2b40ad87ab5b50067d2bf0a8719d6c423330f2293a40fc95ce6c37bec5414ed9 | A circle with radius 1 is located in the first quadrant and is tangent to the $x$-axis and the parabola $y=x^2$ at one point each. Find the coordinates of its center. | [
"\\(\\left(\\frac{1}{4}\\sqrt{\\frac{23+\\sqrt{17}}{2}}+\\frac{1}{2}\\sqrt{\\frac{7+\\sqrt{17}}{2}}, 1\\right)\\)"
] | [
"A circle with radius 1 is located in the first quadrant and is tangent to the $x$-axis and the parabola $y=x^2$ at one point each. Find the coordinates of its center."
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-274eb4d0012989ea0518a653ac64bc81fd91269192ccb85c33b9fd90d6982439 | A circle with radius 1 is randomly placed inside a 15 x 36 rectangle \( ABCD \) such that it lies entirely within the rectangle. Find the probability that the circle does not intersect the diagonal \( AC \). | [
"\\(\\frac{375}{442}\\)"
] | [
"A circle with radius 1 is randomly placed inside a 15 x 36 rectangle \\( ABCD \\) such that it lies entirely within the rectangle. Find the probability that the circle does not intersect the diagonal \\( AC \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 39,
"genselect": 5,
"tir": 14
} | 58 | 0 | 0 |
omr-ws-d6760e02c459c2e5bc0725e886b2a128ed7b020020e85966cce2b0f14868ba59 | A circle with radius 1 is tangent to both sides of a 60-degree angle. A second circle, larger than the first, is tangent to the first circle and both sides of the angle. Find the radius \( r \) of the second circle. | [
"3"
] | [
"A circle with radius 1 is tangent to both sides of a 60-degree angle. A second circle, larger than the first, is tangent to the first circle and both sides of the angle. Find the radius \\( r \\) of the second circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-923660cbe8efda5a44c3520f816520bd4fdb826d382b8ea7b6c3e42cb699e7fb | A circle with radius 1 is tangent to the two axes. Smaller circles are drawn tangent to the large circle and the two axes, continuing infinitely. Find the total area of all the circles. | [
"\\(\\frac{4\\pi}{3}\\)"
] | [
"A circle with radius 1 is tangent to the two axes. Smaller circles are drawn tangent to the large circle and the two axes, continuing infinitely. Find the total area of all the circles."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42ef0d11ad0a5ab00c5482b6a6efd704154184517572e78f2256b2c9608c4c4d | A circle with radius 100 is divided into multiple regions by 32 lines. Find the maximum number of regions that can be created and determine the minimum radius of a circle that can be placed in at least one of these regions without intersecting any of the lines. | [
"\\dfrac{100}{33}"
] | [
"A circle with radius 100 is divided into multiple regions by 32 lines. Find the maximum number of regions that can be created and determine the minimum radius of a circle that can be placed in at least one of these regions without intersecting any of the lines."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-678e9b4a5186381278fb11698f89ee7894e714e3cfc7938308a898c84268ca79 | A circle with radius 2 has parallel and congruent chords \( AB \) and \( CD \), each of length 2. What is the area of quadrilateral \( ABCD \)? | [
"\\(4\\sqrt{3}\\)"
] | [
"A circle with radius 2 has parallel and congruent chords \\( AB \\) and \\( CD \\), each of length 2. What is the area of quadrilateral \\( ABCD \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-b3ea1be1fe22d23c13bfcb48807500f4b39bfe73fe5be97c0f8029d66a145c7e | A circle with radius 2 is inscribed in a square. Let \(Y\) be the area of the grey region (the area outside the circle but inside the square), and \(X\) be the area of the black region (the area of the circle). Find \(Y - X\). | [
"\\(4\\pi - 6 - 4\\sqrt{2}\\)"
] | [
