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-3d1b3b36a47e48a4e615fb2def7755a9d5f1a03baa519cfb2ed7e5ba5f9f586d | A circle of radius \( r \) is inscribed in a sector which is one-sixth of a circle of radius 1. Find \( r \). | [
"\\( r = \\frac{1}{3} \\)"
] | [
"A circle of radius \\( r \\) is inscribed in a sector which is one-sixth of a circle of radius 1. Find \\( r \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-cfe91780acdfd0d817648920c915954b540fec7b35c8d4cb607f9d8c88489472 | A circle of radius \( r \) is inscribed in a semicircle with diameter \( a + b \). If \( a = 4 \) and \( b = 6 \), calculate the value of \( r \). | [
"\\dfrac{12}{5}"
] | [
"A circle of radius \\( r \\) is inscribed in a semicircle with diameter \\( a + b \\). If \\( a = 4 \\) and \\( b = 6 \\), calculate the value of \\( r \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-5c6a57880627b765922417b37160cfae6950353fd86120755357450f08f556d4 | A circle of radius \( r \) is inscribed in a triangle. Straight lines tangent to the circle and parallel to the sides of the triangle are drawn, cutting off three smaller triangles. Let \( g \), \( k \), and \( i \) be the radii of the circles inscribed in these smaller triangles. If the radius \( r \) of the original ... | [
"12"
] | [
"A circle of radius \\( r \\) is inscribed in a triangle. Straight lines tangent to the circle and parallel to the sides of the triangle are drawn, cutting off three smaller triangles. Let \\( g \\), \\( k \\), and \\( i \\) be the radii of the circles inscribed in these smaller triangles. If the radius \\( r \\) o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d14317b18b7ecf0e6229c369f00d9aac063a0c88281293a39606eb84a6f75a1f | A circle of radius \( r \) is inscribed in an isosceles trapezoid with a lateral side of length \( a \). Find the distance between the touchpoints of the circle with the lateral sides of the trapezoid. | [
"\\( 2r \\)"
] | [
"A circle of radius \\( r \\) is inscribed in an isosceles trapezoid with a lateral side of length \\( a \\). Find the distance between the touchpoints of the circle with the lateral sides of the trapezoid."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-36238be94afff95717c3b1260ce6b6702534ef6df54d6b783744064aa256f11d | A circle of radius \( r \) is rolled horizontally until the point \( A \) at the top becomes the bottom \( A' \). The distance \( A'F \) is \( r \). The point \( C \) is the intersection of the vertical line through \( A' \) and the circle with diameter \( FG \). What is the area of the square \( A'CDE \)? | [
"\\(\\pi r^2\\)"
] | [
"A circle of radius \\( r \\) is rolled horizontally until the point \\( A \\) at the top becomes the bottom \\( A' \\). The distance \\( A'F \\) is \\( r \\). The point \\( C \\) is the intersection of the vertical line through \\( A' \\) and the circle with diameter \\( FG \\). What is the area of the square \\( ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc7f6d23c452dbeb5c55245b67e23a1a37170e5c06e7f45b5c4dac455200c48b | A circle of radius \( r \) passes through both foci of, and exactly four points on, the ellipse with equation \( x^2 + 16y^2 = 16 \). The set of all possible values of \( r \) is an interval \([a, b)\). What is \( a + b \)? | [
"\\(\\sqrt{15} + 8\\)"
] | [
"A circle of radius \\( r \\) passes through both foci of, and exactly four points on, the ellipse with equation \\( x^2 + 16y^2 = 16 \\). The set of all possible values of \\( r \\) is an interval \\([a, b)\\). What is \\( a + b \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 22
} | 70 | 0 | 0 |
omr-ws-05329d2dc368be9e74d9a7d331a2584f90e8f8b90c4bb6c2e3cd7f0e79b9409f | A circle of radius \( r \) touches a straight line at point \( M \). Two points \( A \) and \( B \) are chosen on this line on opposite sides of \( M \) such that \( MA = MB = a \). Find the radius of the circle passing through \( A \) and \( B \) and touching the given circle. | [
"\\(\\frac{a^2 + 4r^2}{4r}\\)"
] | [
"A circle of radius \\( r \\) touches a straight line at point \\( M \\). Two points \\( A \\) and \\( B \\) are chosen on this line on opposite sides of \\( M \\) such that \\( MA = MB = a \\). Find the radius of the circle passing through \\( A \\) and \\( B \\) and touching the given circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-eb86cd341d4404e0b8c1d25e03c8bfd814224d6368c9960829989e890d94583b | A circle of radius \( r \) touches the parabola \( x^2 + 4ay = 0 \) (where \( a > 0 \)) at its vertex. The center of the circle lies below the vertex, and the circle lies entirely within the parabola. Find the largest possible value of \( r \). | [
"2a"
] | [
"A circle of radius \\( r \\) touches the parabola \\( x^2 + 4ay = 0 \\) (where \\( a > 0 \\)) at its vertex. The center of the circle lies below the vertex, and the circle lies entirely within the parabola. Find the largest possible value of \\( r \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c04875386852758e1f87a09f546a6f8190582dfa253e6cbf41aa3ad8428eb81d | A circle of radius \(r\) is inscribed inside a triangle whose sides are 13, 14, and 15 units long. Find the radius of the circle. | [
"4"
] | [
"A circle of radius \\(r\\) is inscribed inside a triangle whose sides are 13, 14, and 15 units long. Find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b75e9cf3deaec1953c6cb570ee0a3d5f6abb9bcb14e5605915a3324740848f52 | A circle of the radius of 6 cm lies in the interior of a semicircle of the radius of 24 cm and touches the midpoint of the diameter of the semicircle. Find the radius of the smaller circle which touches the given circles, the semicircle and the diameter of the semicircle. | [
"8 cm"
] | [
"A circle of the radius of 6 cm lies in the interior of a semicircle of the radius of 24 cm and touches the midpoint of the diameter of the semicircle. Find the radius of the smaller circle which touches the given circles, the semicircle and the diameter of the semicircle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bf273f0ea8381e83a5579dcd06903b1e6ae4cd82a9d910cfc00e44940779da62 | A circle on a Cartesian plane has a center at point $(a,b)$ and the origin lies inside the circle. Denote the area of parts of the circle which consists of points with coordinates of the same sign as $S^+$, and the rest as $S^-$. Find the value of $S^+-S^-$. | [
"\\( 4ab \\)"
] | [
"A circle on a Cartesian plane has a center at point $(a,b)$ and the origin lies inside the circle. Denote the area of parts of the circle which consists of points with coordinates of the same sign as $S^+$, and the rest as $S^-$. Find the value of $S^+-S^-$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 7,
"tir": 10
} | 39 | 0 | 0 |
omr-ws-1d7c8f40452630ed99532d6f2fb88403e7d02adbb9d939eefd3806789e4db642 | A circle passes through a fixed point \( O \) and intersects two perpendicular lines through \( O \) at points \( P \) and \( Q \). If the line \( PQ \) always passes through a fixed point \( (m, n) \), find the locus of the center of this circle. | [
"\\(\\frac{m}{x} + \\frac{n}{y} = 2\\)"
] | [
"A circle passes through a fixed point \\( O \\) and intersects two perpendicular lines through \\( O \\) at points \\( P \\) and \\( Q \\). If the line \\( PQ \\) always passes through a fixed point \\( (m, n) \\), find the locus of the center of this circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-36b6a86542ac4aca081e0488f1e2da66d009bf1408c1af783b5ee57957482e26 | A circle passes through a vertex \( C \) of a rectangle \( ABCD \) and touches \( AB \) and \( AD \) at points \( M \) and \( N \) respectively. If the distance from \( C \) to the line segment \( MN \) is 5 units, find the area of rectangle \( ABCD \). | [
