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-a4ff4ef99ab79b35856dca5a72d4c81448c7822675b56ade6ed15bb599c92102 | A circle is inscribed inside a square, which is itself inscribed inside an equilateral triangle with side length \( s \). What is the area of the circle in terms of \( s \)? | [
"\\(\\pi (7s^2 - 4s^2\\sqrt{3})\\)"
] | [
"A circle is inscribed inside a square, which is itself inscribed inside an equilateral triangle with side length \\( s \\). What is the area of the circle in terms of \\( s \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 2
} | 9 | 0 | 0 |
omr-ws-7a326f81510b3464a21fd31b8e60ce6c877062639380f5019b2ffb47cc6fb453 | A circle is inscribed inside a square, which is itself inscribed inside an equilateral triangle with side length \( s \). What is the area of the square in terms of \( s \)? | [
"\\( s^2 (52 - 30 \\sqrt{3}) \\)"
] | [
"A circle is inscribed inside a square, which is itself inscribed inside an equilateral triangle with side length \\( s \\). What is the area of the square in terms of \\( s \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2208096b1e4c64746302730a964f3d23867daa47319a528c5915cd32598b8f23 | A circle is inscribed inside a trapezium ABCD such that it touches all 4 sides. The area of the trapezium is 4 square units and the diameter of the circle is 1 unit. Find the value of the ratio AB/DC. | [
"\\(7 + 4\\sqrt{3}\\)"
] | [
"A circle is inscribed inside a trapezium ABCD such that it touches all 4 sides. The area of the trapezium is 4 square units and the diameter of the circle is 1 unit. Find the value of the ratio AB/DC."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-e0405611a27335408ce4e423afdade460aa6a527a8a586d1f8f4b03bd7cbe8d1 | A circle is inscribed inside of a regular pentagon. Then, another regular pentagon is inscribed inside of that circle, and a circle is inscribed in that pentagon. This process continues until there are $5$ circles and $5$ pentagons. If the side length of the largest pentagon is $20$, find the area of the smallest circl... | [
"100\\pi \\left( \\frac{\\cos^5\\left(\\frac{\\pi}{5}\\right)}{\\sin\\left(\\frac{\\pi}{5}\\right)} \\right)^2"
] | [
"A circle is inscribed inside of a regular pentagon. Then, another regular pentagon is inscribed inside of that circle, and a circle is inscribed in that pentagon. This process continues until there are $5$ circles and $5$ pentagons. If the side length of the largest pentagon is $20$, find the area of the smallest ... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ababb810e4144243845ce5624cac861bc5b89cdc36bd95f6f5f12dcdf00a6076 | A circle is inscribed into an isosceles triangle $ABC$ with base $AC$, the circle touching the lateral side $AB$ at a point $M$. A perpendicular $ML$ is drawn through the point $M$ to the side $AC$ of the triangle $ABC$ (the point $L$ is the foot of the perpendicular). Find the angle $BCA$ if the area of the triangle $... | [
"$\\arccos \\sqrt {2(1-S)}$"
] | [
"A circle is inscribed into an isosceles triangle $ABC$ with base $AC$, the circle touching the lateral side $AB$ at a point $M$. A perpendicular $ML$ is drawn through the point $M$ to the side $AC$ of the triangle $ABC$ (the point $L$ is the foot of the perpendicular). Find the angle $BCA$ if the area of the trian... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b838848dce86c59669bd904efab61e8d19e8402d26a830f741305b2b2b2bd949 | A circle is internally tangent to 6 smaller circles such that each smaller circle is internally tangent to two adjacent ones, and the radii of opposite smaller circles are equal. If the radius of the given circle is 10, calculate the sum of the radii of the 6 smaller circles. | [
"20"
] | [
"A circle is internally tangent to 6 smaller circles such that each smaller circle is internally tangent to two adjacent ones, and the radii of opposite smaller circles are equal. If the radius of the given circle is 10, calculate the sum of the radii of the 6 smaller circles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b546d3a1f5b21609b2d39a13672578955638fd35094ab70bc0adebeb97d5a88 | A circle is marked with the 2013 roots of unity. Each root has a rifle and can shoot at a point in a line. If a root is hit, it is "killed" and does not shoot anymore. Find the least number of shots necessary to eliminate all the roots. | [
"1007"
] | [
"A circle is marked with the 2013 roots of unity. Each root has a rifle and can shoot at a point in a line. If a root is hit, it is \"killed\" and does not shoot anymore. Find the least number of shots necessary to eliminate all the roots."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 1
} | 24 | 0 | 0 |
omr-ws-357efe911864c6288d0be29f1d578abb5cac4e79207427303aaae3995db9125b | A circle is partitioned into $3k$ sectors by $3k$ points. There are $k$ sectors of length $1$, $k$ sectors of length $2$, and $k$ sectors of length $3$. Calculate the number of ways to choose two points such that they lie on a diameter of the circle. | [
"k"
] | [