"A circle with radius 2 is inscribed in a square. Let \\(Y\\) be the area of the grey region (the area outside the circle but inside the square), and \\(X\\) be the area of the black region (the area of the circle). Find \\(Y - X\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b531105e3f5182ba7c8200da5baf198ab7fe32e8f4048d469949a12b54d2f6b | A circle with radius 2 is placed inside a rectangle ABCD with AB=CD=12 and AD=BC=8, such that it is tangent to two of the sides of the rectangle. If the circle is rolled around the rectangle such that it is always tangent to at least one of the sides until it returns to its original position, find the area of the regio... | [
"\\( 80 + 4\\pi \\)"
] | [
"A circle with radius 2 is placed inside a rectangle ABCD with AB=CD=12 and AD=BC=8, such that it is tangent to two of the sides of the rectangle. If the circle is rolled around the rectangle such that it is always tangent to at least one of the sides until it returns to its original position, find the area of the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 4
} | 13 | 0 | 0 |
omr-ws-f6e2f1c8fe22e8eba3486e6ac4fd019143b32dbc997503af6f47fa638e953be7 | A circle with radius 3 cm intersects a circle with radius 4 cm. At the points of intersection, the radii are perpendicular. What is the difference in the areas of the non-overlapping parts of the circles? | [
"\\( 7\\pi \\)"
] | [
"A circle with radius 3 cm intersects a circle with radius 4 cm. At the points of intersection, the radii are perpendicular. What is the difference in the areas of the non-overlapping parts of the circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-7d0ca707a3ff52d7108d22d16ec98bc3824ec2151af6af47859fa1b7e5c937b9 | A circle with radius 5 has its center at $(8, -3)$. Chord $AB$ has its endpoints on the x-axis with $A (4, 0)$. Point $C$ is on the circle such that $AB = AC$. What is the sum of the coordinates of $C$? | [
"\\(-\\frac{36}{25}\\)"
] | [
"A circle with radius 5 has its center at $(8, -3)$. Chord $AB$ has its endpoints on the x-axis with $A (4, 0)$. Point $C$ is on the circle such that $AB = AC$. What is the sum of the coordinates of $C$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-21803e404192b0db6572d6db0534c7de4307d23df1d342b0c5409b7b0048aef5 | A circle with radius 5 is drawn on a gridded paper where each grid square has sides of length 1. The center of the circle is placed in the middle of one of the squares. All the squares through which the circle passes are colored. How many squares are colored? | [
"40"
] | [
"A circle with radius 5 is drawn on a gridded paper where each grid square has sides of length 1. The center of the circle is placed in the middle of one of the squares. All the squares through which the circle passes are colored. How many squares are colored?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-6827ec4f4f5e0ae92f22145f272c4508bfcafbb5547cf8692ac4fe1b08f84430 | A circle with radius 5 is drawn. A diameter is drawn, intersecting the circle at A and B. Two points are chosen on the circle, such that C is length 5 from A, and D is length 5 from B. Find the sum of all possible values of CD. | [
"15"
] | [
"A circle with radius 5 is drawn. A diameter is drawn, intersecting the circle at A and B. Two points are chosen on the circle, such that C is length 5 from A, and D is length 5 from B. Find the sum of all possible values of CD."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-3e4686763c73229cf92ef1f095359ebb954033cfda96a0443032e0e04fa83a1b | A circle with radius 7 is inscribed in an equilateral triangle. Three smaller circles are inscribed in the triangle, each tangent to two sides of the triangle and the large circle. This process is repeated infinitely with smaller circles. What is the sum of the circumferences of all the circles? | [
"\\(35\\pi\\)"
] | [
"A circle with radius 7 is inscribed in an equilateral triangle. Three smaller circles are inscribed in the triangle, each tangent to two sides of the triangle and the large circle. This process is repeated infinitely with smaller circles. What is the sum of the circumferences of all the circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 17