"25"
] | [
"A circle passes through a vertex \\( C \\) of a rectangle \\( ABCD \\) and touches \\( AB \\) and \\( AD \\) at points \\( M \\) and \\( N \\) respectively. If the distance from \\( C \\) to the line segment \\( MN \\) is 5 units, find the area of rectangle \\( ABCD \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 2
} | 24 | 0 | 0 |
omr-ws-6dca28c3d1e474fd2c8523986827e81f2059ed247d64afdf4dc86af51448447b | A circle passes through both trisection points of side $\overline{AB}$ of square $ABCD$ and intersects $\overline{BC}$ at points $P$ and $Q$. Compute the greatest possible value of $\tan\angle PAQ$. | [
"\\(\\frac{7}{11}\\)"
] | [
"A circle passes through both trisection points of side $\\overline{AB}$ of square $ABCD$ and intersects $\\overline{BC}$ at points $P$ and $Q$. Compute the greatest possible value of $\\tan\\angle PAQ$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-9a288767fdb9e7e94e07bd1222f78bdebd81683c5f67fe3c7c0ef823e72e2882 | A circle passes through point C of a rectangle ABCD and is tangent to AB and AD at points M and N, respectively. If the distance from C to line segment MN is 25 cm, find the area of rectangle ABCD. | [
"625"
] | [
"A circle passes through point C of a rectangle ABCD and is tangent to AB and AD at points M and N, respectively. If the distance from C to line segment MN is 25 cm, find the area of rectangle ABCD."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9af663799fb7c08f00887e575c39371e9135a70d5bec3d1f0c03bbc12d5eb5ca | A circle passes through points \( A \) and \( C \) of triangle \( ABC \) and intersects sides \( AB \) and \( BC \) at points \( D \) and \( E \) respectively. Given that \( \frac{BD}{CE} = \frac{3}{2} \), \( BE = 4 \), \( AD = 5 \), and \( AC = 2\sqrt{7} \), find the measure of angle \( \angle BDC \). | [
"\\( 180^\\circ - \\arccos\\left(-\\frac{1}{2\\sqrt{2}}\\right) \\)"
] | [
"A circle passes through points \\( A \\) and \\( C \\) of triangle \\( ABC \\) and intersects sides \\( AB \\) and \\( BC \\) at points \\( D \\) and \\( E \\) respectively. Given that \\( \\frac{BD}{CE} = \\frac{3}{2} \\), \\( BE = 4 \\), \\( AD = 5 \\), and \\( AC = 2\\sqrt{7} \\), find the measure of angle \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 7,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-98008771ec1ff3649697001b94a775d0573e2e12803469194c9b0306e813a807 | A circle passes through points \( A(0,4) \) and \( B(8,0) \) and has its center on the \( x \)-axis. If point \( C \) lies on the circumference of the circle, what is the greatest possible area of triangle \( ABC \)? | [
"\\( 10(\\sqrt{5}+1) \\)"
] | [
"A circle passes through points \\( A(0,4) \\) and \\( B(8,0) \\) and has its center on the \\( x \\)-axis. If point \\( C \\) lies on the circumference of the circle, what is the greatest possible area of triangle \\( ABC \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-86c3e1166f7bc53f0c6e2e2f5f2700bafdc7ddc9a015405d405337919b975f0b | A circle passes through the origin and the point \(\left(\frac{5}{2}, \frac{1}{2}\right)\) and has \(3x - 2y = 0\) as its diameter. Find the equation of the circle. | [
"\\((x-1)^2+\\left(y-\\frac{3}{2}\\right)^2=\\frac{13}{4}\\)"
] | [
"A circle passes through the origin and the point \\(\\left(\\frac{5}{2}, \\frac{1}{2}\\right)\\) and has \\(3x - 2y = 0\\) as its diameter. Find the equation of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-498317ce92e2db68a2960c9509d7bfb1fe65c892cfb47efaa4a4d459ee101ee9 | A circle passes through the points $(2,0)$ and $(18,0)$ and is tangent to the parabola $y=x^2$. What are the coordinates of the center of the circle? | [
"(10, 2)"
] | [
"A circle passes through the points $(2,0)$ and $(18,0)$ and is tangent to the parabola $y=x^2$. What are the coordinates of the center of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-1f22967b093efecadb830d0d5e1ea1baa2dbe78b60cc2fdc012261c2260f2c20 | A circle passes through the three vertices of an isosceles triangle that has two sides of length 3 and a base of length 2. What is the area of this circle? Express your answer in terms of $\pi$. | [
"\\frac{81}{32}\\pi"
] | [
"A circle passes through the three vertices of an isosceles triangle that has two sides of length 3 and a base of length 2. What is the area of this circle? Express your answer in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-15566e6f782469c12b2c0423c03527dfd2314cd6964cb4115d2ac2ee3db38e65 | A circle passes through the vertex $C$ of a rectangle $ABCD$ and touches its sides $AB$ and $AD$ at points $M$ and $N$ respectively. The distance from $C$ to the line segment $MN$ is $2$. Find the area of rectangle $ABCD$. | [
"4"
] | [
"A circle passes through the vertex $C$ of a rectangle $ABCD$ and touches its sides $AB$ and $AD$ at points $M$ and $N$ respectively. The distance from $C$ to the line segment $MN$ is $2$. Find the area of rectangle $ABCD$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e0e3b71d20db26e83483a6fffde8e2d25b6b03cc32cc46f82837726a60fd3e7 | A circle passes through the vertex of a rectangle \(ABCD\) and touches its sides \(AB\) and \(AD\) at points \(M\) and \(N\) respectively. If the distance from \(C\) to the line segment \(MN\) is equal to 5 units, find the area of the rectangle \(ABCD\). | [
"25"
] | [
"A circle passes through the vertex of a rectangle \\(ABCD\\) and touches its sides \\(AB\\) and \\(AD\\) at points \\(M\\) and \\(N\\) respectively. If the distance from \\(C\\) to the line segment \\(MN\\) is equal to 5 units, find the area of the rectangle \\(ABCD\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f1e59c52255e8c7a8586f608db013462e5bcfad4a75b4be7ce9e6d330c120680 | A circle passes through the vertex of a rectangle \(ABCD\) and touches its sides \(AB\) and \(AD\) at points \(M\) and \(N\) respectively. If the distance from point \(C\) to the line segment \(MN\) is 5 units, find the area of rectangle \(ABCD\). | [
"25"
] | [
"A circle passes through the vertex of a rectangle \\(ABCD\\) and touches its sides \\(AB\\) and \\(AD\\) at points \\(M\\) and \\(N\\) respectively. If the distance from point \\(C\\) to the line segment \\(MN\\) is 5 units, find the area of rectangle \\(ABCD\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 8,
"tir": 1
} | 56 | 0 | 0 |
omr-ws-fd417ceafc98e1d247308bcf4ddf88ebce9dcd3ffabcbbb6a15bd6c82035c1bc | A circle passes through the vertex of a rectangle \(ABCD\) and touches its sides \(AB\) and \(AD\) at points \(M\) and \(N\) respectively. If the distance from point \(C\) to the line segment \(MN\) is equal to 5 units, find the area of rectangle \(ABCD\). | [
"25"
] | [
"A circle passes through the vertex of a rectangle \\(ABCD\\) and touches its sides \\(AB\\) and \\(AD\\) at points \\(M\\) and \\(N\\) respectively. If the distance from point \\(C\\) to the line segment \\(MN\\) is equal to 5 units, find the area of rectangle \\(ABCD\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 2
} | 50 | 0 | 0 |
omr-ws-03bae887ee1c7679bfedbdb11c0834309a906991adb8e347bc88a4c2b28307e0 | A circle passes through the vertices $A$ and $D$ of a square and touches the side $BC$. If the side of the square is $2$ units, find the radius of the circle. | [
"\\(\\frac{5}{4}\\)"
] | [
