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-9fe63ebd573c5cfbe588004d064ddd8ce134afe68cd518ad679ba8f0acf0a31c | A circle centered at the incenter \( I \) of a triangle \( ABC \) meets all three sides of the triangle: side \( BC \) at \( D \) and \( P \) (with \( D \) nearer to \( B \)), side \( CA \) at \( E \) and \( Q \) (with \( E \) nearer to \( C \)), and side \( AB \) at \( F \) and \( R \) (with \( F \) nearer to \( A \))... | [
"2"
] | [
"A circle centered at the incenter \\( I \\) of a triangle \\( ABC \\) meets all three sides of the triangle: side \\( BC \\) at \\( D \\) and \\( P \\) (with \\( D \\) nearer to \\( B \\)), side \\( CA \\) at \\( E \\) and \\( Q \\) (with \\( E \\) nearer to \\( C \\)), and side \\( AB \\) at \\( F \\) and \\( R \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-f54ccf4a4634a727af3f1bd7ab9c17b63600bc0ac20f2a82856462e69e13a992 | A circle centered at the origin is tangent to the graph \( 25(x-1)^2 - 36y^2 = 900 \). Find the minimum radius of this circle. | [
"5"
] | [
"A circle centered at the origin is tangent to the graph \\( 25(x-1)^2 - 36y^2 = 900 \\). Find the minimum radius of this circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e2a6ae0b85b0dbd045839b816ac2a65faa20df45c96bf7d4b2c1d1ef24391340 | A circle centered at the origin is tangent to the graph \(25(x-1) - 36y = 900\). Find the minimum radius of this circle. | [
"\\dfrac{925}{\\sqrt{1921}}"
] | [
"A circle centered at the origin is tangent to the graph \\(25(x-1) - 36y = 900\\). Find the minimum radius of this circle."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ef271e16c65367146991973e4e65df0dfcccfed101b63d990fb4d2954175837 | A circle centered at the origin is tangent to the parabola $y = k - x^2$ at two distinct points which are $k$ units apart. Find all possible values for the positive constant $k$. | [
"2 - \\sqrt{2}"
] | [
"A circle centered at the origin is tangent to the parabola $y = k - x^2$ at two distinct points which are $k$ units apart. Find all possible values for the positive constant $k$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-585385aab6abe8041b942972e2253970c60f3792223a8dc5305cc2fcbbf35807 | A circle centered at the origin passes through the point $(99,101)$ and intersects the positive coordinate axes at points $X$ and $Y$. What is the area of the circle with diameter $\overline{XY}$? | [
"10001"
] | [
"A circle centered at the origin passes through the point $(99,101)$ and intersects the positive coordinate axes at points $X$ and $Y$. What is the area of the circle with diameter $\\overline{XY}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-467b25a057aac50dd1d7e2cc5f389cd561afeaa3bdba45c2c039aa2da9a3faae | A circle circumscribes the rectangle ABCD. Vertex A is at the origin (0,0), vertex B is on the positive x-axis, vertex C is on the parabola \( y = x^2 \), and vertex D is on the positive y-axis. If the perimeter of the rectangle is 24, what is the area of the circle? | [
"\\(\\frac{45\\pi}{2}\\)"
] | [
"A circle circumscribes the rectangle ABCD. Vertex A is at the origin (0,0), vertex B is on the positive x-axis, vertex C is on the parabola \\( y = x^2 \\), and vertex D is on the positive y-axis. If the perimeter of the rectangle is 24, what is the area of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-8578ac62880ee7c1041de12b3dd8760800c9fc89707d4b646a5d7338f884a5c5 | A circle constructed on the base $AD$ of a trapezoid $ABCD$ as a diameter passes through the midpoints of the nonparallel sides $AB$ and $CD$ of the trapezoid and touches the base $BC$. Find the angles of the trapezoid. | [
"The trapezoid is isosceles; $75^\\circ$ and $105^\\circ$"
] | [
"A circle constructed on the base $AD$ of a trapezoid $ABCD$ as a diameter passes through the midpoints of the nonparallel sides $AB$ and $CD$ of the trapezoid and touches the base $BC$. Find the angles of the trapezoid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e1526b5681583f0e7715bdbc362f5ea351539bdc708a0fd4afcfbf7e0609b58 | A circle contains two parallel chords AB and CD. The line passing through point C and the midpoint M of chord AB intersects the circle again at point E. Let K be the midpoint of DE. If the measure of angle <AKE is represented by $\theta$, find the value of $\sin(\theta) + \cos(\theta)$. | [
"\\sqrt{2}"
] | [
"A circle contains two parallel chords AB and CD. The line passing through point C and the midpoint M of chord AB intersects the circle again at point E. Let K be the midpoint of DE. If the measure of angle <AKE is represented by $\\theta$, find the value of $\\sin(\\theta) + \\cos(\\theta)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-44ec33f5163a21cce8a739db77be6288fd0087d0e5d049698100bbe3785b3a9b | A circle exists such that for a certain quadrilateral $ABCD$ it is tangent to all $4$ sides. $AC$ and $BD$ have the property that they split the circle into two regions of equal area. $AC$ and $BD$ intersect at a point $E$, and the foot of the altitude from $E$ to $BC$ is $F$. Given that $AB=6$ and $DF=3\sqrt{5}$, comp... | [
"605"
] | [
"A circle exists such that for a certain quadrilateral $ABCD$ it is tangent to all $4$ sides. $AC$ and $BD$ have the property that they split the circle into two regions of equal area. $AC$ and $BD$ intersect at a point $E$, and the foot of the altitude from $E$ to $BC$ is $F$. Given that $AB=6$ and $DF=3\\sqrt{5}$... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b74cf4d39269a603115490b0cf8ee5477630ca8fdaf2b0a72c69c5a743545bc | A circle has 2006 points chosen such that the arcs between any two adjacent points are equal. Three of these points are chosen at random. Let the probability that the triangle formed is right be \( R \), and the probability that the triangle formed is isosceles be \( I \). Find \( |R - I| \). | [
"0"
] | [
"A circle has 2006 points chosen such that the arcs between any two adjacent points are equal. Three of these points are chosen at random. Let the probability that the triangle formed is right be \\( R \\), and the probability that the triangle formed is isosceles be \\( I \\). Find \\( |R - I| \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-053c48abc332006942101925cb97746a17740828f54e7c25f0f5b85f97402b4a | A circle has \( n \) evenly spaced points on its circumference. Two (not necessarily distinct) chords are drawn, so that all chord endpoints are one of the \( n \) points. The probability that these two chords intersect (possibly at one of the \( n \) points) is greater than \(\frac{1}{2}\). Find the maximum possible v... | [
"15"
] | [
"A circle has \\( n \\) evenly spaced points on its circumference. Two (not necessarily distinct) chords are drawn, so that all chord endpoints are one of the \\( n \\) points. The probability that these two chords intersect (possibly at one of the \\( n \\) points) is greater than \\(\\frac{1}{2}\\). Find the maxi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-d074cb479363b4124232dc1608396cff18df932d4ed53c824a4a0d39334e9934 | A circle has a circumference of 10. Find the area of an inscribed hexagon. | [
"\\(\\frac{75}{2\\pi^2}\\)"
] | [
"A circle has a circumference of 10. Find the area of an inscribed hexagon."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1cdbff0fba9cda29f1faea16f2a97eb6ea322e989f973a64ff6bf8c062eaff86 | A circle has a diameter $AB$ whose length is a $2$-digit integer. The arc string $CD$ is perpendicular to $AB$ and intersects $AB$ at point $H$. The length of $CD$ is equal to the number obtained by changing the position of the two digits from the length of $AB$. If the distance from $H$ to the center of the circle is ... | [
"65"
] | [