} | 39 | 0 | 0 |
omr-ws-272cff2c57b2b9e11042a53ec5f3d6d87cd8dea3ebd6e441cbf4297f02ef9b95 | A circle with radius \( 2 \) and an equilateral triangle with side length \( s \) are concentric, with the triangle entirely inside the circle. The three circles whose diameters are the sides of the triangle intersect the larger circle at \( 6 \) points equally spaced around its circumference. Find \( s \). | [
"\\(\\sqrt{33} - 3\\)"
] | [
"A circle with radius \\( 2 \\) and an equilateral triangle with side length \\( s \\) are concentric, with the triangle entirely inside the circle. The three circles whose diameters are the sides of the triangle intersect the larger circle at \\( 6 \\) points equally spaced around its circumference. Find \\( s \\)... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-d826204f1c5d13ca6aa6fc69e84c6aad9fa690e56bee528551fd79d4dd9937d9 | A circle with radius \( R \) is inscribed in an equilateral triangle. Three smaller circles, each with radius \( r \), are inscribed in the corners of the triangle such that each smaller circle touches two sides of the triangle and also touches the larger circle. Find the value of \( \frac{R}{r} \). | [
"3"
] | [
"A circle with radius \\( R \\) is inscribed in an equilateral triangle. Three smaller circles, each with radius \\( r \\), are inscribed in the corners of the triangle such that each smaller circle touches two sides of the triangle and also touches the larger circle. Find the value of \\( \\frac{R}{r} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f188024f37dd6fec63d7821ea2fee172b733a3540014e29a49ffd17ce46e645c | A circle with radius \( r \) is contained within a region bounded by a circle with radius \( R \). The area bounded by the larger circle is \(\frac{a}{b}\) times the area of the region outside the smaller circle and inside the larger circle. Find the ratio \( R : r \). | [
"\\( R : r = \\sqrt{\\frac{a}{a-b}} \\)"
] | [
"A circle with radius \\( r \\) is contained within a region bounded by a circle with radius \\( R \\). The area bounded by the larger circle is \\(\\frac{a}{b}\\) times the area of the region outside the smaller circle and inside the larger circle. Find the ratio \\( R : r \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-1b2c3d885ebf3a5ff86532060f9ba896e1516dff17efca1d83f13df171291df5 | A circle with radius \( r \) is inscribed in quadrilateral \( ABCD \). It touches \( AB \) at \( P \) and \( CD \) at \( Q \). Given \( AP = 19 \), \( PB = 26 \), \( CQ = 37 \), and \( QD = 23 \), find \( r \). | [
"\\( \\sqrt{647} \\)",
"\\(\\sqrt{647}\\)"
] | [
"A circle with radius \\( r \\) is inscribed in quadrilateral \\( ABCD \\). It touches \\( AB \\) at \\( P \\) and \\( CD \\) at \\( Q \\). Given \\( AP = 19 \\), \\( PB = 26 \\), \\( CQ = 37 \\), and \\( QD = 23 \\), find \\( r \\)."
] | [
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-91daa064f1885ad19a77c38263d4a1c8f319314dd1d1f22d902069984378c416 | A circle with radius \( r \) is inscribed in quadrilateral \( ABCD \). The circle touches \( AB \) at \( P \) and \( CD \) at \( Q \). Given \( AP = 19 \), \( PB = 26 \), \( CQ = 37 \), and \( QD = 23 \), find the radius \( r \). | [
"\\( \\sqrt{647} \\)",
"\\(\\sqrt{647}\\)"
] | [
"A circle with radius \\( r \\) is inscribed in quadrilateral \\( ABCD \\). The circle touches \\( AB \\) at \\( P \\) and \\( CD \\) at \\( Q \\). Given \\( AP = 19 \\), \\( PB = 26 \\), \\( CQ = 37 \\), and \\( QD = 23 \\), find the radius \\( r \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 16
} | 26 | 0 | 0 |
omr-ws-59c0c6319f8744a6344ca5ed80435f688403b5345adf70b3fd9de9328106fe7b | A circle with radius \( r \) is rolling inside a circle of radius \( 2r \). What is the locus of a fixed point on the rolling circle as it rolls? | [
"A diameter of the larger circle."