"A circle passes through the vertices $A$ and $D$ of a square and touches the side $BC$. If the side of the square is $2$ units, find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-9363eecf82130e0d21d88744bef87b2ec17c9afccb2e338b3dd95c5e0a3f9d4d | A circle passes through the vertices of a triangle with side-lengths of \(7\tfrac{1}{2}, 10, 12\tfrac{1}{2}\). What is the radius of the circle? | [
"\\(\\frac{25}{4}\\)"
] | [
"A circle passes through the vertices of a triangle with side-lengths of \\(7\\tfrac{1}{2}, 10, 12\\tfrac{1}{2}\\). What is the radius of the circle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-50354167251a6d23ca25c27248d84320dfb25b41e92eaaa02ce6b57c9296b38a | A circle passes through three non-collinear points \((a, b)\), \((c, d)\), and \((e, f)\). Determine the coordinates of the center of the circle without using trigonometric functions. | [
"\\(\\left( \\frac{b(d^2-f^2+c^2-e^2)+d(f^2-b^2+e^2-a^2)+f(b^2-d^2+a^2-c^2)}{2a(f-d)+2c(b-f)+2e(d-b)}, \\frac{a(c^2-e^2+d^2-f^2)+c(e^2-a^2+f^2-b^2)+e(a^2-c^2+b^2-d^2)}{2b(e-c)+2d(a-e)+2f(c-a)} \\right)\\)"
] | [
"A circle passes through three non-collinear points \\((a, b)\\), \\((c, d)\\), and \\((e, f)\\). Determine the coordinates of the center of the circle without using trigonometric functions."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 1,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-e7c1570e9a3d2482391e64124340e21c824577a53773969dff0251aaa010a66e | A circle passes through two adjacent vertices of a square and is tangent to one side of the square. If the side length of the square is 2, what is the radius of the circle? | [
"\\( \\frac{5}{4} \\)"
] | [
"A circle passes through two adjacent vertices of a square and is tangent to one side of the square. If the side length of the square is 2, what is the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-d00165ee576dc12f13dc9ba7d3132e9fbcf1cc8e54ae7e62b1991b478b00d4d6 | A circle passes through two adjacent vertices of a square and is tangent to the opposite side of the square. If the side length of the square is 3, what is the area of the circle? | [
"\\(\\frac{225\\pi}{64}\\)"
] | [
"A circle passes through two adjacent vertices of a square and is tangent to the opposite side of the square. If the side length of the square is 3, what is the area of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-11f0ec996dcd7238b21863f9860733bd4ab36c4c1df8937c81bb5012e1983351 | A circle passes through two diagonally opposite vertices of a 3-inch by 4-inch rectangle. What is the least possible distance between the center of the circle and a vertex of the rectangle? Express your answer as a common fraction. | [
"\\(\\frac{7}{10}\\)"
] | [
"A circle passes through two diagonally opposite vertices of a 3-inch by 4-inch rectangle. What is the least possible distance between the center of the circle and a vertex of the rectangle? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 37
} | 69 | 0 | 0 |
omr-ws-f06f699888ab96505684d61702980cf643c203200c4fa2fddfe7a4be18b28a7c | A circle passes through vertex $B$ of triangle $ABC$, intersects sides $AB$ and $BC$ at points $K$ and $L$, respectively, and touches side $AC$ at its midpoint $M$. Point $N$ on the arc $BL$ (which does not contain $K$) is such that $\angle LKN = \angle ACB$. Given that triangle $CKN$ is equilateral, find $\angle BAC$. | [
"75 degrees"
] | [
"A circle passes through vertex $B$ of triangle $ABC$, intersects sides $AB$ and $BC$ at points $K$ and $L$, respectively, and touches side $AC$ at its midpoint $M$. Point $N$ on the arc $BL$ (which does not contain $K$) is such that $\\angle LKN = \\angle ACB$. Given that triangle $CKN$ is equilateral, find $\\ang... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8272c8f13f896f53580ccabf3085adbcfb615832755111a842f314726b8acf6c | A circle passes through vertex \( C \) of a rectangle \( ABCD \) and touches sides \( AB \) and \( AD \) at points \( M \) and \( N \) respectively. If the distance between \( C \) and line segment \( MN \) is 5 cm, find the area of the rectangle. | [
"25"
] | [
"A circle passes through vertex \\( C \\) of a rectangle \\( ABCD \\) and touches sides \\( AB \\) and \\( AD \\) at points \\( M \\) and \\( N \\) respectively. If the distance between \\( C \\) and line segment \\( MN \\) is 5 cm, find the area of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9e3ec41d820ca18eff2fd13fdc83ca88008d665f1a33ce2572c5360d6caac6f1 | A circle passes through vertex \( C \) of rectangle \( ABCD \) and touches sides \( AB \) and \( AD \) at points \( M \) and \( N \) respectively. If the distance from \( C \) to the line segment \( MN \) is 5 units, what is the area of rectangle \( ABCD \)? | [
"25"
] | [
"A circle passes through vertex \\( C \\) of rectangle \\( ABCD \\) and touches sides \\( AB \\) and \\( AD \\) at points \\( M \\) and \\( N \\) respectively. If the distance from \\( C \\) to the line segment \\( MN \\) is 5 units, what is the area of rectangle \\( ABCD \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-d0f39f81afe15b7d7c2c3f762f4cd3a0f0ce45ee1f42aca4f8998e48a1ae42dd | A circle passes through vertices A and D and touches side BC of a square with side length 2. What is the radius of the circle? | [
"1.25"
] | [
"A circle passes through vertices A and D and touches side BC of a square with side length 2. What is the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-1889496a6591690caaf542e91cf51d76025fc49f969cd32df521e4942e1f7216 | A circle passing through points \( M \) and \( O \) of \( \triangle MNO \) intersects sides \( MN \) and \( NO \) at points \( P \) and \( Q \) respectively. Given \( \frac{NP}{OQ} = \frac{3}{2} \), \( NQ = 4 \), \( MP = 5 \), and \( MO = 2\sqrt{7} \), find \( \angle NPO \). | [
"90^\\circ + \\arctan\\left(\\frac{1}{\\sqrt{7}}\\right)"
] | [
"A circle passing through points \\( M \\) and \\( O \\) of \\( \\triangle MNO \\) intersects sides \\( MN \\) and \\( NO \\) at points \\( P \\) and \\( Q \\) respectively. Given \\( \\frac{NP}{OQ} = \\frac{3}{2} \\), \\( NQ = 4 \\), \\( MP = 5 \\), and \\( MO = 2\\sqrt{7} \\), find \\( \\angle NPO \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-cf92f5c5e95dae121899ce42bc4e4cfc74e614c066450a1a6465038d3496d7ce | A circle passing through the points $(2, 0)$ and $(1, 7)$ is tangent to the $y$-axis at $(0, r)$. Find all possible values of $r$. | [
"24"
] | [
"A circle passing through the points $(2, 0)$ and $(1, 7)$ is tangent to the $y$-axis at $(0, r)$. Find all possible values of $r$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f5f408f7bd50dc431c0e2d9c67fb620b231e985a92ecc079b8a24381b39df91 | A circle rests in the interior of the parabola \( y = x^2 \) and is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency? | [
"\\(\\frac{1}{2}\\)"
] | [
"A circle rests in the interior of the parabola \\( y = x^2 \\) and is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-a178352ce0299c85c592bbe85488d9af02534c6100ef255936ceb23e3223f01a | A circle rests in the interior of the parabola with equation $y = x^2,$ so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency? | [
"\\frac{1}{2}"
] | [
"A circle rests in the interior of the parabola with equation $y = x^2,$ so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a5d8259f0aaff9b5837299cc9f7b616f16cecc05749a9c95bbb15c03618142c | A circle stands in a plane perpendicular to the ground and a point $A$ lies in this plane exterior to the circle and higher than its bottom. A particle starting from rest at $A$ slides without friction down an inclined straight line until it reaches the circle. Which straight line allows descent in the shortest time? | [
"AM"
] | [