"A circle has a diameter $AB$ whose length is a $2$-digit integer. The arc string $CD$ is perpendicular to $AB$ and intersects $AB$ at point $H$. The length of $CD$ is equal to the number obtained by changing the position of the two digits from the length of $AB$. If the distance from $H$ to the center of the circl... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-97fb5d8eeaca8f4a3e707695545fc9d82490a03355e2b97b2cc0b7ce548249be | A circle has a diameter of 1. Find the distance from the center of the circle to the chord where the segment corresponding to the chord has an area of $\frac{\pi}{12}$. | [
"\\dfrac{\\sqrt{3}}{12}"
] | [
"A circle has a diameter of 1. Find the distance from the center of the circle to the chord where the segment corresponding to the chord has an area of $\\frac{\\pi}{12}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-7f27a16d595f6e53d06af8d440bddd9e324311354ae1ff6e428b994b340ed246 | A circle has a diameter with endpoints at (1,0) and (b,0) where b > 1. A tangent to the circle has the equation \( y = \frac{4}{3}x \). Find the value of b. | [
"9"
] | [
"A circle has a diameter with endpoints at (1,0) and (b,0) where b > 1. A tangent to the circle has the equation \\( y = \\frac{4}{3}x \\). Find the value of b."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0dd80207c45ce591289c84cffa54ca69de4f4d6f69e6f90438abf7add2d85cd0 | A circle has a radius of 10 and its center is in the second quadrant. If the circle has y-intercepts at 0 and 12, compute the equation of the circle. | [
"\\((x+8)^2+(y-6)^2=100\\)\n\nExplanation: The solution provided in Post 3 gives the equation of the circle as \\((x-8)^2+(k-6)^2=100\\). However, since the center of the circle is in the second quadrant, the x-coordinate of the center should be negative. Therefore, the correct equation of the circle is \\((x+8)^2+... | [
"A circle has a radius of 10 and its center is in the second quadrant. If the circle has y-intercepts at 0 and 12, compute the equation of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-0ff65ab17d15a5a0d96702dc2137db3bf391bbca486fbb1fc6e38674ecef0751 | A circle has a radius of 10 cm. Using \(\pi = 3.14\), find: a) The circumference of the circle to the nearest integer. b) The area of the circle to the nearest integer. | [
"314"
] | [
"A circle has a radius of 10 cm. Using \\(\\pi = 3.14\\), find:\na) The circumference of the circle to the nearest integer.\nb) The area of the circle to the nearest integer."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2047c8fb614ba920490dfd64b9e7e4ddf5063b6a7b5b582b611699f1ec2f6c50 | A circle has a radius of \( \log_{10}(a^2) \) and a circumference of \( \log_{10}(b^4) \). What is \( \log_ab \)? | [
"\\(\\pi\\)"
] | [
"A circle has a radius of \\( \\log_{10}(a^2) \\) and a circumference of \\( \\log_{10}(b^4) \\). What is \\( \\log_ab \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-b348454599ee3eec55d44f75b6dc9c5e4f1b34d7e60e9431379cce5d6f981798 | A circle has a radius of three inches. The distance from the center of the circle to chord $CD$ is two inches. How many inches long is chord $CD$? Express your answer in simplest radical form. | [
"2\\sqrt{5}"
] | [
"A circle has a radius of three inches. The distance from the center of the circle to chord $CD$ is two inches. How many inches long is chord $CD$? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-894f4c044193539c9c817e0f4e7b657f8870ddd8a8b63a17ac2b8e29782bc7aa | A circle has an area of $16\pi$, and it has perpendicular diameters PQ and RS. PRQS creates a square. M and N are the midpoints of PR and SQ, respectively. MN, RS, and PQ all intersect at the center of the square, L. SM intersects PQ at J. What is the area of $\Delta$LJS? | [
"\\(\\frac{8}{3}\\)"
] | [
"A circle has an area of $16\\pi$, and it has perpendicular diameters PQ and RS. PRQS creates a square. M and N are the midpoints of PR and SQ, respectively. MN, RS, and PQ all intersect at the center of the square, L. SM intersects PQ at J. What is the area of $\\Delta$LJS?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-489b8ef42bbd79464bc0518e058e9c3ddd484d0f2239daaf76656d7152962d9c | A circle has an area of $2023$. An equilateral triangle is inscribed in that circle, and another circle is inscribed in that triangle. Find the area of the second circle. | [
"\\(\\frac{2023}{4}\\)"
] | [
"A circle has an area of $2023$. An equilateral triangle is inscribed in that circle, and another circle is inscribed in that triangle. Find the area of the second circle."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-fcfd6b3a41a37a07ac410498d53d7d0ec8ace8e6f05565b3aa795ff6b871c1ab | A circle has an area of $M\text{ cm}^2$ and a circumference of $N\text{ cm}$. If $\dfrac{M}{N}=20$, what is the radius of the circle, in cm? | [
"40"
] | [
"A circle has an area of $M\\text{ cm}^2$ and a circumference of $N\\text{ cm}$. If $\\dfrac{M}{N}=20$, what is the radius of the circle, in cm?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8f43a48254c76a4fc4aefdb4b00aa6adf051a32524946f8cb51064d6f691bdf7 | A circle has an area of $\pi$ square units. What is the length of the circle's diameter, in units? | [
"2"
] | [
"A circle has an area of $\\pi$ square units. What is the length of the circle's diameter, in units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-33d307df24ef838450ae7a04d1e68beeaf377ccd9c1f8b44012ea3fb76790ca7 | A circle has an inscribed triangle with sides of lengths $5\sqrt{3}$, $10\sqrt{3}$, and $15$. What is the measure of the angle subtended at the center of the circle by the shortest side? | [
"60"
] | [
"A circle has an inscribed triangle with sides of lengths $5\\sqrt{3}$, $10\\sqrt{3}$, and $15$. What is the measure of the angle subtended at the center of the circle by the shortest side?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-f8facb33ddbd30d540b585733a7915d3e4f9cbc8eb8b0dce116efe7728636243 | A circle has center $(-10, -4)$ and radius $13$. Another circle has center $(3, 9)$ and radius $\sqrt{65}$. The line passing through the two points of intersection of the two circles has equation $x + y = c$. What is $c$? | [
"3"
] | [
"A circle has center $(-10, -4)$ and radius $13$. Another circle has center $(3, 9)$ and radius $\\sqrt{65}$. The line passing through the two points of intersection of the two circles has equation $x + y = c$. What is $c$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-542a42b49c4c135d63f666df3a3484df55eae94ae5417f6e4a5b2aab63a31499 | A circle has center $O$ and radius $OA = 5$. Line $m$ passes through $O$, is perpendicular to $OA$, and intersects the circle at points $X$ and $Y$. Another line passes through point $A$ and the midpoint of arc $AX$, intersecting line $m$ at point $B$. What is the length of $OB$? | [
"\\(5\\sqrt{2}\\)"
] | [
"A circle has center $O$ and radius $OA = 5$. Line $m$ passes through $O$, is perpendicular to $OA$, and intersects the circle at points $X$ and $Y$. Another line passes through point $A$ and the midpoint of arc $AX$, intersecting line $m$ at point $B$. What is the length of $OB$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f87b44b69b8800c80f5cabacdce118fa81d0fa93f8c5461272a235dfd518a3f0 | A circle has center $O$. Lines $PA$ and $PB$ are tangent to the circle respectively at $A$ and $B$. Points $E$ and $F$ belong respectively to segments $PA$ and $PB$ and the line $EF$ is tangent to the circle. Knowing that $\angle EPF = \angle EOF$, find the measure of $\angle EPF$ (in radians). | [
"\\dfrac{\\pi}{3}"
] | [
"A circle has center $O$. Lines $PA$ and $PB$ are tangent to the circle respectively at $A$ and $B$. Points $E$ and $F$ belong respectively to segments $PA$ and $PB$ and the line $EF$ is tangent to the circle. Knowing that $\\angle EPF = \\angle EOF$, find the measure of $\\angle EPF$ (in radians)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ea694d19f3484c8663eb04d083f5129328adfa1037f05c7cbf0b397467c01fef | A circle has center A. Two lines are drawn tangent to the circle at points B and D, intersecting at point C at an angle of 120 degrees. Line BD has length \(6\sqrt{3}\). Find the length of BC. | [
"6"
] | [