] | [
"A circle with radius \\( r \\) is rolling inside a circle of radius \\( 2r \\). What is the locus of a fixed point on the rolling circle as it rolls?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 9
} | 30 | 0 | 0 |
omr-ws-35eae349ff6c65e021e672ccdb277c3457d762bbc8d8cc695b9f46020b710202 | A circle with radius \( r \) is tangent to sides \( AB \), \( AD \), and \( CD \) of rectangle \( ABCD \) and passes through the midpoint of diagonal \( AC \). What is the area of the rectangle in terms of \( r \)? | [
"\\( 8r^2 \\)"
] | [
"A circle with radius \\( r \\) is tangent to sides \\( AB \\), \\( AD \\), and \\( CD \\) of rectangle \\( ABCD \\) and passes through the midpoint of diagonal \\( AC \\). What is the area of the rectangle in terms of \\( r \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-42ee1fe17091b2ca2696f6dc36c8ce76d4f672c10d3b601cc702068fd5d80a22 | A circle with radius \( r \) touches the sides \( AB, BC, CD, DA \) of a convex quadrilateral \( ABCD \) at points \( E, F, G, H \), respectively. The inradii of the triangles \( EBF, FCG, GDH, HAE \) are equal to \( r_1, r_2, r_3, r_4 \). Calculate the minimum possible value of \( \frac{r_1 + r_2 + r_3 + r_4}{r} \). | [
"4 - 2\\sqrt{2}"
] | [
"A circle with radius \\( r \\) touches the sides \\( AB, BC, CD, DA \\) of a convex quadrilateral \\( ABCD \\) at points \\( E, F, G, H \\), respectively. The inradii of the triangles \\( EBF, FCG, GDH, HAE \\) are equal to \\( r_1, r_2, r_3, r_4 \\). Calculate the minimum possible value of \\( \\frac{r_1 + r_2 + ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-13e10b4c3a85bd6aafce6f7cc7d610786f17df452399a16d41512d470c24cd98 | A circle with radius \(r\) is inscribed in quadrilateral \(ABCD\). It touches \(AM\) at \(P\) and \(CD\) at \(Q\). Given \(AP = 19\), \(PB = 26\), \(CQ = 37\), and \(QD = 23\), find \(r\). | [
"\\( r = 25.42857142857143 \\)"
] | [
"A circle with radius \\(r\\) is inscribed in quadrilateral \\(ABCD\\). It touches \\(AM\\) at \\(P\\) and \\(CD\\) at \\(Q\\). Given \\(AP = 19\\), \\(PB = 26\\), \\(CQ = 37\\), and \\(QD = 23\\), find \\(r\\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-624610f0b6984f220b818cb7ca13283bba2483edbd73312b14b91eff8b10f344 | A circle with radius one is centered at the origin. Four points, $A, B, C$, and $D$, are chosen on the circle so that $AC$ and $BD$ intersect at $\left(\frac{1}{2}, 0\right)$ and $AC \perp BD$. Find the maximum possible area of quadrilateral $ABCD$. | [
"$\\sqrt{3}$"
] | [
"A circle with radius one is centered at the origin. Four points, $A, B, C$, and $D$, are chosen on the circle so that $AC$ and $BD$ intersect at $\\left(\\frac{1}{2}, 0\\right)$ and $AC \\perp BD$. Find the maximum possible area of quadrilateral $ABCD$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ed04e5a579d035629d1e38880bb8aa18ea2ec1a520f7c30ba33ec4597d1897dd | A circle, inscribed in a square \( ABCD \) with side \( a \), touches \( \overline{CD} \) at point \( E \). Find the length of the chord connecting points at which the circle intersects with \( \overline{AE} \). | [
"\\(\\frac{2a\\sqrt{5}}{5}\\)"
] | [
"A circle, inscribed in a square \\( ABCD \\) with side \\( a \\), touches \\( \\overline{CD} \\) at point \\( E \\). Find the length of the chord connecting points at which the circle intersects with \\( \\overline{AE} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-36c12fc5c3f1df40c98bc2a416d62114d220bde826a6ab5c2e22747a3567a974 | A circle-shaped pizza is divided into 31 equal slices. Each slice can be flavored with a combination of tomato, salt, cheese, pineapple, and pepper, where each flavor must include at least one of these components. If no two adjacent slices can have the same flavor, compute the number of ways Jessen can flavor the pizza... | [
"970"
] | [