"A circle stands in a plane perpendicular to the ground and a point $A$ lies in this plane exterior to the circle and higher than its bottom. A particle starting from rest at $A$ slides without friction down an inclined straight line until it reaches the circle. Which straight line allows descent in the shortest ti... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 3
} | 6 | 0 | 0 |
omr-ws-37047b037d9704ab4ef597f4572cc8e97588cb623bf46838d316a141048ed493 | A circle that passes through the origin has a rightmost point at $(2,8)$. Compute its radius. | [
"17"
] | [
"A circle that passes through the origin has a rightmost point at $(2,8)$. Compute its radius."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f707a1314b15ebd3565193df67ff0c9a0dde0226868b99a8709a089f216ef789 | A circle through the vertices $A$ and $B$ of $\triangle ABC$ intersects segments $AC$ and $BC$ at points $P$ and $Q$ respectively. Given that $AQ=AC$, $\angle BAQ=\angle CBP$, and $AB=(\sqrt{3}+1)PQ$, find the measures of the angles of $\triangle ABC$. | [
"30^\\circ"
] | [
"A circle through the vertices $A$ and $B$ of $\\triangle ABC$ intersects segments $AC$ and $BC$ at points $P$ and $Q$ respectively. Given that $AQ=AC$, $\\angle BAQ=\\angle CBP$, and $AB=(\\sqrt{3}+1)PQ$, find the measures of the angles of $\\triangle ABC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-f0cc58c3595309c8ca468f39eb4e87aab22cca54e56f352c60a4dd870fb9d141 | A circle through vertices \( A \) and \( B \) of triangle \( ABC \) meets side \( BC \) again at \( D \). A circle through \( B \) and \( C \) meets side \( AB \) at \( E \) and the first circle again at \( F \). The points \( A \), \( E \), \( D \), \( C \) lie on a circle with center \( O \), and the radical center o... | [
"10"
] | [
"A circle through vertices \\( A \\) and \\( B \\) of triangle \\( ABC \\) meets side \\( BC \\) again at \\( D \\). A circle through \\( B \\) and \\( C \\) meets side \\( AB \\) at \\( E \\) and the first circle again at \\( F \\). The points \\( A \\), \\( E \\), \\( D \\), \\( C \\) lie on a circle with center ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-b6c7ed1177b9223aa055daf830dc17d88c5856878c1169a96d0e4cf984166cca | A circle touches a square and goes through two of its vertices. The distance from the midpoint of one side of the square to the point where the circle touches the square is given. Find the area of the square. | [
"16"
] | [
"A circle touches a square and goes through two of its vertices. The distance from the midpoint of one side of the square to the point where the circle touches the square is given. Find the area of the square."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-436f03fa38b711e4e1dce22a98f0384efdeaca4e44169a21a1662233e572a848 | A circle touches sides $DA$, $AB$, $BC$, $CD$ of a quadrilateral $ABCD$ at points $K$, $L$, $M$, $N$, respectively. Let $S_1$, $S_2$, $S_3$, $S_4$ be the incircles of triangles $AKL$, $BLM$, $CMN$, $DNK$. The external common tangents distinct from the sides of $ABCD$ are drawn to $S_1$ and $S_2$, $S_2$ and $S_3$, $S_3$... | [
"3 - 2\\sqrt{2}"
] | [
"A circle touches sides $DA$, $AB$, $BC$, $CD$ of a quadrilateral $ABCD$ at points $K$, $L$, $M$, $N$, respectively. Let $S_1$, $S_2$, $S_3$, $S_4$ be the incircles of triangles $AKL$, $BLM$, $CMN$, $DNK$. The external common tangents distinct from the sides of $ABCD$ are drawn to $S_1$ and $S_2$, $S_2$ and $S_3$, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-b77b9d38ca5ccfed499299dad774e53e1b379a2dd7322f19ae30ef75bf065bd0 | A circle touches the \(x\)-axis and also touches the circle \(x^2 + y^2 = 4\). Find the equation of the locus of its center. | [
"\\( x^2 = 4(y + 1) \\)"
] | [
"A circle touches the \\(x\\)-axis and also touches the circle \\(x^2 + y^2 = 4\\). Find the equation of the locus of its center."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a85369d96ea34c013d36be29a485f591941e54d74b4a407e6fa8aa83fd2b1bff | A circle touches the diameter \(AB\) of a unit circle with center \(O\) at \(T\) where \(OT > 1\). These circles intersect at two different points \(C\) and \(D\). The circle through \(O\), \(D\), and \(C\) meets the line \(AB\) at \(P\) different from \(O\). If \(OT = 2\), calculate the value of \(|PA| \cdot |PB| \cdo... | [
"\\dfrac{36}{25}"
] | [
"A circle touches the diameter \\(AB\\) of a unit circle with center \\(O\\) at \\(T\\) where \\(OT > 1\\). These circles intersect at two different points \\(C\\) and \\(D\\). The circle through \\(O\\), \\(D\\), and \\(C\\) meets the line \\(AB\\) at \\(P\\) different from \\(O\\). If \\(OT = 2\\), calculate the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-281d5c35f082feb20e450d80d108466e1f8e6d349fef5b9881c59d670b48d432 | A circle touches the lateral sides of a trapezoid $ABCD$ at points $B$ and $C$, and its center lies on $AD$. If $AB = 10$, $CD = 14$, and the height of the trapezoid is $12$, calculate the length of the diameter of the circle. | [
"12"
] | [
"A circle touches the lateral sides of a trapezoid $ABCD$ at points $B$ and $C$, and its center lies on $AD$. If $AB = 10$, $CD = 14$, and the height of the trapezoid is $12$, calculate the length of the diameter of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-7b947c9b0288886f0edd5591a3316dd6e8a08b95f5f8b333b9a892a6236e269d | A circle touches the parabola \( y^2 = 4x \) at \( (1,2) \) and also touches its directrix. The \( y \)-coordinate of the point of contact of the circle and the directrix is? | [
"2\\sqrt{2}"
] | [
"A circle touches the parabola \\( y^2 = 4x \\) at \\( (1,2) \\) and also touches its directrix. The \\( y \\)-coordinate of the point of contact of the circle and the directrix is?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-6bc3bef9b0e90e824faf7a9dfda865f56f913cf83f01ac5713c2df348e0a001c | A circle touches the parabola \( y^2 = x \) at the point \((1,1)\) and its directrix is normal to the circle. Determine the radius of the circle. | [
"\\(\\frac{5\\sqrt{5}}{4}\\)"
] | [
"A circle touches the parabola \\( y^2 = x \\) at the point \\((1,1)\\) and its directrix is normal to the circle. Determine the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 18
} | 63 | 0 | 0 |
omr-ws-9554a095f4c40c2d879ba215bce9a2d7ed94b269682411ee9b8dcdc38f0095b1 | A circle touches the side $BC$ and the extensions of the other two sides of the triangle $A BC$. Find the radius of the circle if $|AB|=c$, $\angle BAC=\alpha $, $\angle ABC=\beta $. | [
"$R=c\\frac{\\sin (\\alpha /2)\\cos (\\beta /2)}{\\cos (\\alpha /2+\\beta /2)}$"
] | [
"A circle touches the side $BC$ and the extensions of the other two sides of the triangle $A BC$. Find the radius of the circle if $|AB|=c$, $\\angle BAC=\\alpha $, $\\angle ABC=\\beta $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e5f258a7fcfaf5bf4f3cf308c2a5fbdb9805b67915c7bd794c3081a4cd62a535 | A circle touches the sides $\overline{AB}$ and $\overline{AD}$ of rectangle $ABCD$, passes through the vertex $C$, and intersects $\overline{DC}$ at point $N$. Find the area of trapezoid $ABND$ if $AB=9$ and $AD=8$. | [
"40"
] | [
"A circle touches the sides $\\overline{AB}$ and $\\overline{AD}$ of rectangle $ABCD$, passes through the vertex $C$, and intersects $\\overline{DC}$ at point $N$. Find the area of trapezoid $ABND$ if $AB=9$ and $AD=8$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-a08891b735cf03b8276e8a5eba58c3b3c6610521ac972516258d7fa90348163b | A circle touches the sides AB and AD of a rectangle ABCD at points P and Q, respectively, and passes through vertex C. If the distance from C to the line PQ is 97 units, find the area of the rectangle. | [
"9409",
"\\(97^2\\) or 9409."