"A circle has center A. Two lines are drawn tangent to the circle at points B and D, intersecting at point C at an angle of 120 degrees. Line BD has length \\(6\\sqrt{3}\\). Find the length of BC."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-994a90c5f944c30d9e42c846fbbbfe9394790979f9baa2e5e9f546ad93952213 | A circle has chords of lengths \(2\sqrt{3}\), 12, and \(12\sqrt{3}\). The chords can be positioned such that the distance between the shortest chord and the center is the sum of the distances between the longest and the center and the middle and the center. Find the radius of the circle. | [
"\\sqrt{111}"
] | [
"A circle has chords of lengths \\(2\\sqrt{3}\\), 12, and \\(12\\sqrt{3}\\). The chords can be positioned such that the distance between the shortest chord and the center is the sum of the distances between the longest and the center and the middle and the center. Find the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-416b99f97a610483fd3cf05c58f253a8a98a7aa3c094d3df86da905820b97f87 | A circle has diameter \(AB\) and a fixed point \(X\) on \(AB\) between \(A\) and \(B\). A point \(P\) lies on the circumference of the circle, distinct from \(A\) and \(B\). If \(AX = 3\) and \(XB = 5\), find the value of \(\frac{\tan \angle APX}{\tan \angle PAX}\). | [
"\\dfrac{3}{5}"
] | [
"A circle has diameter \\(AB\\) and a fixed point \\(X\\) on \\(AB\\) between \\(A\\) and \\(B\\). A point \\(P\\) lies on the circumference of the circle, distinct from \\(A\\) and \\(B\\). If \\(AX = 3\\) and \\(XB = 5\\), find the value of \\(\\frac{\\tan \\angle APX}{\\tan \\angle PAX}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-87778c2115618b2e8b6204a487ac1ad70d74584435faefff54004e08f2673e4f | A circle has perpendicular chords $AB$ and $CD$ which meet at $M$. Given $CM = 6$, $MD = 2$, $BM = 4$, and $MA = 3$, find the diameter of the circle. | [
"$\\sqrt{65}$"
] | [
"A circle has perpendicular chords $AB$ and $CD$ which meet at $M$. Given $CM = 6$, $MD = 2$, $BM = 4$, and $MA = 3$, find the diameter of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-7d9f11bc9de2e1d18b1805dd9c847cfff90a9dafc58540694c58122d907464ee | A circle has radius 3. Find the sum of the squares of the distances from any point on the circle to the vertices of an inscribed square. | [
"72"
] | [
"A circle has radius 3. Find the sum of the squares of the distances from any point on the circle to the vertices of an inscribed square."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-abb33b8ca60090e3a6b104fd9932dc274e886004d49ff3e870f3d1981566d0a7 | A circle has the equation \( x^2 + y^2 - 4\sqrt{3}x + 2y - 3 = 0 \). Find the larger of the two areas into which the y-axis divides the circle. | [
"\\(4\\sqrt{3}+\\frac{40\\pi}{3}\\)"
] | [
"A circle has the equation \\( x^2 + y^2 - 4\\sqrt{3}x + 2y - 3 = 0 \\). Find the larger of the two areas into which the y-axis divides the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-167e8bced353679d6585b9e16b1ee2174f86dbe8be6836ccb03d12ec3bd14633 | A circle has the equation \((x-a)^{2} + (y - a)^{2} = a^{2}\) where \(a\) is a constant. The line \(y + x - a = 0\) splits the area of the circle into two parts, \(A_{1}\) and \(A_{2}\) where \(A_{1} > A_{2}\). Find the area of \(A_{2}\) in the form \(\frac{a^{2}}{b}(c \pi + d)\) where \(b, c,\) and \(d\) are integers. | [
"\\(\\frac{a^{2}}{4}(\\pi - 2)\\)"
] | [
"A circle has the equation \\((x-a)^{2} + (y - a)^{2} = a^{2}\\) where \\(a\\) is a constant. The line \\(y + x - a = 0\\) splits the area of the circle into two parts, \\(A_{1}\\) and \\(A_{2}\\) where \\(A_{1} > A_{2}\\). Find the area of \\(A_{2}\\) in the form \\(\\frac{a^{2}}{b}(c \\pi + d)\\) where \\(b, c,\\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-ee5eb665df7321101a98f634773f1b68f6fb49915fb002f404548fce820307db | A circle has the equation \(x^2 + y^2 = 36\). A chord is drawn from the point \(A\) where the negative x-axis meets the circle and passes through the y-axis at \(y = 2\) to point \(B\). A vertical line from the positive x-axis at point \(C\) meets point \(B\) to form a triangle \(ABC\). Show that the area enclosed by t... | [
"2.94"
] | [
"A circle has the equation \\(x^2 + y^2 = 36\\). A chord is drawn from the point \\(A\\) where the negative x-axis meets the circle and passes through the y-axis at \\(y = 2\\) to point \\(B\\). A vertical line from the positive x-axis at point \\(C\\) meets point \\(B\\) to form a triangle \\(ABC\\). Show that the... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-e9dfb0807b5809413faf7f41896df8c8016adc567f52a5a0c2ec497b5b858cb0 | A circle has three parallel chords A, B, and C on the same side, with lengths 8, 16, and 20 respectively. The distance between chords A and B is the same as the distance between chords B and C. What is the radius of the circle? | [
"\\(\\sqrt{\\frac{275}{2}}\\)"
] | [
"A circle has three parallel chords A, B, and C on the same side, with lengths 8, 16, and 20 respectively. The distance between chords A and B is the same as the distance between chords B and C. What is the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-24f7c1392a6c2040d7d53388dfef11b58dc62d76be086c070b8dc01f40e24f3e | A circle having center $(0,k)$, with $k>6$, is tangent to the lines $y=x$, $y=-x$ and $y=6$. What is the radius of this circle? | [
"6\\sqrt{2}+6"
] | [
"A circle having center $(0,k)$, with $k>6$, is tangent to the lines $y=x$, $y=-x$ and $y=6$. What is the radius of this circle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d1a34ac3690cb3077cdf2abb541f7bf3c9c59d75e84c8c1d9a48610586b9574c | A circle in the 1st quadrant touches the x-axis at point A and the y-axis at point B. It also touches the line \(x + 2y - 3 - \sqrt{5} = 0\) at point C(a, b). Find the value of \(a + b\). | [
"\\(\\frac{10 + 3\\sqrt{5}}{5}\\)"
] | [
"A circle in the 1st quadrant touches the x-axis at point A and the y-axis at point B. It also touches the line \\(x + 2y - 3 - \\sqrt{5} = 0\\) at point C(a, b). Find the value of \\(a + b\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-6b9cf7a3b17bae391e86fff5675c7f35026eff72f161230bcf634ec134cbe3a5 | A circle in the first quadrant with its center on the curve $y = 2x^2 - 27$ is tangent to the $y$-axis and the line $4x = 3y$. Find the radius of the circle. | [
"\\(\\frac{9}{2}\\)"
] | [
"A circle in the first quadrant with its center on the curve $y = 2x^2 - 27$ is tangent to the $y$-axis and the line $4x = 3y$. Find the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-6f84aaf8c18483be0da06e0cd8740e60076fbcb0052c4ad6e771c0c46c304bac | A circle inscribed in a square $ABCD$ with side length $10$ cm intersects sides $BC$ and $AD$ at points $M$ and $N$ respectively. The point $I$ is the intersection of $AM$ with the circle different from $M$, and $P$ is the orthogonal projection of $I$ onto $MN$. Find the length of segment $PI$. | [
"4"
] | [
"A circle inscribed in a square $ABCD$ with side length $10$ cm intersects sides $BC$ and $AD$ at points $M$ and $N$ respectively. The point $I$ is the intersection of $AM$ with the circle different from $M$, and $P$ is the orthogonal projection of $I$ onto $MN$. Find the length of segment $PI$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-9c61cddbdb9221a3b226359202bdfdbe4734aac17c9f233620c15bce69d6b200 | A circle inscribed in a square $ABCD$ with side length $a$ touches the side $[CD]$ at a point $E$. Find the length of the chord connecting the points at which the circle intersects the line $AE$. | [
"\\(\\frac{2a\\sqrt{5}}{5}\\)"
] | [
"A circle inscribed in a square $ABCD$ with side length $a$ touches the side $[CD]$ at a point $E$. Find the length of the chord connecting the points at which the circle intersects the line $AE$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-9e4a44131f6e352208d36bd42a23b8ba33c2ef5f1a94ec3415514a131778c89f | A circle inscribed in a triangle \(ABC\) touches the sides \(AB\), \(BC\), and \(AC\) at points \(X\), \(Y\), and \(Z\) respectively. Perpendiculars \(YK \perp AB\), \(XP \perp AC\), and \(ZQ \perp BC\) are constructed. Find the area of triangle \(XYZ\) in terms of the lengths of \(XP\), \(YK\), and \(ZQ\). | [