"A circle-shaped pizza is divided into 31 equal slices. Each slice can be flavored with a combination of tomato, salt, cheese, pineapple, and pepper, where each flavor must include at least one of these components. If no two adjacent slices can have the same flavor, compute the number of ways Jessen can flavor the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-a0067910f43660263d66b045fe37edb5a9afba5dad02059f7611f0cb8f8ed773 | A circular bracelet consists of 420 segments, each colored either red or blue. The majority sequence of the bracelet is formed by taking every group of 3 consecutive segments and taking the color that is the majority for those three. Find the expected number of switches in color in the majority sequence. A switch in co... | [
"105"
] | [
"A circular bracelet consists of 420 segments, each colored either red or blue. The majority sequence of the bracelet is formed by taking every group of 3 consecutive segments and taking the color that is the majority for those three. Find the expected number of switches in color in the majority sequence. A switch ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb6527fe62e20ab86a0e7fb324f116b6826523da2c910ec36bc28f65c92a6634 | A circular castle of radius \(r\) is surrounded by a circular moat of width \(m\) (\(m\) is the shortest distance from each point of the castle wall to its nearest point on shore outside the moat). Life guards are to be placed around the outer edge of the moat, so that at least one life guard can see anyone swimming in... | [
"\\left\\lceil \\dfrac{\\pi}{\\arcsin\\left( \\dfrac{r}{r + m} \\right)} \\right\\rceil"
] | [
"A circular castle of radius \\(r\\) is surrounded by a circular moat of width \\(m\\) (\\(m\\) is the shortest distance from each point of the castle wall to its nearest point on shore outside the moat). Life guards are to be placed around the outer edge of the moat, so that at least one life guard can see anyone ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-749dc06abfbccdc46c2c4b4693ac36536af9f7f0fe1cf7e737dec56da48ae563 | A circular ceiling fan rotates at a constant speed of 80 revolutions per minute. A point halfway between the center of the fan and the outer edge of the fan travels 97,968 inches in 15 minutes. How far does a point on the outer edge of the fan travel in 30 minutes? | [
"391872"
] | [
"A circular ceiling fan rotates at a constant speed of 80 revolutions per minute. A point halfway between the center of the fan and the outer edge of the fan travels 97,968 inches in 15 minutes. How far does a point on the outer edge of the fan travel in 30 minutes?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-5367645e0e724d48dc34e8bf3a5509c3fb50b5cb6ac7ee14a412415989352025 | A circular clock face with radius 20 cm has a smaller disk with radius 10 cm externally tangent to it at the 12 o'clock position. The disk rolls clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow on the disk is next pointing in the upward vertical direction? | [
"4"
] | [
"A circular clock face with radius 20 cm has a smaller disk with radius 10 cm externally tangent to it at the 12 o'clock position. The disk rolls clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow on the disk is next pointing in the upward vertical direction?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 11
} | 30 | 0 | 0 |
omr-ws-5d533311780f2502870883b51412bb3b5d0f178b82ad1ab5af1c62d034f96004 | A circular clock has a radius of $20$ cm, and a series of disks with radii $15$ cm, $10$ cm, $5$ cm, and $1$ cm are stacked on top of each other, each externally tangent to the one below it. The smallest disk has an arrow pointing upwards. The disks roll around each other with the following speeds: the $15$ cm disk at ... | [
"$\\frac{4\\pi}{83s}$"
] | [
"A circular clock has a radius of $20$ cm, and a series of disks with radii $15$ cm, $10$ cm, $5$ cm, and $1$ cm are stacked on top of each other, each externally tangent to the one below it. The smallest disk has an arrow pointing upwards. The disks roll around each other with the following speeds: the $15$ cm dis... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0349a3c543b648df33ef5829afb0f840d127ddb4af56f19f322804c6222d434f | A circular coin of radius 1 cm is dropped near the center of a chessboard with 8 squares on each side, where each square has sides of 2 cm. What is the probability that the coin lands entirely within one square, touching but not crossing any of the lines? Assume the lines have zero width. | [