] | [
"A circle touches the sides AB and AD of a rectangle ABCD at points P and Q, respectively, and passes through vertex C. If the distance from C to the line PQ is 97 units, find the area of the rectangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-f61d55fa61b3ce3d98791c52a3b6ed15018e462977efa0b79d20a0fc8f5e5a0d | A circle touches the sides \(AB\) and \(AD\) of the rectangle \(ABCD\) and intersects the side \(DC\) at a single point \(F\) and the side \(BC\) at a single point \(E\). Find the area of the trapezoid \(AFCB\) if \(AB = 32\), \(AD = 40\), and \(BE = 1\). | [
"1180"
] | [
"A circle touches the sides \\(AB\\) and \\(AD\\) of the rectangle \\(ABCD\\) and intersects the side \\(DC\\) at a single point \\(F\\) and the side \\(BC\\) at a single point \\(E\\). Find the area of the trapezoid \\(AFCB\\) if \\(AB = 32\\), \\(AD = 40\\), and \\(BE = 1\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ea9668c225d726f41dbc3efb0b284bdec51429c8fa97d72b32654d824eddbc1c | A circle touches the sides of an angle at points A and B. The distance from a point C on the circle to the line AB is 4. Find the sum of the distances from point C to the sides of the angle, given that one of these distances is four times larger than the other. | [
"10"
] | [
"A circle touches the sides of an angle at points A and B. The distance from a point C on the circle to the line AB is 4. Find the sum of the distances from point C to the sides of the angle, given that one of these distances is four times larger than the other."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 1
} | 19 | 0 | 0 |
omr-ws-34fd8a80a8377e87ca08cf456d404cfd5075a98edd5e8b3bda3b54f986bb91f3 | A circle touches the sides of an angle with vertex $A$ at points $B$ and $C$. A line passing through $A$ intersects this circle at points $D$ and $E$. A chord $BX$ is parallel to $DE$. If $M$ is the midpoint of segment $DE$, calculate the ratio $\frac{XC}{XM}$. | [
"3"
] | [
"A circle touches the sides of an angle with vertex $A$ at points $B$ and $C$. A line passing through $A$ intersects this circle at points $D$ and $E$. A chord $BX$ is parallel to $DE$. If $M$ is the midpoint of segment $DE$, calculate the ratio $\\frac{XC}{XM}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b80b179e24e07bb3ab0d8455a7922cb99eded5f0a793bdc08b2629345e522713 | A circle touches the sides of an angle with vertex \(A\) at points \(B\) and \(C\). A line passing through \(A\) intersects this circle at points \(D\) and \(E\). A chord \(BX\) is parallel to \(DE\). If \(M\) is the midpoint of \(DE\), find the ratio \(\frac{XC}{XM}\). | [
"no_valid_answer_found"
] | [
"A circle touches the sides of an angle with vertex \\(A\\) at points \\(B\\) and \\(C\\). A line passing through \\(A\\) intersects this circle at points \\(D\\) and \\(E\\). A chord \\(BX\\) is parallel to \\(DE\\). If \\(M\\) is the midpoint of \\(DE\\), find the ratio \\(\\frac{XC}{XM}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8c0bf24b4cf6002ae19fd3f8bf0d67ee1d5e7703a213725099a9a61124d4c67e | A circle touches the sides of an angle with vertex at point \( M \) at points \( A \) and \( B \). The segment \( AC \) is the diameter of this circle. A line passing through point \( M \) parallel to the side \( MA \) of the angle intersects the diameter \( AC \) at point \( D \) and the segment \( MC \) at point \( N... | [
"8"
] | [
"A circle touches the sides of an angle with vertex at point \\( M \\) at points \\( A \\) and \\( B \\). The segment \\( AC \\) is the diameter of this circle. A line passing through point \\( M \\) parallel to the side \\( MA \\) of the angle intersects the diameter \\( AC \\) at point \\( D \\) and the segment \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-7cb2c21ba767b010604a4ddeefbf56046b0e1aa02bfa61bc8e9c9f55ec584f72 | A circle touches the sides of the angle \(ABC\) at points \(A\) and \(C\). The straight line passing through point \(B\) intersects the circle at points \(D\) and \(E\), and \(AE \parallel BC\). Lines \(AD\) and \(BC\) intersect at point \(F\). Find \(BF\) if \(AB = 1\). | [
"\\(BF = 1\\)."
] | [
"A circle touches the sides of the angle \\(ABC\\) at points \\(A\\) and \\(C\\). The straight line passing through point \\(B\\) intersects the circle at points \\(D\\) and \\(E\\), and \\(AE \\parallel BC\\). Lines \\(AD\\) and \\(BC\\) intersect at point \\(F\\). Find \\(BF\\) if \\(AB = 1\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ed66f4ab64cc941dad3b15028d1021119b1d13db4b77c4fd5f4d68d30f2e687 | A circle touches the sides of the quadrilateral \( ABCD \). The points of tangency are \( R \) with \( AB \), \( S \) with \( BC \), \( T \) with \( CD \), and \( U \) with \( DA \). It is known that \( AU = 1 \), \( DU = 2 \), \( BS = 2 \), and \( CS = 4 \). Calculate the length of \( SU \). | [
"4"
] | [
"A circle touches the sides of the quadrilateral \\( ABCD \\). The points of tangency are \\( R \\) with \\( AB \\), \\( S \\) with \\( BC \\), \\( T \\) with \\( CD \\), and \\( U \\) with \\( DA \\). It is known that \\( AU = 1 \\), \\( DU = 2 \\), \\( BS = 2 \\), and \\( CS = 4 \\). Calculate the length of \\( S... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-288f3bd02035b4b4589fcc179077d7b515ead1f380517ff5404da5efae80b0ba | A circle touches two adjacent sides of a square ABCD of unit area and passes through exactly one of its vertices. Find the radius \( x \) of the circle. | [
"\\( 2 - \\sqrt{2} \\)"
] | [
"A circle touches two adjacent sides of a square ABCD of unit area and passes through exactly one of its vertices. Find the radius \\( x \\) of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f23592babb2f492eb4b975c660c32991b27e6163565ae7fdc98c1d7741d4f19f | A circle with a diameter of $10$ cm is drawn on paper. A cylinder with a circumference of $3$ cm is rolled over the paper, imprinting a line from the circle onto the cylinder. How many self-intersection points does this line have on the cylinder? | [
"\\(0\\)"
] | [
"A circle with a diameter of $10$ cm is drawn on paper. A cylinder with a circumference of $3$ cm is rolled over the paper, imprinting a line from the circle onto the cylinder. How many self-intersection points does this line have on the cylinder?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-884fc6380a101991c12471020778e38e917c733fbb7c86cba1eb3d496872cbbe | A circle with a diameter of $10$ is circumscribed by a right-angled isosceles triangle. Determine the perimeter of the triangle. | [
"\\(30 + 20\\sqrt{2}\\)"
] | [
"A circle with a diameter of $10$ is circumscribed by a right-angled isosceles triangle. Determine the perimeter of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-1fce7026dc85b2473db06a64bd020abd06b836a73447198b36824db39308c865 | A circle with a diameter of 24 inches has a sector of 120 degrees cut out and is then folded into a cone. What is the radius and height of the cone? | [
"Radius = 4, Height = 4√3"
] | [
"A circle with a diameter of 24 inches has a sector of 120 degrees cut out and is then folded into a cone. What is the radius and height of the cone?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-445f50e57c24aa92ad2ca7af0c0cef246a7a4366f5e3a05a330e8525183b9c7b | A circle with a radius of 2 units has its center at $(0, 0)$. A circle with a radius of 7 units has its center at $(15, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction. | [
"\\frac{10}{3}"
] | [
"A circle with a radius of 2 units has its center at $(0, 0)$. A circle with a radius of 7 units has its center at $(15, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-833cf2f810c64f015564d24a528921334199d701c2d6ea58713237149965dd1b | A circle with a radius of 4 is inscribed in a circle with a radius of 5. What is the area of the part of the larger circle that is not covered by the smaller circle? | [
"\\( 9\\pi \\)"
] | [
"A circle with a radius of 4 is inscribed in a circle with a radius of 5. What is the area of the part of the larger circle that is not covered by the smaller circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
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} | 46 | 0 | 0 |