"\\(\\frac{XP^2}{2R}\\)"
] | [
"A circle inscribed in a triangle \\(ABC\\) touches the sides \\(AB\\), \\(BC\\), and \\(AC\\) at points \\(X\\), \\(Y\\), and \\(Z\\) respectively. Perpendiculars \\(YK \\perp AB\\), \\(XP \\perp AC\\), and \\(ZQ \\perp BC\\) are constructed. Find the area of triangle \\(XYZ\\) in terms of the lengths of \\(XP\\),... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f6c36b85e474715631153813aa4efade4d4de7191ac4dd81c8eb1163d624f119 | A circle inscribed in a triangle \(ABC\) touches the sides \(AB\), \(BC\), and \(CA\) at points \(X\), \(Y\), and \(Z\) respectively. Perpendiculars \(YK\), \(XP\), and \(ZQ\) are constructed to \(AB\), \(AC\), and \(BC\) respectively, where \(K\), \(P\), and \(Q\) lie on \(AB\), \(AC\), and \(BC\) respectively. Find t... | [
"\\sqrt{\\dfrac{YK \\cdot XP \\cdot ZQ}{2}}"
] | [
"A circle inscribed in a triangle \\(ABC\\) touches the sides \\(AB\\), \\(BC\\), and \\(CA\\) at points \\(X\\), \\(Y\\), and \\(Z\\) respectively. Perpendiculars \\(YK\\), \\(XP\\), and \\(ZQ\\) are constructed to \\(AB\\), \\(AC\\), and \\(BC\\) respectively, where \\(K\\), \\(P\\), and \\(Q\\) lie on \\(AB\\), ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-7ea2bbdfe203fc5fee92e2fdbfb4c8b9d7b99fdb7e82cfaca115ca915a37f149 | A circle inscribed in an angle with a vertex \( O \) touches its sides at points \( A \) and \( B \). Let \( K \) be an arbitrary point on the smaller of the two arcs \( AB \) of this circle. On line \( OB \), a point \( L \) is taken such that lines \( OA \) and \( KL \) are parallel. Let \( M \) be the intersection p... | [
"2"
] | [
"A circle inscribed in an angle with a vertex \\( O \\) touches its sides at points \\( A \\) and \\( B \\). Let \\( K \\) be an arbitrary point on the smaller of the two arcs \\( AB \\) of this circle. On line \\( OB \\), a point \\( L \\) is taken such that lines \\( OA \\) and \\( KL \\) are parallel. Let \\( M ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-87c37e5e17a6ab97d253720c540a4d0f41941c4b7d0664bb63484fa39266e30a | A circle inscribed in the right angle of an isosceles right triangle divides the hypotenuse into three equal parts. The leg of the triangle is of length $1$. Find the radius of the circle. | [
"\\(\\frac{1}{3}\\)"
] | [
"A circle inscribed in the right angle of an isosceles right triangle divides the hypotenuse into three equal parts. The leg of the triangle is of length $1$. Find the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 26
} | 74 | 0 | 0 |
omr-ws-3933f675362b7a0ecea721456ed10df3b36899f07633c035339566cc13bed648 | A circle inscribed in triangle $ABC$ divides side $AB$ into segments $AD$ and $DB$ with lengths 5 cm and 3 cm, respectively. The value of angle $A$ is $60^\circ$. Find the length of side $BC$. | [
"13"
] | [
"A circle inscribed in triangle $ABC$ divides side $AB$ into segments $AD$ and $DB$ with lengths 5 cm and 3 cm, respectively. The value of angle $A$ is $60^\\circ$. Find the length of side $BC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4ff919f2b9772f3e3047aef4dc5ca480a07e20812a2d7c4d9c72a447583aacce | A circle intersects $\triangle ABC$ at 6 points $A_1, A_2, B_1, B_2, C_1, C_2$ in the order $A, C_1, C_2, B, A_1, A_2, C, B_1, B_2, A$. Suppose $B_1C_1$ and $B_2C_2$ meet at $X$, $C_1A_1$ and $C_2A_2$ meet at $Y$, and $A_1B_1$ and $A_2B_2$ meet at $Z$. If $AX$, $BY$, and $CZ$ are concurrent at point $P$, and $AX = 12$,... | [
"24"
] | [
"A circle intersects $\\triangle ABC$ at 6 points $A_1, A_2, B_1, B_2, C_1, C_2$ in the order $A, C_1, C_2, B, A_1, A_2, C, B_1, B_2, A$. Suppose $B_1C_1$ and $B_2C_2$ meet at $X$, $C_1A_1$ and $C_2A_2$ meet at $Y$, and $A_1B_1$ and $A_2B_2$ meet at $Z$. If $AX$, $BY$, and $CZ$ are concurrent at point $P$, and $AX ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-a61680f9fe7e33694f3c0d3737492b2aba944fca0106d91195e33377faf0c978 | A circle intersects a parabola at four distinct points. Let $M$ and $N$ be the midpoints of the arcs of the circle which are outside the parabola. If the equation of the parabola is $y = x^2$, and the center of the circle is at $(0, c)$, find the value of $c$ such that the line $MN$ is perpendicular to the axis of the ... | [
"2"
] | [
"A circle intersects a parabola at four distinct points. Let $M$ and $N$ be the midpoints of the arcs of the circle which are outside the parabola. If the equation of the parabola is $y = x^2$, and the center of the circle is at $(0, c)$, find the value of $c$ such that the line $MN$ is perpendicular to the axis of... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-517a439c1f17f5529a9c69ea44ffb5a428673da9d574dfa4fd888c2e6722fcac | A circle intersects a triangle \(ABC\) at six points \(A_1\), \(A_2\), \(B_1\), \(B_2\), \(C_1\), \(C_2\), where the order of appearance along the triangle is \(A\), \(C_1\), \(C_2\), \(B\), \(A_1\), \(A_2\), \(C\), \(B_1\), \(B_2\), \(A\). Suppose \(B_1C_1\) and \(B_2C_2\) meet at \(X\), \(C_1A_1\) and \(C_2A_2\) meet... | [
"\\dfrac{169}{225}"
] | [
"A circle intersects a triangle \\(ABC\\) at six points \\(A_1\\), \\(A_2\\), \\(B_1\\), \\(B_2\\), \\(C_1\\), \\(C_2\\), where the order of appearance along the triangle is \\(A\\), \\(C_1\\), \\(C_2\\), \\(B\\), \\(A_1\\), \\(A_2\\), \\(C\\), \\(B_1\\), \\(B_2\\), \\(A\\). Suppose \\(B_1C_1\\) and \\(B_2C_2\\) me... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-736e8f08cf9143403d9314fe2769ef19b5c6a64531391f87a372f7e2e5fb7423 | A circle intersects an equilateral triangle \( ABC \) at six points such that each pair of intersection points is on a different side of the triangle. Given \( AG = 2 \), \( GF = 13 \), \( FC = 1 \), and \( HJ = 7 \), find the length of \( DE \). | [
"14"
] | [
"A circle intersects an equilateral triangle \\( ABC \\) at six points such that each pair of intersection points is on a different side of the triangle. Given \\( AG = 2 \\), \\( GF = 13 \\), \\( FC = 1 \\), and \\( HJ = 7 \\), find the length of \\( DE \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-998b41ce5263d1a6ca2600d6299057bb29ba9bec369937b84b0384ac8c518c3d | A circle intersects each side of a rectangle at two points, creating two trapezoids by connecting crossing points with opposite sides. Let $A$, $B$, $C$, and $D$ be the intersection points of the sides of these trapezoids. If the coordinates of the rectangle's vertices are $(0,0)$, $(a,0)$, $(a,b)$, and $(0,b)$, and th... | [
"20"
] | [
"A circle intersects each side of a rectangle at two points, creating two trapezoids by connecting crossing points with opposite sides. Let $A$, $B$, $C$, and $D$ be the intersection points of the sides of these trapezoids. If the coordinates of the rectangle's vertices are $(0,0)$, $(a,0)$, $(a,b)$, and $(0,b)$, a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-efe0954336a4a99e47db67ca69628b11e98489676f7f5906b8d470238134f5bd | A circle intersects each side of a rhombus twice, dividing each side into three parts. Starting from any vertex, go around the rhombus in a fixed direction and color the segments red, white, blue, red, white, blue, red, white, blue, red, white, blue. If the lengths of the red segments are $r_1, r_2, r_3, r_4$ and the l... | [
"0"
] | [
"A circle intersects each side of a rhombus twice, dividing each side into three parts. Starting from any vertex, go around the rhombus in a fixed direction and color the segments red, white, blue, red, white, blue, red, white, blue, red, white, blue. If the lengths of the red segments are $r_1, r_2, r_3, r_4$ and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-029cf70bab609b78b03402f2220de68a7d8c829d7c8b1a70c97628eaf6fed91b | A circle intersects each side of a rhombus twice, dividing each side into three segments. Starting from a vertex and moving clockwise around the perimeter of the rhombus, the segments are painted red, white, and blue in that order. If the lengths of the red segments are $r_1, r_2, r_3, r_4$, the lengths of the white se... | [