"A circle is partitioned into $3k$ sectors by $3k$ points. There are $k$ sectors of length $1$, $k$ sectors of length $2$, and $k$ sectors of length $3$. Calculate the number of ways to choose two points such that they lie on a diameter of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-3a0f99324a86392451a9450d8fbbbacd2d02076e3fe944abcb125b51235e48c1 | A circle is placed in the $xy$-plane and a line $L$ is drawn through the center of the circle. Suppose $P$ is a point interior to the circle which is $6$ units from the circle, $6$ units from the line $L$, and $10$ units from the closest intersection point of the line $L$ with the circle. What is the area of the circle... | [
"\\( 256\\pi \\)"
] | [
"A circle is placed in the $xy$-plane and a line $L$ is drawn through the center of the circle. Suppose $P$ is a point interior to the circle which is $6$ units from the circle, $6$ units from the line $L$, and $10$ units from the closest intersection point of the line $L$ with the circle. What is the area of the c... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-63a5c4c54bcaef7db3211f20dbb72bf09dcffa142c4d03bfee1f985f81a0a58d | A circle is randomly chosen within a circle of radius $1$ by selecting a random point for its center and a random radius such that the new circle is contained within the original circle. What is the probability that the new circle contains the center of the original circle? | [
"\\(\\frac{1}{2}\\)"
] | [
"A circle is randomly chosen within a circle of radius $1$ by selecting a random point for its center and a random radius such that the new circle is contained within the original circle. What is the probability that the new circle contains the center of the original circle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e7363cd422154fafe71d4b80053bb2eba41bcce2c76eff3ebb1d689ea6b9d69 | A circle is randomly chosen within a circle of radius 1 by selecting a random point for its center and a random radius such that the new circle is contained within the original circle. What is the probability that the new circle contains the center of the original circle? | [
"\\(\\frac{3}{2} - 2\\ln 2\\)"
] | [
"A circle is randomly chosen within a circle of radius 1 by selecting a random point for its center and a random radius such that the new circle is contained within the original circle. What is the probability that the new circle contains the center of the original circle?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 25
} | 37 | 0 | 0 |
omr-ws-5fd88cd1058732febdf974d9382e5da7a806b0a83997cb4bff0c200d9a21fdfa | A circle is surrounded by 6 congruent circles. If each circle has a radius of 3, find the total area between the seven circles. | [
"18\\pi"
] | [
"A circle is surrounded by 6 congruent circles. If each circle has a radius of 3, find the total area between the seven circles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-235f414aa134b23e305f4bbebc8c27e7bb3506ff83441a700b83b46c35e1eb10 | A circle is tangent to a line at a point $A$. The point $P$ on the circle is projected vertically onto the line at $Q$. If $\left | PQ \right |=3$ and $\left | AQ \right |=4$, what is the radius of the circle? | [
"\\(\\frac{25}{6}\\)"
] | [
"A circle is tangent to a line at a point $A$. The point $P$ on the circle is projected vertically onto the line at $Q$. If $\\left | PQ \\right |=3$ and $\\left | AQ \\right |=4$, what is the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-04f73aaf6ba075d19bba7ca895955f6b185e99a013577ef9d49b22987619ddee | A circle is tangent to all sides of a square. Let \( P \) be a point on the circle such that the distances from \( P \) to sides \( AB \) and \( AD \) are 2 cm and 1 cm, respectively. What is the sum of the possible values for the radius of the circle? | [
"6"
] | [
"A circle is tangent to all sides of a square. Let \\( P \\) be a point on the circle such that the distances from \\( P \\) to sides \\( AB \\) and \\( AD \\) are 2 cm and 1 cm, respectively. What is the sum of the possible values for the radius of the circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-15e5990a4aa950938d04ea0ee57998bd7d1fa29d7c4b90f1f0f6a6d66d69db54 | A circle is tangent to one of the sides of a square and passes through two of its vertices. Determine the ratio between the side of the square and the radius of the circle. | [
"\\(\\frac{4}{5}\\)"
] | [
"A circle is tangent to one of the sides of a square and passes through two of its vertices. Determine the ratio between the side of the square and the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-465f5963e9b3eb5c10ffff73b499fa98a77448a2c0984384c9cf0fe0e8731790 | A circle is tangent to rays \(OA\) and \(OB\) at points \(A\) and \(B\), respectively. Let \(K\) be a point on the minor arc \(AB\) of this circle. Let \(L\) be a point on \(OB\) such that \(OA \parallel KL\). Let \(M\) be the intersection (distinct from \(K\)) of \(AK\) and the circumcircle \(\omega\) of \(\triangle K... | [
"10"
] | [