omr-ws-2bbb72e5c2a18d624e6ec9b3e1713a7e4698105cd5a6d75984e39658cdd2b9d6 | A circle with a radius of 6 cm is inscribed in a regular hexagon. Another circle with a radius of 6 cm is circumscribed about another regular hexagon. Find the ratio of the area of the smaller hexagon to the area of the larger hexagon. | [
"\\(\\frac{3}{4}\\)"
] | [
"A circle with a radius of 6 cm is inscribed in a regular hexagon. Another circle with a radius of 6 cm is circumscribed about another regular hexagon. Find the ratio of the area of the smaller hexagon to the area of the larger hexagon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-89380ca1663163e0f947898fad4a2feca3eeebdc001ca6e425d55c31ede5e00d | A circle with center $(0, 0)$ has a diameter of $2122416$ units. Three points on the circle with integer coordinates are chosen to form a triangle. What is the minimum possible distance, in units, between any pair of these three points? | [
"6 \\cdot 1061208"
] | [
"A circle with center $(0, 0)$ has a diameter of $2122416$ units. Three points on the circle with integer coordinates are chosen to form a triangle. What is the minimum possible distance, in units, between any pair of these three points?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d64837d064b82a2d66a68ba74f242419a5da9563a7f81f2d043e92b9e20d1902 | A circle with center $A$ has radius $10$. Point $B$ is in the same plane such that $AB=20$. Let $k$ and $l$ be lines tangent to the circle at points $P$ and $Q$ respectively, which intersect at $B$. Segment $CD$ with $C$ on $k$ and $D$ on $l$ is formed such that $CD \perp AB$, and $CD$ is tangent to the circle. If $E$ ... | [
"400\\sqrt{3}"
] | [
"A circle with center $A$ has radius $10$. Point $B$ is in the same plane such that $AB=20$. Let $k$ and $l$ be lines tangent to the circle at points $P$ and $Q$ respectively, which intersect at $B$. Segment $CD$ with $C$ on $k$ and $D$ on $l$ is formed such that $CD \\perp AB$, and $CD$ is tangent to the circle. I... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b9c725e94954e731445ea29625ab78a0adbd3527712311b719be4cacd6f1a905 | A circle with center $C$ and radius $r$ intersects the square $EFGH$ at $H$ and at $M$, the midpoint of $EF$. If $C$, $E$, and $F$ are collinear and $E$ lies between $C$ and $F$, what is the area of the region outside the circle and inside the square in terms of $r$? | [
"\\(\\frac{44-25\\sin^{-1}\\frac{4}{5}}{50}r^2\\)"
] | [
"A circle with center $C$ and radius $r$ intersects the square $EFGH$ at $H$ and at $M$, the midpoint of $EF$. If $C$, $E$, and $F$ are collinear and $E$ lies between $C$ and $F$, what is the area of the region outside the circle and inside the square in terms of $r$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-cf2b6b7aca28f6836178312e5711efcf1e7ae98efeb0871d6c157fc1a5ccafab | A circle with center $C$ and radius $r$ intersects the square $EFGH$ at $H$ and at $M$, the midpoint of $EF$. If $C, E$ and $F$ are collinear and $E$ lies between $C$ and $F$, what is the area of the region outside the circle and inside the square in terms of $r$? | [
"\\( r^2 \\left( \\frac{22}{25} - \\frac{\\tan^{-1} \\frac{4}{3}}{2} \\right) \\)"
] | [
"A circle with center $C$ and radius $r$ intersects the square $EFGH$ at $H$ and at $M$, the midpoint of $EF$. If $C, E$ and $F$ are collinear and $E$ lies between $C$ and $F$, what is the area of the region outside the circle and inside the square in terms of $r$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 3
} | 9 | 0 | 0 |
omr-ws-0d8aa3668551cdec47fdbe4158eab9bb8456a926c44b4bb3e01ac3b233674861 | A circle with center $C$ is tangent to the positive $x$ and $y$-axes and externally tangent to the circle centered at $(3,0)$ with radius $1$. What is the sum of all possible radii of the circle with center $C$? | [
"8"
] | [
"A circle with center $C$ is tangent to the positive $x$ and $y$-axes and externally tangent to the circle centered at $(3,0)$ with radius $1$. What is the sum of all possible radii of the circle with center $C$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bad60434df9289941d5527c7b0d7fa29c3d987f94f4a851a053e83e27e446113 | A circle with center $O$ and diameter $AB$ has radius $2$. The perpendicular bisector of $AB$ intersects the circle at a point $P_1$, and let $M$ be the midpoint of $OP_1$. The perpendicular bisector of $OP_1$ intersects the circle at a point $P_2$, where $P_2A > P_2B$. The perpendicular bisector of $OB$ intersects the... | [
"18"
] | [
"A circle with center $O$ and diameter $AB$ has radius $2$. The perpendicular bisector of $AB$ intersects the circle at a point $P_1$, and let $M$ be the midpoint of $OP_1$. The perpendicular bisector of $OP_1$ intersects the circle at a point $P_2$, where $P_2A > P_2B$. The perpendicular bisector of $OB$ intersect... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 6
} | 28 | 0 | 0 |
omr-ws-f3c22fef184b32ea530ca0b3a7516508f0ca9b3855fe08f3d953a7948c2e1258 | A circle with center $O$ and radius $1$ contains chord $\overline{AB}$ of length $1$, and point $M$ is the midpoint of $\overline{AB}$. If the perpendicular to $\overline{AO}$ through $M$ intersects $\overline{AO}$ at $P$, compute the area of triangle $MAP$. | [
"\\(\\frac{\\sqrt{3}}{32}\\)"
] | [
"A circle with center $O$ and radius $1$ contains chord $\\overline{AB}$ of length $1$, and point $M$ is the midpoint of $\\overline{AB}$. If the perpendicular to $\\overline{AO}$ through $M$ intersects $\\overline{AO}$ at $P$, compute the area of triangle $MAP$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-45e0137193077788c4e9025c6bc9b7c18221ed90199b977bc798715a9e3b864a | A circle with center $O$ and radius $15$ is given. Let $P$ be a point such that $|OP|=9$. How many of the chords of the circle pass through $P$ and have integer length? | [
"12"
] | [
"A circle with center $O$ and radius $15$ is given. Let $P$ be a point such that $|OP|=9$. How many of the chords of the circle pass through $P$ and have integer length?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 9
} | 32 | 0 | 0 |
omr-ws-edac37801881ea82ff68142ad70be517f71903717e8257457bca5e78671ed16b | A circle with center $O$ and radius $r$ touches the sides $BA$ and $BC$ of an acute triangle $ABC$ at points $M$ and $N$, respectively. The straight line passing through point $M$ parallel to $BC$ intersects the ray $BO$ at point $K$. On the ray $MN$, point $T$ is chosen such that $\angle MTK = \frac{1}{2} \angle ABC$.... | [
"2a"
] | [
"A circle with center $O$ and radius $r$ touches the sides $BA$ and $BC$ of an acute triangle $ABC$ at points $M$ and $N$, respectively. The straight line passing through point $M$ parallel to $BC$ intersects the ray $BO$ at point $K$. On the ray $MN$, point $T$ is chosen such that $\\angle MTK = \\frac{1}{2} \\ang... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-44dff3616e1d726b59f2fdf85840f6531d509f61dc7330611023e70474aa42b3 | A circle with center $O$ has an integer diameter. A point $P$ inside the circle is such that $OP = 12$, and there are exactly 5 chords of integer length passing through $P$. The sum of the radii of all such circles can be expressed as $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime. Find the remainder when $m + ... | [
"687"
] | [
"A circle with center $O$ has an integer diameter. A point $P$ inside the circle is such that $OP = 12$, and there are exactly 5 chords of integer length passing through $P$. The sum of the radii of all such circles can be expressed as $\\dfrac{m}{n}$, where $m$ and $n$ are relatively prime. Find the remainder when... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-84ad3a3137c9af2c9d44ba38a84cf1e4aae99fba0e7557be7f1961e05e27672b | A circle with center $O$ has radius $8$ units and circle $P$ has radius $2$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. What is the length of segment $OS$? Express your answer in simpl... | [
"8\\sqrt{2}"
] | [