"0"
] | [
"A circle intersects each side of a rhombus twice, dividing each side into three segments. Starting from a vertex and moving clockwise around the perimeter of the rhombus, the segments are painted red, white, and blue in that order. If the lengths of the red segments are $r_1, r_2, r_3, r_4$, the lengths of the whi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e7d74eea938d872aa013be3b0e4ecf9ef5979bfb1a26f2d197a02fbc9444fda | A circle intersects sides $BC$, $CA$, and $AB$ of triangle $ABC$ at two points for each side in the following order: $\{D_{1}, D_{2}\}$, $\{E_{1}, E_{2}\}$, $\{F_{1}, F_{2}\}$. Line segments $D_{1}E_{1}$ and $D_{2}E_{2}$ intersect at point $L$, $E_{1}F_{1}$ and $E_{2}F_{2}$ intersect at point $M$, and $F_{1}D_{1}$ and ... | [
"2"
] | [
"A circle intersects sides $BC$, $CA$, and $AB$ of triangle $ABC$ at two points for each side in the following order: $\\{D_{1}, D_{2}\\}$, $\\{E_{1}, E_{2}\\}$, $\\{F_{1}, F_{2}\\}$. Line segments $D_{1}E_{1}$ and $D_{2}E_{2}$ intersect at point $L$, $E_{1}F_{1}$ and $E_{2}F_{2}$ intersect at point $M$, and $F_{1}... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d60b5faa2cb51c25683addaab2a6fc3bcea6b06983cc551d0beebf829f687e47 | A circle intersects square \(ABCD\) at points \(A\), \(E\), and \(F\), where \(E\) lies on \(AB\) and \(F\) lies on \(AD\), such that \(AE + AF = 2(BE + DF)\). If the square and the circle each have area \(1\), determine the area of the union of the circle and square. | [
"1"
] | [
"A circle intersects square \\(ABCD\\) at points \\(A\\), \\(E\\), and \\(F\\), where \\(E\\) lies on \\(AB\\) and \\(F\\) lies on \\(AD\\), such that \\(AE + AF = 2(BE + DF)\\). If the square and the circle each have area \\(1\\), determine the area of the union of the circle and square."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b6a26dccc7673fa5be8f7ee502ccc579378e9db4c76f40ffbb24c00fd0011b58 | A circle intersects the $y$-axis at two points $(0, a)$ and $(0, b)$ and is tangent to the line $x + 100y = 100$ at $(100, 0)$. Compute the sum of all possible values of $ab - a - b$. | [
"10000"
] | [
"A circle intersects the $y$-axis at two points $(0, a)$ and $(0, b)$ and is tangent to the line $x + 100y = 100$ at $(100, 0)$. Compute the sum of all possible values of $ab - a - b$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-52077f88019c0e8e696185dc98b1a29dada619ece1afaedfa48839a36f9ef365 | A circle intersects the graph of \( y = ax^2 \) at \( x = 1 \) and \( x = 4 \), and it is tangent to \( y = ax^2 \) at a third point. Compute the \( x \)-value of the third point. | [
"\\(-\\frac{5}{2}\\)"
] | [
"A circle intersects the graph of \\( y = ax^2 \\) at \\( x = 1 \\) and \\( x = 4 \\), and it is tangent to \\( y = ax^2 \\) at a third point. Compute the \\( x \\)-value of the third point."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-5630b84cdaf8f927b6dbb315c6bd65d7db8ca6e0bbdf24918ba3ace49418f06e | A circle intersects the graph of the function $y=x^3-2009x$ at six points. What can be the sum of the abscissas of these points? | [
"0"
] | [
"A circle intersects the graph of the function $y=x^3-2009x$ at six points. What can be the sum of the abscissas of these points?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9a7f33b7c571c724d2ab7d54062e14fe086e7eadf7fdf3249e58f273819be0c | A circle intersects the sides $AB$, $BC$, and $CA$ of triangle $ABC$ at points $D_1$ and $D_2$, $E_1$ and $E_2$, and $F_1$ and $F_2$ respectively. Let $L$ be the intersection of segments $D_1E_1$ and $D_2E_2$, $M$ be the intersection of segments $E_1F_1$ and $E_2F_2$, and $N$ be the intersection of segments $F_1D_1$ an... | [
"60"
] | [
"A circle intersects the sides $AB$, $BC$, and $CA$ of triangle $ABC$ at points $D_1$ and $D_2$, $E_1$ and $E_2$, and $F_1$ and $F_2$ respectively. Let $L$ be the intersection of segments $D_1E_1$ and $D_2E_2$, $M$ be the intersection of segments $E_1F_1$ and $E_2F_2$, and $N$ be the intersection of segments $F_1D_... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-7b96aaa4f82154d312c2e42a1a87805d1ee3bae552353a9c122e95f0c4973bc4 | A circle intersects the sides $BC$, $CA$, and $AB$ of triangle $ABC$ at points $D_1, D_2$, $E_1, E_2$, and $F_1, F_2$ respectively. The line segments $D_1E_1$ and $D_2F_2$ intersect at point $L$, the line segments $E_1F_1$ and $E_2D_2$ intersect at point $M$, and the line segments $F_1D_1$ and $F_2E_2$ intersect at poi... | [
"\\left( \\dfrac{1}{3}, \\dfrac{1}{3} \\right)"
] | [
"A circle intersects the sides $BC$, $CA$, and $AB$ of triangle $ABC$ at points $D_1, D_2$, $E_1, E_2$, and $F_1, F_2$ respectively. The line segments $D_1E_1$ and $D_2F_2$ intersect at point $L$, the line segments $E_1F_1$ and $E_2D_2$ intersect at point $M$, and the line segments $F_1D_1$ and $F_2E_2$ intersect a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-e67e0803eb0540abbbf42f2724e6b3100ef6ec5a3862d0f0994976e9b4036898 | A circle intersects the sides \( BC, CA, AB \) of triangle \( ABC \) at points \( D_1, D_2 \); \( E_1, E_2 \); and \( F_1, F_2 \) respectively. The line segments \( D_1E_1 \) and \( D_2F_2 \) intersect at point \( L \), line segments \( E_1F_1 \) and \( E_2D_2 \) intersect at point \( M \), and line segments \( F_1D_1 ... | [
"\\left( \\dfrac{1}{3}, \\dfrac{1}{3} \\right)"
] | [
"A circle intersects the sides \\( BC, CA, AB \\) of triangle \\( ABC \\) at points \\( D_1, D_2 \\); \\( E_1, E_2 \\); and \\( F_1, F_2 \\) respectively. The line segments \\( D_1E_1 \\) and \\( D_2F_2 \\) intersect at point \\( L \\), line segments \\( E_1F_1 \\) and \\( E_2D_2 \\) intersect at point \\( M \\), a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-b2a443361e371d67050292ec75a900369ff7bafde10af9e3b2f0c5d6155f04f9 | A circle intersects the sides \(BC\), \(CA\), and \(AB\) of a triangle \(ABC\) at points \(D_1, D_2\), \(E_1, E_2\), and \(F_1, F_2\) respectively. The line segments \(D_1E_1\) and \(D_2F_2\) intersect at point \(L\), \(E_1F_1\) and \(E_2D_2\) intersect at point \(M\), and \(F_1D_1\) and \(F_2E_2\) intersect at point \... | [
"\\dfrac{1}{4}"
] | [
"A circle intersects the sides \\(BC\\), \\(CA\\), and \\(AB\\) of a triangle \\(ABC\\) at points \\(D_1, D_2\\), \\(E_1, E_2\\), and \\(F_1, F_2\\) respectively. The line segments \\(D_1E_1\\) and \\(D_2F_2\\) intersect at point \\(L\\), \\(E_1F_1\\) and \\(E_2D_2\\) intersect at point \\(M\\), and \\(F_1D_1\\) an... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-f83a4c48311c1b2b01578846b03273a975f3883d49a13bb23cc4cfce41c940bd | A circle intersects the sides \(BC\), \(CA\), and \(AB\) of triangle \(ABC\) at points \(D_1, D_2\), \(E_1, E_2\), and \(F_1, F_2\) respectively. The line segments \(D_1E_1\) and \(D_2F_2\) intersect at point \(L\), the line segments \(E_1F_1\) and \(E_2D_2\) intersect at point \(M\), and the line segments \(F_1D_1\) a... | [
"5"
] | [
"A circle intersects the sides \\(BC\\), \\(CA\\), and \\(AB\\) of triangle \\(ABC\\) at points \\(D_1, D_2\\), \\(E_1, E_2\\), and \\(F_1, F_2\\) respectively. The line segments \\(D_1E_1\\) and \\(D_2F_2\\) intersect at point \\(L\\), the line segments \\(E_1F_1\\) and \\(E_2D_2\\) intersect at point \\(M\\), and... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-6876c8732e6c7a1e70582870eed2d88dd00d8de6182e094c553332dc6ed0d796 | A circle intersects the sides of triangle \(ABC\) at six different points. Let \(\ell_A\) be the line determined by the points from the sides \(AB\) and \(AC\) farthest from \(A\). Define \(\ell_B\) and \(\ell_C\) similarly. Let \(A'\) be the intersection of \(\ell_B\) and \(\ell_C\). Define \(B'\) and \(C'\) similarly... | [
"1"
] | [