"0"
] | [
"A circular coin of radius 1 cm is dropped near the center of a chessboard with 8 squares on each side, where each square has sides of 2 cm. What is the probability that the coin lands entirely within one square, touching but not crossing any of the lines? Assume the lines have zero width."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c637abbd8b1bf559f7d6f35af7f6f23759585066da005723f39dddf840914ea1 | A circular coin of radius 1 cm is dropped near the center of a chessboard with 8 squares on each side, where each square has sides of 2 cm. What is the probability that the coin lands so that it is entirely within one square? (The coin may touch but not cross any of the lines, and a line is considered to have zero widt... | [
"0"
] | [
"A circular coin of radius 1 cm is dropped near the center of a chessboard with 8 squares on each side, where each square has sides of 2 cm. What is the probability that the coin lands so that it is entirely within one square? (The coin may touch but not cross any of the lines, and a line is considered to have zero... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61c1a163fa03ba2ac8c5d51dc11acdd5d82032406b3cb8e95dff9c3a06dcdacc | A circular coin of radius 1 cm is dropped near the center of an 8x8 chessboard where each square has a side length of 2 cm. What is the probability that the coin lands entirely within one square? | [
"0"
] | [
"A circular coin of radius 1 cm is dropped near the center of an 8x8 chessboard where each square has a side length of 2 cm. What is the probability that the coin lands entirely within one square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b554f7846b037ba68bfe7408148aecf614452a002741e25b98cadd5ab214ec4e | A circular coin of radius 1 cm is dropped near the center of an 8x8 chessboard, where each square of the chessboard has a side length of 2 cm. What is the probability that the coin lands entirely within one of the squares? | [
"\\( 1 - \\pi \\left(\\frac{13}{48}\\right) \\)"
] | [
"A circular coin of radius 1 cm is dropped near the center of an 8x8 chessboard, where each square of the chessboard has a side length of 2 cm. What is the probability that the coin lands entirely within one of the squares?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3782db60b1bc5738065e7d32ffd223239b1daf72f07134eb6608e632bad67ec5 | A circular dart board is hung on a wall by a nail at the top edge of the board. You throw a dart and hit the board. Assuming you have a uniform probability of hitting any spot on the dart board, what is the probability that the dart lands closer to the nail than to the center of the board? | [
"\\(\\frac{4\\pi-3\\sqrt{3}}{4\\pi}\\)"
] | [
"A circular dart board is hung on a wall by a nail at the top edge of the board. You throw a dart and hit the board. Assuming you have a uniform probability of hitting any spot on the dart board, what is the probability that the dart lands closer to the nail than to the center of the board?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7f3904e4c07adf0e18a9044b8b310dff3283af6f5c3e6289ad4d8a5e6b1de0d4 | A circular dartboard with radius 4 inches has an equilateral triangular bullseye with edge length of 2 inches. What is the probability that 5 out of 50 darts thrown will hit the bullseye? (Round your answer to four decimal places.) | [
"0.0212"
] | [
"A circular dartboard with radius 4 inches has an equilateral triangular bullseye with edge length of 2 inches. What is the probability that 5 out of 50 darts thrown will hit the bullseye? (Round your answer to four decimal places.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-ddc1b2cad788ec977be3caf9441a3d7903c6d21298da8cc1a3fb8db83f03a83e | A circular disc is divided into 12 equal sectors. Each sector is to be colored using one of 6 different colors, with the condition that no two adjacent sectors can have the same color. Find the number of such distinct colorings possible. | [