"A circle with center $O$ has radius $8$ units and circle $P$ has radius $2$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. What is the length of segment $OS$? Express your answer... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ec841f41c537cd8fef870eb6af18b4e4185511053765c0ac1cb0d56dafb1bb2 | A circle with center $O$ has radius 25. Chord $\overline{AB}$ has length 30 and chord $\overline{CD}$ has length 14. These chords intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive i... | [
"057"
] | [
"A circle with center $O$ has radius 25. Chord $\\overline{AB}$ has length 30 and chord $\\overline{CD}$ has length 14. These chords intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime po... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 16
} | 63 | 0 | 0 |
omr-ws-6caededfd9d04b321eb02031aa4ec3b3df060bdf9c83c7f353856e351cd5e1e3 | A circle with center $O$ has radius 25. Chord $\overline{AB}$ of length 30 and chord $\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find th... | [
"057",
"57"
] | [
"A circle with center $O$ has radius 25. Chord $\\overline{AB}$ of length 30 and chord $\\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive inte... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 63,
"tir": 21
} | 84 | 0 | 0 |
omr-ws-f6ad76fa4afa82f15ebefb939385dfc7ea5f865816e9dc46f48c38ea3d5617dd | A circle with center $O$ is circumscribed around a quadrilateral $ABCD$ with perpendicular diagonals. Points $P$ and $Q$ are the midpoints of arcs $ABC$ and $ADC$ of this circle. Line $BO$ intersects segment $CQ$ at point $R$. On side $AB$, point $S$ is chosen such that the lines $AQ$ and $RS$ are parallel. If the leng... | [
"13"
] | [
"A circle with center $O$ is circumscribed around a quadrilateral $ABCD$ with perpendicular diagonals. Points $P$ and $Q$ are the midpoints of arcs $ABC$ and $ADC$ of this circle. Line $BO$ intersects segment $CQ$ at point $R$. On side $AB$, point $S$ is chosen such that the lines $AQ$ and $RS$ are parallel. If the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-5ba613b3477e7db34d4ca18d6782774b1ee3f9c0ef9cb8e551dfc4a5b5a41b26 | A circle with center $O$ is inscribed in a quadrilateral $ABCD$. Given $OA=5$, $OB=6$, $OC=7$, and $OD=8$, where $M$ and $N$ are the midpoints of $AC$ and $BD$ respectively, compute the ratio $OM:ON$. | [
"\\dfrac{6}{7}"
] | [
"A circle with center $O$ is inscribed in a quadrilateral $ABCD$. Given $OA=5$, $OB=6$, $OC=7$, and $OD=8$, where $M$ and $N$ are the midpoints of $AC$ and $BD$ respectively, compute the ratio $OM:ON$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-748f64de573f0c89badefd15a9ee411203f91fcb2835539bae71613fdad39d96 | A circle with center $O$ is internally tangent to two circles inside it, with centers $O_1$ and $O_2$, at points $S$ and $T$ respectively. Suppose the two circles inside intersect at points $M$ and $N$, with $N$ closer to $ST$. If $S, N, T$ are collinear, calculate the value of $\frac{SO_1}{OO_1} + \frac{OO_2}{TO_2}$. | [
"2"
] | [
"A circle with center $O$ is internally tangent to two circles inside it, with centers $O_1$ and $O_2$, at points $S$ and $T$ respectively. Suppose the two circles inside intersect at points $M$ and $N$, with $N$ closer to $ST$. If $S, N, T$ are collinear, calculate the value of $\\frac{SO_1}{OO_1} + \\frac{OO_2}{T... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-030afdd3928452209ef1e039e508ee3b6d4600a767fd6976a69cd648966bea3a | A circle with center $O$ passes through the ends of the hypotenuse $AB$ of a right-angled triangle $ABC$ and intersects legs $AC$ and $BC$ at points $M$ and $K$, respectively. If the length of the hypotenuse $AB$ is $20$, find the distance from point $O$ to line $MK$. | [
"10"
] | [
"A circle with center $O$ passes through the ends of the hypotenuse $AB$ of a right-angled triangle $ABC$ and intersects legs $AC$ and $BC$ at points $M$ and $K$, respectively. If the length of the hypotenuse $AB$ is $20$, find the distance from point $O$ to line $MK$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-63369e8d1bb2e31f1ca56f65fe46ab366cfa6317e24d207c5956d692187d0ba5 | A circle with center O contains two internally tangent circles at points S and T. The radii of these two circles are 3 cm and 4 cm, respectively. The two circles intersect at points M and N such that S, N, and T are collinear. If $\sin(\angle OMN) = \frac{\sqrt{a}}{b}$, where $(a, b) = 1$ and $\{a, b\} \in \mathbb{N}$,... | [
"440"
] | [
"A circle with center O contains two internally tangent circles at points S and T. The radii of these two circles are 3 cm and 4 cm, respectively. The two circles intersect at points M and N such that S, N, and T are collinear. If $\\sin(\\angle OMN) = \\frac{\\sqrt{a}}{b}$, where $(a, b) = 1$ and $\\{a, b\\} \\in ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-73a2628a387806d37b0ce0ef506f82b54be0ba5b8270ec23c1c92254100e3776 | A circle with center O contains two internally tangent circles at points S and T. The two circles intersect each other at points M and N such that S, N, T are collinear. The radii of the two circles are 3 cm and 4 cm. If \(\sin(\angle OMN) = \frac{\sqrt{a}}{b}\), where \((a, b) = 1\) and \(\{a, b\} \in \mathbb{N}\), fi... | [
"1"
] | [
"A circle with center O contains two internally tangent circles at points S and T. The two circles intersect each other at points M and N such that S, N, T are collinear. The radii of the two circles are 3 cm and 4 cm. If \\(\\sin(\\angle OMN) = \\frac{\\sqrt{a}}{b}\\), where \\((a, b) = 1\\) and \\(\\{a, b\\} \\in... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-e3cfa225109056b05adf32dc70ab4a3d925e65cca1b3869f63565093a938430b | A circle with center O has an area of $156\pi$. Triangle ABC is equilateral, with BC as a chord of the circle, OA = $4\sqrt{3}$, and point O is outside 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, with BC as a chord of the circle, OA = $4\\sqrt{3}$, and point O is outside triangle ABC. What is the side length of triangle ABC?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-e7dde9fc4bec9d5246ab0e81705e8774f194c8aa830987863bcd16d0e4eab154 | A circle with center O has an area of \(156\pi\). Triangle ABC is equilateral, with BC as a chord of the circle, and \(OA = 4\sqrt{3}\). Point O is outside of 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, with BC as a chord of the circle, and \\(OA = 4\\sqrt{3}\\). Point O is outside of triangle ABC. What is the side length of triangle ABC?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-85786ad01ec08d3350c1f36de716a75a8f4b67eb4cd897d6a42b800dfd272551 | A circle with center O is inscribed in a 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. If AD = 337, BE = 62, FC = 225, and DG = AE + 175, then DO:AO = k:w, where k and w are two relatively prime positive integers. Find t... | [
"25"
] | [
"A circle with center O is inscribed in a 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. If AD = 337, BE = 62, FC = 225, and DG = AE + 175, then DO:AO = k:w, where k and w are two relatively prime positive integers. F... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-6958f9f347b2281ec4ea747d688c9ae9ff11c52d3decf2d31cc96da3ea90b64d | A circle with center O is tangent to the sides AB, BC, CA, and DA of a convex quadrilateral ABCD at points K, L, M, and N respectively. The lines KL and MN intersect at point S. If the coordinates of B, D, and O are (0,0), (4,0), and (2,2) respectively, find the slope of line OS. | [
"1"
] | [
"A circle with center O is tangent to the sides AB, BC, CA, and DA of a convex quadrilateral ABCD at points K, L, M, and N respectively. The lines KL and MN intersect at point S. If the coordinates of B, D, and O are (0,0), (4,0), and (2,2) respectively, find the slope of line OS."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-d8764a9e3a417a46c08866a61bdc6c848a49d461317606fbc426cb8d7f985165 | A circle with center \( C \) and radius 36 and a circle with center \( D \) and radius 9 touch externally. They lie above a common horizontal tangent which meets the first and second circle at points \( A \) and \( B \) respectively. A circle with center \( E \) is tangent to these two circles and tangent to the segmen... | [
"252"
] | [