"A circle intersects the sides of triangle \\(ABC\\) at six different points. Let \\(\\ell_A\\) be the line determined by the points from the sides \\(AB\\) and \\(AC\\) farthest from \\(A\\). Define \\(\\ell_B\\) and \\(\\ell_C\\) similarly. Let \\(A'\\) be the intersection of \\(\\ell_B\\) and \\(\\ell_C\\). Defi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72d5bbc0a27af41cf8fdb1b6fc6b9675589a192ccc5a6b9569d4a6fb398ebb19 | A circle is called "interesting" if its center lies on the Y-axis and is tangent to the hyperbola \(x^2 - y^2 = 1\), whose vertex on the right branch is \(A\). When two "interesting" circles \(O_1\) and \(O_2\) are externally tangent at \(P\), there exists a constant \(k\) such that \(O_1O_2 = k \cdot AP\) always holds... | [
"2"
] | [
"A circle is called \"interesting\" if its center lies on the Y-axis and is tangent to the hyperbola \\(x^2 - y^2 = 1\\), whose vertex on the right branch is \\(A\\). When two \"interesting\" circles \\(O_1\\) and \\(O_2\\) are externally tangent at \\(P\\), there exists a constant \\(k\\) such that \\(O_1O_2 = k \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-36affd48301aa31bc36250414f4465656f5d5bdfce63a94b8223be10d427a270 | A circle is centered at \( O \) with a diameter of \( \overline{AC} \). Let \( B \) be a point on the tangent line at \( C \). Let \( H \) be the intersection of \( \overline{AB} \) with the circle, and let \( P \) be the intersection of \( \overline{CH} \) and \( \overline{BO} \). Let \( AP \) intersect the circle aga... | [
"no_valid_answer_found"
] | [
"A circle is centered at \\( O \\) with a diameter of \\( \\overline{AC} \\). Let \\( B \\) be a point on the tangent line at \\( C \\). Let \\( H \\) be the intersection of \\( \\overline{AB} \\) with the circle, and let \\( P \\) be the intersection of \\( \\overline{CH} \\) and \\( \\overline{BO} \\). Let \\( AP... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7d7374aefa4d20ac813be044535facd5b858632633f4a3a73aad3f38b6b3b9e4 | A circle is centered at \( O \), \( \overline{AB} \) is a diameter, and \( C \) is a point on the circle with \( \angle COB = 50^\circ \). What is the degree measure of \( \angle CAB \)? | [
"25"
] | [
"A circle is centered at \\( O \\), \\( \\overline{AB} \\) is a diameter, and \\( C \\) is a point on the circle with \\( \\angle COB = 50^\\circ \\). What is the degree measure of \\( \\angle CAB \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1dab53cabb3a70fccf985f5e839f743102546634e8e51f8d0e0423f46deaa9ee | A circle is centered at point $O$ in the plane. Distinct pairs of points $A, B$ and $C, D$ are diametrically opposite on this circle. Point $P$ is chosen on line segment $AD$ such that line $BP$ intersects the circle again at $M$ and line $AC$ at $X$, with $M$ being the midpoint of $PX$. Point $Y \neq X$ is taken such ... | [
"35^\\circ"
] | [
"A circle is centered at point $O$ in the plane. Distinct pairs of points $A, B$ and $C, D$ are diametrically opposite on this circle. Point $P$ is chosen on line segment $AD$ such that line $BP$ intersects the circle again at $M$ and line $AC$ at $X$, with $M$ being the midpoint of $PX$. Point $Y \\neq X$ is taken... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31dd5c9c6bb379c46ea76348d36f4096081d42056da2dae860909d89fba4546c | A circle is centered at the origin with a radius larger than one. The arc length of the arc of the circle above the line \( y = 1 \) is \( \frac{4\pi}{3} \). Find the radius of the circle. | [
"2"
] | [
"A circle is centered at the origin with a radius larger than one. The arc length of the arc of the circle above the line \\( y = 1 \\) is \\( \\frac{4\\pi}{3} \\). Find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c110c25e7085e7b3dc1ae9c59a42d149cc83834f1c03c99dce4021b74fefbb64 | A circle is centered at the origin with radius larger than one. The arc length of the arc of the circle above the line \( y = 1 \) is \(\frac{4}{3}\pi\). Find the radius of the circle. | [
"2"
] | [
"A circle is centered at the origin with radius larger than one. The arc length of the arc of the circle above the line \\( y = 1 \\) is \\(\\frac{4}{3}\\pi\\). Find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e17c24d53ca1c26385d98b30af2df066ac2417b89448b9afb17fac113deb74f2 | A circle is circumscribed about a regular $n$-gon with side $a$, another circle is inscribed in it. Determine the area of the annulus bounded by the circles and its width. | [
"$Q=\\frac{\\pi a^2}{4};\\ d=\\frac{a}{2}\\tan\\frac{90^\\circ}{n}$"
] | [
"A circle is circumscribed about a regular $n$-gon with side $a$, another circle is inscribed in it. Determine the area of the annulus bounded by the circles and its width."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f14d66b825791feefd35fe309c2205e59e5deca0f030dd010c67ffe865084008 | A circle is circumscribed about a right triangle with legs 7 and 24. What is the area of the circle? | [
"\\( 156.25\\pi \\)"
] | [
"A circle is circumscribed about a right triangle with legs 7 and 24. What is the area of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-dcd005afcbf6739084896664eb7a4e8dd79d749f418ffe1d8019629e2084da1b | A circle is circumscribed about a right triangle. Another circle of the same radius touches the legs of that triangle so that one of the points of tangency is the vertex of the triangle. Find the ratio of the area of the triangle to the area of the common part of the two given circles. | [
"$\\frac{3\\sqrt 3}{5\\pi -3}$"
] | [
"A circle is circumscribed about a right triangle. Another circle of the same radius touches the legs of that triangle so that one of the points of tangency is the vertex of the triangle. Find the ratio of the area of the triangle to the area of the common part of the two given circles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0c1c136f55f4185b81d7b2175da54b84a885c58b0704144e1afed9ed04de90b1 | A circle is circumscribed about a triangle with side lengths 20, 21, and 29. The circumference of the circle is split into 3 arcs. Find the length of the longest arc. | [
"\\(\\frac{29\\pi}{2}\\)"
] | [
"A circle is circumscribed about a triangle with side lengths 20, 21, and 29. The circumference of the circle is split into 3 arcs. Find the length of the longest arc."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-a1871d45494881dfe99c7c1f67a1e95ddcb5856657f6955324aebbe9465ce188 | A circle is circumscribed about a triangle with side lengths 20, 21, and 29. The interior of the circle is split into 4 regions. Let A, B, and C be the non-triangular regions. Find A + B - C. | [
"-210"
] | [
"A circle is circumscribed about a triangle with side lengths 20, 21, and 29. The interior of the circle is split into 4 regions. Let A, B, and C be the non-triangular regions. Find A + B - C."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 7,
"tir": 16
} | 30 | 0 | 0 |
omr-ws-8bb746477aaa9cdbdf179cead4bce5595a7e4e5a56124b1c6bfc02df9c85f5d5 | A circle is circumscribed about a triangle with sides $30$, $34$, and $16$. The circle is divided into $4$ regions, with the non-triangular regions having areas $A$, $B$, and $C$, where $C$ is the largest. Find the value of $C - A - B$. | [
"240"
] | [
"A circle is circumscribed about a triangle with sides $30$, $34$, and $16$. The circle is divided into $4$ regions, with the non-triangular regions having areas $A$, $B$, and $C$, where $C$ is the largest. Find the value of $C - A - B$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 23
} | 44 | 0 | 0 |
omr-ws-f10dad7b2953da71e5f8c3811dbab7e672ff98010dea878189e99fcaa9df424e | A circle is circumscribed about an equilateral triangle with side lengths of $9$ units each. What is the area of the circle, in square units? Express your answer in terms of $\pi$. | [
"27\\pi"
] | [
"A circle is circumscribed about an equilateral triangle with side lengths of $9$ units each. What is the area of the circle, in square units? Express your answer in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9a45dae476eb784331a373e2d4d83914112958b2cb3c59c86f9ff49c59eaf8a | A circle is circumscribed about an isosceles triangle $ABC$. A chord of length $m$, cutting the base $BC$ at a point $D$ is drawn through the vertex $A$. Given a ratio $| BD | : | DC | = k$ and an angle $A$ $(A<\pi /2)$. Find the radius of the circle. | [