"244140630"
] | [
"A circular disc is divided into 12 equal sectors. Each sector is to be colored using one of 6 different colors, with the condition that no two adjacent sectors can have the same color. Find the number of such distinct colorings possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-702fb3224ef3cf47ed5d08dffc3e9b104fa35610a22c56e0597dadeb4ddce1ff | A circular disc is divided into 12 equal sectors. Each sector is to be colored using one of six different colors such that no two adjacent sectors have the same color. Find the number of such distinct colorings possible. | [
"\\(5^{12} + 5\\)"
] | [
"A circular disc is divided into 12 equal sectors. Each sector is to be colored using one of six different colors such that no two adjacent sectors have the same color. Find the number of such distinct colorings possible."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f5a1e02548eb73c55c3d7b06d3703f0221c0f474b65794a331f48ea9823256c | A circular disc is partitioned into $n$ equal sectors. Each sector is to be colored with one of four different colors such that no two adjacent sectors are of the same color. Let $a_n$ be the number of ways to color the disc with $n$ sectors. Find an explicit formula for $a_n$. | [
"3^n + 3(-1)^n",
"\\( a_n = \\begin{cases} \n3^n + 3 & \\text{if } n \\text{ is even} \\\\\n3^n + 21 & \\text{if } n \\text{ is odd} \n\\end{cases} \\)"
] | [
"A circular disc is partitioned into $n$ equal sectors. Each sector is to be colored with one of four different colors such that no two adjacent sectors are of the same color. Let $a_n$ be the number of ways to color the disc with $n$ sectors. Find an explicit formula for $a_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-5a9ffd643f427e87aea3eb16c9886ea180bf5fed28f8ec986399b06d2a326af3 | A circular disc is partitioned into 2 equal sectors. Each sector is to be coloured with one of the four different colours provided, and no two adjacent sectors are to be of the same colour. Find the number of ways to colour the disc, denoted as $a_{2}$. | [
"12"
] | [
"A circular disc is partitioned into 2 equal sectors. Each sector is to be coloured with one of the four different colours provided, and no two adjacent sectors are to be of the same colour. Find the number of ways to colour the disc, denoted as $a_{2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-ef28649c1d01c0287862c85a8417d16bedd869a00ef697f226b938053473194b | A circular disc is partitioned into 3 equal sectors. Each sector is to be coloured with one of the four different colours provided, and no two adjacent sectors are to be of the same colour. Find the number of ways to colour the disc, denoted as $a_{3}$. | [
"24"
] | [
"A circular disc is partitioned into 3 equal sectors. Each sector is to be coloured with one of the four different colours provided, and no two adjacent sectors are to be of the same colour. Find the number of ways to colour the disc, denoted as $a_{3}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b3d05a57914ffa5c38b3ee3937198ace533c7af367b0c183b2c43836953127b9 | A circular disc is partitioned into 4 equal sectors. Each sector is to be coloured with one of the four different colours provided, and no two adjacent sectors are to be of the same colour. Find the number of ways to colour the disc, denoted as $a_{4}$. | [
"84"
] | [
"A circular disc is partitioned into 4 equal sectors. Each sector is to be coloured with one of the four different colours provided, and no two adjacent sectors are to be of the same colour. Find the number of ways to colour the disc, denoted as $a_{4}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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