"A circle with center \\( C \\) and radius 36 and a circle with center \\( D \\) and radius 9 touch externally. They lie above a common horizontal tangent which meets the first and second circle at points \\( A \\) and \\( B \\) respectively. A circle with center \\( E \\) is tangent to these two circles and tangen... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-940ab64a6b21e099e3bfc82234226b76e7ded8c2c2e2bf2d4ca7fe1b5a89aacd | A circle with center \( I \) is entirely inside a circle with center \( O \). Consider all possible chords \( AB \) of the larger circle that are tangent to the smaller circle. Find the locus of the centers of the circumscribed circles about triangle \( AIB \). | [
"\\( OQ = \\frac{R^2 - d^2}{2r} \\)"
] | [
"A circle with center \\( I \\) is entirely inside a circle with center \\( O \\). Consider all possible chords \\( AB \\) of the larger circle that are tangent to the smaller circle. Find the locus of the centers of the circumscribed circles about triangle \\( AIB \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 26,
"tir": 3
} | 29 | 0 | 0 |
omr-ws-8b5eacc4d4b7c31fb2c90d8716fd3a39f6aea3812baf9d8107e03d9bd22d92ca | A circle with center \( I \) is inscribed in \( \triangle ABC \). Extend \( AI \) to intersect side \( BC \) at point \( D \), and let \( P \) be the intersection of \( IC \) with the angle bisector of \( \angle IAC \). If \( \angle A = 40^\circ \) and \( \angle B = 110^\circ \), find the measure of \( \angle APD \) in... | [
"105"
] | [
"A circle with center \\( I \\) is inscribed in \\( \\triangle ABC \\). Extend \\( AI \\) to intersect side \\( BC \\) at point \\( D \\), and let \\( P \\) be the intersection of \\( IC \\) with the angle bisector of \\( \\angle IAC \\). If \\( \\angle A = 40^\\circ \\) and \\( \\angle B = 110^\\circ \\), find the... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 26
} | 45 | 0 | 0 |
omr-ws-582c312eae1710b55fe73fc2089b1acfdb36adbf4e1ab6f23bf6b33a605f7cb8 | A circle with center \( I \) is inscribed in a quadrilateral \( ABCD \) with \( \angle BAD + \angle ADC > 180^\circ \). A line through \( I \) meets \( AB \) and \( CD \) at points \( X \) and \( Y \), respectively. Given that \( IX = IY \), calculate the value of \( \frac{AX \cdot DY}{BX \cdot CY} \). | [
"1"
] | [
"A circle with center \\( I \\) is inscribed in a quadrilateral \\( ABCD \\) with \\( \\angle BAD + \\angle ADC > 180^\\circ \\). A line through \\( I \\) meets \\( AB \\) and \\( CD \\) at points \\( X \\) and \\( Y \\), respectively. Given that \\( IX = IY \\), calculate the value of \\( \\frac{AX \\cdot DY}{BX \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8fcd5b1c3afb1b37e39b8f7e0f36aa0c06fe476b438d0f4158b0056812adae0f | A circle with center \( O \) and radius \( R \) is given. Let \( P \) be an exterior point according to the circle. Draw a tangent from \( P \) to the circle at \( T \). Let \( OP \) intersect the circle at \( A \) and \( B \). If the radius of the Apollonius circle of \( \angle OTP \) equals the radius of the Apolloni... | [
"\\( 2R \\)"
] | [
"A circle with center \\( O \\) and radius \\( R \\) is given. Let \\( P \\) be an exterior point according to the circle. Draw a tangent from \\( P \\) to the circle at \\( T \\). Let \\( OP \\) intersect the circle at \\( A \\) and \\( B \\). If the radius of the Apollonius circle of \\( \\angle OTP \\) equals th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 37
} | 37 | 0 | 0 |
omr-ws-de401a49cdb3015a966ba50f9e3e7dc05cf810d056933530e7783bc6cd11d8db | A circle with center \( O \) has an area of \( 156\pi \). Triangle \( ABC \) is equilateral, with \( \overline{BC} \) as a chord of the circle, \( OA = 4\sqrt{3} \), and point \( O \) is outside \( \triangle ABC \). What is the side length of \( \triangle ABC \)? | [
"6"
] | [
"A circle with center \\( O \\) has an area of \\( 156\\pi \\). Triangle \\( ABC \\) is equilateral, with \\( \\overline{BC} \\) as a chord of the circle, \\( OA = 4\\sqrt{3} \\), and point \\( O \\) is outside \\( \\triangle ABC \\). What is the side length of \\( \\triangle ABC \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 42,
"genselect": 8
} | 50 | 0 | 0 |
omr-ws-dfc97570f25e85b7d7103a0045e4b2a3a85d8818832e235058403ccac7935c3c | A circle with center \( O \) intersects a circle with center \( O' \) at points \( A \) and \( B \). The line \( TT' \) is tangent to the circle centered at \( O \) at point \( T \) and to the circle centered at \( O' \) at point \( T' \). The perpendiculars from \( T \) and \( T' \) to the line \( OO' \) meet \( OO' \... | [
"\\dfrac{1}{2}"
] | [
"A circle with center \\( O \\) intersects a circle with center \\( O' \\) at points \\( A \\) and \\( B \\). The line \\( TT' \\) is tangent to the circle centered at \\( O \\) at point \\( T \\) and to the circle centered at \\( O' \\) at point \\( T' \\). The perpendiculars from \\( T \\) and \\( T' \\) to the l... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-893bca7b7941d7c5333aa112d1f74bf2c736a9bec60ec00788ce639b2a75f384 | A circle with center \( O \) intersects a circle with center \( O' \) at two points \( A \) and \( B \). A line touches the first circle at \( T \) and the second circle at \( T' \). The perpendiculars from points \( T \) and \( T' \) to the line \( OO' \) meet \( OO' \) at \( S \) and \( S' \). If the length of \( OO'... | [
"\\dfrac{48}{5}"
] | [
"A circle with center \\( O \\) intersects a circle with center \\( O' \\) at two points \\( A \\) and \\( B \\). A line touches the first circle at \\( T \\) and the second circle at \\( T' \\). The perpendiculars from points \\( T \\) and \\( T' \\) to the line \\( OO' \\) meet \\( OO' \\) at \\( S \\) and \\( S'... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae62506af59f40e19edca41c74b506694f22dc6706743e2589cb424c7e3a4431 | A circle with center \( O \) is inscribed in a convex quadrilateral \( ABCD \) and is tangent to the sides \( AB, BC, CD, \) and \( DA \) at points \( K, L, M, \) and \( N \) respectively. Assume that the lines \( KL \) and \( MN \) intersect at point \( S \). If \( BD \) and \( OS \) intersect at point \( T \), calcul... | [
"90"
] | [
"A circle with center \\( O \\) is inscribed in a convex quadrilateral \\( ABCD \\) and is tangent to the sides \\( AB, BC, CD, \\) and \\( DA \\) at points \\( K, L, M, \\) and \\( N \\) respectively. Assume that the lines \\( KL \\) and \\( MN \\) intersect at point \\( S \\). If \\( BD \\) and \\( OS \\) interse... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1c564ebd0e149a56cd8d9ec31cd4e4cc13cb075616fa2873563598ef7b626583 | A circle with center \( O \) is inscribed in a quadrilateral \( ABCD \) and touches its non-parallel sides \( BC \) and \( AD \) at \( E \) and \( F \) respectively. The lines \( AO \) and \( DO \) meet the segment \( EF \) at \( K \) and \( N \) respectively, and the lines \( BK \) and \( CN \) meet at \( M \). If \( ... | [
"\\dfrac{1}{2}"
] | [
"A circle with center \\( O \\) is inscribed in a quadrilateral \\( ABCD \\) and touches its non-parallel sides \\( BC \\) and \\( AD \\) at \\( E \\) and \\( F \\) respectively. The lines \\( AO \\) and \\( DO \\) meet the segment \\( EF \\) at \\( K \\) and \\( N \\) respectively, and the lines \\( BK \\) and \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-7a9eac1d18e18b047d8e91334fa86619d2f2f3b9e00f2b43be559447b2390bf4 | A circle with center \( O \) is inscribed in an angle. Let \( A \) be the reflection of \( O \) across one side of the angle. Tangents to the circle from \( A \) intersect the other side of the angle at points \( B \) and \( C \). If the measure of the original angle is \( 2\theta \), find the measure of \( \angle BAC ... | [
"60^\\circ"
] | [
"A circle with center \\( O \\) is inscribed in an angle. Let \\( A \\) be the reflection of \\( O \\) across one side of the angle. Tangents to the circle from \\( A \\) intersect the other side of the angle at points \\( B \\) and \\( C \\). If the measure of the original angle is \\( 2\\theta \\), find the measu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
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