"$\\frac{m\\sqrt {k^2+2k\\cos \\alpha +1}}{2(k+1)\\cos (A/2)}$"
] | [
"A circle is circumscribed about an isosceles triangle $ABC$. A chord of length $m$, cutting the base $BC$ at a point $D$ is drawn through the vertex $A$. Given a ratio $| BD | : | DC | = k$ and an angle $A$ $(A<\\pi /2)$. Find the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a7cbbf29f2ba6ddd46be915a4b49086b4322f587bbb56e1d6ef966617fb32394 | A circle is circumscribed about the triangle $ABC$ with sides $| AB | = 5 \left(1+\sqrt 3\right)$, $| BC | = 5 \sqrt 6$, $| AC | = 10$. A tangent to the circle is drawn through the point $C$ and a straight line parallel to the side $AC$ is drawn through the point $B$. The tangent and the straight line meet at a point $... | [
"$\\frac{125}{4}\\left(3+\\sqrt 3\\right)$"
] | [
"A circle is circumscribed about the triangle $ABC$ with sides $| AB | = 5 \\left(1+\\sqrt 3\\right)$, $| BC | = 5 \\sqrt 6$, $| AC | = 10$. A tangent to the circle is drawn through the point $C$ and a straight line parallel to the side $AC$ is drawn through the point $B$. The tangent and the straight line meet at ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3bd2917653b1d0ff944c8ca26a297d1c02aa8e9a9270a33bb79395404f4a673 | A circle is circumscribed about the triangle $ABC$ with sides\linebreak $| AB | =10\sqrt 2$, $ | AC | = 20$ and $\angle B=45^\circ$. A tangent is drawn to the circle through the point $C$ which cuts the extension of the side $AB$ at a point~$D$. Find the area of the triangle $BCD$. | [
"$150+\\frac{250}{\\sqrt 3}$"
] | [
"A circle is circumscribed about the triangle $ABC$ with sides\\linebreak $| AB | =10\\sqrt 2$, $ | AC | = 20$ and $\\angle B=45^\\circ$. A tangent is drawn to the circle through the point $C$ which cuts the extension of the side $AB$ at a point~$D$. Find the area of the triangle $BCD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d64637b3a25ae57300cea547707a351ca5800701cb4adb38f252ef024e136690 | A circle is circumscribed about triangle PQR. Take a point S on the circle. Construct three lines through S, each perpendicular to lines PQ, PR, and QR, and let the points of intersection with PQ, PR, and QR be A, B, and C respectively. If the coordinates of P, Q, R, and S are (0,0), (4,0), (0,3), and (2,2) respectivel... | [
"2x + y = 4"
] | [
"A circle is circumscribed about triangle PQR. Take a point S on the circle. Construct three lines through S, each perpendicular to lines PQ, PR, and QR, and let the points of intersection with PQ, PR, and QR be A, B, and C respectively. If the coordinates of P, Q, R, and S are (0,0), (4,0), (0,3), and (2,2) respec... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-667fe65088cbf2b2e4a9b51d24014ab647147272cb42d275288868a3d3e9e20e | A circle is circumscribed around $ABCD$ as follows: [asy] pair pA, pB, pC, pD, pO; pO = (0, 0); pA = pO + dir(-40); pB = pO + dir(40); pC = pO + dir(130); pD = pO + dir(190); draw(pA--pB--pC--pA); draw(pA--pD--pC--pA); label("$A$", pA, SE); label("$B$", pB, NE); label("$C$", pC, NW); label("$D$", pD, SW); draw(circle(p... | [
"110^\\circ"
] | [
"A circle is circumscribed around $ABCD$ as follows: [asy]\npair pA, pB, pC, pD, pO;\npO = (0, 0);\npA = pO + dir(-40);\npB = pO + dir(40);\npC = pO + dir(130);\npD = pO + dir(190);\ndraw(pA--pB--pC--pA);\ndraw(pA--pD--pC--pA);\nlabel(\"$A$\", pA, SE);\nlabel(\"$B$\", pB, NE);\nlabel(\"$C$\", pC, NW);\nlabel(\"$D$\... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-b4c40ea300264cb60d6ecd3b29ba22c8d7843244b3fb52e3a4638894bf9c9a1a | A circle is circumscribed around a square. A square is then circumscribed around the circle. What is the ratio of the area of the smaller square to the area of the larger square? | [
"\\dfrac{1}{2}"
] | [
"A circle is circumscribed around a square. A square is then circumscribed around the circle. What is the ratio of the area of the smaller square to the area of the larger square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6414e7e0c136677a5eec4651479dd8f6e2e39434e65c661cd6caf142b97fd884 | A circle is circumscribed around an isosceles triangle \(ABC\) with base \(AC\). On the arc \(AB\) that does not contain point \(C\), an arbitrary point \(D\) is marked. Let point \(E\) be the symmetric point to \(C\) with respect to the line \(BD\). If \(\angle BAC = 30^\circ\) and the circumradius of triangle \(ABC\)... | [
"R \\sqrt{2 + \\sqrt{3}}"
] | [
"A circle is circumscribed around an isosceles triangle \\(ABC\\) with base \\(AC\\). On the arc \\(AB\\) that does not contain point \\(C\\), an arbitrary point \\(D\\) is marked. Let point \\(E\\) be the symmetric point to \\(C\\) with respect to the line \\(BD\\). If \\(\\angle BAC = 30^\\circ\\) and the circumr... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae6eca4447b5d2962e3aa70bb76569f6a1620f7059caff2cfbda1b5cb827177e | A circle is circumscribed around an isosceles triangle whose two base angles are equal to $x^{\circ}$. Two points are chosen independently and randomly on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}.$ Find the sum of the largest and smallest po... | [
"120"
] | [
"A circle is circumscribed around an isosceles triangle whose two base angles are equal to $x^{\\circ}$. Two points are chosen independently and randomly on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\\frac{14}{25}.$ Find the sum of the largest and smal... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 5
} | 52 | 0 | 0 |
omr-ws-1114b11f8e4fed31b3675418928abc5332ad55a0fbaed6165d1774fce58b1df3 | A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x$. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}$. Find the difference between ... | [
"048",
"48"
] | [
"A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x$. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\\frac{14}{25}$. Find the difference be... | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 95,
"tir": 32
} | 127 | 0 | 0 |
omr-ws-cb0f887b449ecb3204eabdb0861371e4d1e1d997f45a1395bb6fe1f22e40b53e | A circle is circumscribed around triangle \(ABC\) and another circle is inscribed in it, touching the sides \(BC\), \(CA\), and \(AB\) at points \(A_1\), \(B_1\), and \(C_1\), respectively. The line \(B_1C_1\) intersects the line \(BC\) at point \(P\), and \(M\) is the midpoint of segment \(PA_1\). If the tangents from... | [
"58"
] | [
"A circle is circumscribed around triangle \\(ABC\\) and another circle is inscribed in it, touching the sides \\(BC\\), \\(CA\\), and \\(AB\\) at points \\(A_1\\), \\(B_1\\), and \\(C_1\\), respectively. The line \\(B_1C_1\\) intersects the line \\(BC\\) at point \\(P\\), and \\(M\\) is the midpoint of segment \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-f2393d62e5a52dfb954aa482c14a56431140eaf3f7b22550f152ab97681700ba | A circle is circumscribed around triangle \(ABC\) with \(\angle B = 60^\circ\). Tangents to the circle at points \(A\) and \(C\) intersect at point \(B_1\). On rays \(AB\) and \(CB\), points \(A_0\) and \(C_0\) are noted respectively, such that \(AA_0 = AC = CC_0\). If \(A_0C_0\) intersects the line \(B_1A\) at point \... | [
"\\dfrac{\\sqrt{2}}{2}"
] | [
"A circle is circumscribed around triangle \\(ABC\\) with \\(\\angle B = 60^\\circ\\). Tangents to the circle at points \\(A\\) and \\(C\\) intersect at point \\(B_1\\). On rays \\(AB\\) and \\(CB\\), points \\(A_0\\) and \\(C_0\\) are noted respectively, such that \\(AA_0 = AC = CC_0\\). If \\(A_0C_0\\) intersects... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-1f1a75c71b06407332dc2f3b30484e4ac81a9121cf1e810dc344c02806f9bb9e | A circle is circumscribed around triangle \(ABC\) with center \(O\), and \(\angle BAO = 20^\circ\). Find the measure of the angle formed by the lines containing the altitudes \(AP\) and \(BH\) of this triangle. | [
"110"
] | [
"A circle is circumscribed around triangle \\(ABC\\) with center \\(O\\), and \\(\\angle BAO = 20^\\circ\\). Find the measure of the angle formed by the lines containing the altitudes \\(AP\\) and \\(BH\\) of this triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-9615864860986236ba8593d79934470f196a47b20ef096576025fd1d9b647090 | A circle is colored with two colors. Let $N$ be the smallest number of equally spaced points on the circle such that there are guaranteed to be three points of the same color that are equally spaced from each other. Find $N$. | [
"9"
] | [
"A circle is colored with two colors. Let $N$ be the smallest number of equally spaced points on the circle such that there are guaranteed to be three points of the same color that are equally spaced from each other. Find $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-fb99063bfd848734f87eace1bd165aaa4e89afc220331c9673aa36421448ab92 | A circle is constructed on a line segment \(AB\) of length \(R\) as its diameter. A second circle of the same radius \(R\) is centered at point \(A\). A third circle touches the first circle internally, the second circle externally, and also touches the line segment \(AB\). Find the radius of the third circle. | [
"\\dfrac{4R}{9}"
] | [
"A circle is constructed on a line segment \\(AB\\) of length \\(R\\) as its diameter. A second circle of the same radius \\(R\\) is centered at point \\(A\\). A third circle touches the first circle internally, the second circle externally, and also touches the line segment \\(AB\\). Find the radius of the third c... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-184a6373e13880a614f5c0be9a96d36deec95ce5fcd13b7387ff6fc2812c0702 | A circle is constructed on the leg $BC$ of the right triangle $ABC$ as a diameter, the circle cutting the hypotenuse at a point $D$ so that\linebreak $| AD | : | DB | = 1 : 4$. Find the length of the altitude dropped from the vertex $C$ of the right angle to the hypotenuse if the length of the leg $BC$ is known to be e... | [
"$2\\sqrt 5$"
] | [
"A circle is constructed on the leg $BC$ of the right triangle $ABC$ as a diameter, the circle cutting the hypotenuse at a point $D$ so that\\linebreak $| AD | : | DB | = 1 : 4$. Find the length of the altitude dropped from the vertex $C$ of the right angle to the hypotenuse if the length of the leg $BC$ is known t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e071fc20f0bd257a0cb6aa4a37c3866e9b307cdfd165e3e11997d11cec8444e | A circle is divided by $2N$ points into $2N$ arcs of length $1$. These points are joined in pairs to form $N$ chords. Each chord divides the circle into two arcs, the length of each being an even integer. If $N$ is the smallest integer for which this is possible, find the sum of all possible values of $N$. | [
"2"
] | [
"A circle is divided by $2N$ points into $2N$ arcs of length $1$. These points are joined in pairs to form $N$ chords. Each chord divides the circle into two arcs, the length of each being an even integer. If $N$ is the smallest integer for which this is possible, find the sum of all possible values of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-621af3b0e534a468cc37ead0d62b0ef13c12a30c9b33d990d7e85a3d9d85b9a1 | A circle is divided by $2n$ points into $2n$ equal arcs. Let $P_1, P_2, \ldots, P_{2n}$ be an arbitrary permutation of the $2n$ division points. Consider the polygonal line $P_1 P_2 \cdots P_{2n} P_1$. Calculate the minimum number of pairs of parallel segments that must exist in this polygonal line for $n=6$. | [
"6"
] | [
"A circle is divided by $2n$ points into $2n$ equal arcs. Let $P_1, P_2, \\ldots, P_{2n}$ be an arbitrary permutation of the $2n$ division points. Consider the polygonal line $P_1 P_2 \\cdots P_{2n} P_1$. Calculate the minimum number of pairs of parallel segments that must exist in this polygonal line for $n=6$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-0afd35d5a163a1d60e8544bc8301f6d132ad743dbdd82d93dc649098121fba4e | A circle is divided by $3k$ points into $k$ arcs of lengths $1$, $2$, and $3$ each. Calculate the number of ways to place these points such that no two points are diametrically opposite. | [
"0"
] | [
"A circle is divided by $3k$ points into $k$ arcs of lengths $1$, $2$, and $3$ each. Calculate the number of ways to place these points such that no two points are diametrically opposite."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cb594e4ce4cc0ca8ae7113504381f111c8e4c03325241f4114d798b761b1e487 | A circle is divided into $2n$ equal arcs by $2n$ points. Find all $n>1$ such that these points can be joined in pairs using $n$ segments, all of different lengths, and such that each point is the endpoint of exactly one segment. | [
"n \\geq 4"
] | [
"A circle is divided into $2n$ equal arcs by $2n$ points. Find all $n>1$ such that these points can be joined in pairs using $n$ segments, all of different lengths, and such that each point is the endpoint of exactly one segment."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-1f93a1a1470670111897ff7cf3dc586f1e161fbf11a6a05defd82c4293060026 | A circle is divided into $2n$ equal parts by $2n$ points. Ali draws $n+1$ arcs of lengths $1, 2, \ldots, n+1$. Let $k$ be the number of pairs of arcs such that one arc is completely inside the other. Find the maximum possible value of $k$ for $n=5$. | [
"15"
] | [
"A circle is divided into $2n$ equal parts by $2n$ points. Ali draws $n+1$ arcs of lengths $1, 2, \\ldots, n+1$. Let $k$ be the number of pairs of arcs such that one arc is completely inside the other. Find the maximum possible value of $k$ for $n=5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-784c41d675fd5280da2000882b37291a59c6457763d688be060ba25dbce7593c | A circle is divided into $2n$ equal parts with $2n$ points. Ali draws $n+1$ arcs with lengths $1, 2, \ldots, n+1$. Let $k$ be the number of pairs of arcs such that one is completely contained within the other. Find the sum of all possible values of $k$ for $n=5$. | [
"120"
] | [
"A circle is divided into $2n$ equal parts with $2n$ points. Ali draws $n+1$ arcs with lengths $1, 2, \\ldots, n+1$. Let $k$ be the number of pairs of arcs such that one is completely contained within the other. Find the sum of all possible values of $k$ for $n=5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6ef7ce5c5685f6d2812e52a87a0b706d58321a26eb29e5ec0b5b74be17fdfa37 | A circle is divided into $360$ equal parts, with points numbered clockwise from $1$ to $360$. Alice and Bob both start at point $1$. In each turn, Alice moves $35$ points clockwise, while Bob moves $13$ points counterclockwise. How many turns at least are needed for Alice and Bob to meet at the same point? | [
"15"
] | [
"A circle is divided into $360$ equal parts, with points numbered clockwise from $1$ to $360$. Alice and Bob both start at point $1$. In each turn, Alice moves $35$ points clockwise, while Bob moves $13$ points counterclockwise. How many turns at least are needed for Alice and Bob to meet at the same point?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-a1a3a7736b09febf2f269f8d654361dc6bbb7551e4f29654145595a369272a2d | A circle is divided into $432$ congruent arcs by $432$ points. The points are colored in four colors such that $108$ points are colored Red, $108$ points are colored Green, $108$ points are colored Blue, and $108$ points are colored Yellow. Let $N$ be the number of ways to choose three points of each color in such a wa... | [
"156"
] | [
"A circle is divided into $432$ congruent arcs by $432$ points. The points are colored in four colors such that $108$ points are colored Red, $108$ points are colored Green, $108$ points are colored Blue, and $108$ points are colored Yellow. Let $N$ be the number of ways to choose three points of each color in such... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.