"A circle is tangent to rays \\(OA\\) and \\(OB\\) at points \\(A\\) and \\(B\\), respectively. Let \\(K\\) be a point on the minor arc \\(AB\\) of this circle. Let \\(L\\) be a point on \\(OB\\) such that \\(OA \\parallel KL\\). Let \\(M\\) be the intersection (distinct from \\(K\\)) of \\(AK\\) and the circumcirc... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-8a928f0511c7b96741ef42bc2c0ca3d7d9e16b81ced45ab21fb5336ffda9214b | A circle is tangent to rays \(OA\) and \(OB\). Let \(K\) be a point on the minor arc \(AB\) of this circle. Let \(L\) be a point on line \(OB\) such that \(OA \parallel KL\). Let \(M\) be the intersection (distinct from \(K\)) of line \(AK\) and the circumcircle \(\omega\) of triangle \(KLB\). If the radius of the circ... | [
"no_valid_answer_found"
] | [
"A circle is tangent to rays \\(OA\\) and \\(OB\\). Let \\(K\\) be a point on the minor arc \\(AB\\) of this circle. Let \\(L\\) be a point on line \\(OB\\) such that \\(OA \\parallel KL\\). Let \\(M\\) be the intersection (distinct from \\(K\\)) of line \\(AK\\) and the circumcircle \\(\\omega\\) of triangle \\(KL... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d88ca0100e54f10732dfb96a8f1f6766e1088ad626b2c7c5f2cbf7cbd551b74f | A circle is tangent to sides \(AB\) and \(AD\) of square \(ABCD\) and also tangent to a semicircle with diameter \(DC = 4\). Find the radius of the circle. | [
"\\( 8 - 4\\sqrt{3} \\)"
] | [
"A circle is tangent to sides \\(AB\\) and \\(AD\\) of square \\(ABCD\\) and also tangent to a semicircle with diameter \\(DC = 4\\). Find the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 26
} | 73 | 0 | 0 |
omr-ws-648621c6be8847a46e46a3f029bc0a41ee38cc43ce48c860d8576428967d60b7 | A circle is tangent to the lines $4x - 3y = 30$ and $4x - 3y = -10.$ The center of the circle lies on the line $2x + y = 0.$ Find the center of the circle. | [
"(1,-2)"
] | [
"A circle is tangent to the lines $4x - 3y = 30$ and $4x - 3y = -10.$ The center of the circle lies on the line $2x + y = 0.$ Find the center of the circle."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-13c8ec56b3f84519226240b18fbdc37fddd80c0f599c5fe3ff79f9d63eebcf78 | A circle is tangent to the parabola with equation \( y = x^2 \) at two points. How much higher is the center of the circle than the points of tangency? | [
"\\(\\frac{1}{2}\\)"
] | [
"A circle is tangent to the parabola with equation \\( y = x^2 \\) 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": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-dbce8db1ed595c9b2b14a1acb93c5bcb67be06958e7cb3835f789262ff3df9ee | A circle is tangent to the sides of a square AB, BC, CD, DA at points K, L, M, N respectively. Let U be a point on segment DM and V be a point on segment DN such that BU is parallel to KV. If the side length of the square is 10 units, calculate the length of segment UV. | [
"5"
] | [
"A circle is tangent to the sides of a square AB, BC, CD, DA at points K, L, M, N respectively. Let U be a point on segment DM and V be a point on segment DN such that BU is parallel to KV. If the side length of the square is 10 units, calculate the length of segment UV."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-2b041916a522413aa76d862d423035162c62f65efc4ef71cf0005d2cbc257362 | A circle is tangent to the y-axis at \((0, 2)\) and the larger of its x-intercepts is 8. Calculate the area of the circle. | [
"\\(\\frac{289}{16} \\pi\\)"
] | [
"A circle is tangent to the y-axis at \\((0, 2)\\) and the larger of its x-intercepts is 8. Calculate the area of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-2d3f226bd23f67963cb1dedb3fbca38ecd31767c0ed8d3fdcd5248c0d503f38a | A circle is tangent to three semicircles with radii 1, 2, and 3, where the semicircles with radii 1 and 2 are externally tangent to each other and both lie on the diameter of the larger semicircle of radius 3. What is the radius \( r \) of the circle that is tangent to all three semicircles and is completely contained ... | [
"\\(\\frac{6}{7}\\)"
] | [
"A circle is tangent to three semicircles with radii 1, 2, and 3, where the semicircles with radii 1 and 2 are externally tangent to each other and both lie on the diameter of the larger semicircle of radius 3. What is the radius \\( r \\) of the circle that is tangent to all three semicircles and is completely con... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-99e996cfbd920027c9e3cb92275f742699e2d00c8a1bc5592cc913d8d468121c | A circle is tangent to two perpendicular lines. There is a point on the circle that is 8 units from one line and 9 units from the other. What is the radius of the circle? | [
"5 or 29"
] | [
"A circle is tangent to two perpendicular lines. There is a point on the circle that is 8 units from one line and 9 units from the other. What is the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af33fca3a6f16584e5e835e119e6edd1073794ae58f888f02c6efc040ee54844 | A circle is tangential to sides \(AB\) and \(AC\) of convex quadrilateral \(ABCD\) at \(G\) and \(H\) respectively, and cuts diagonal \(AC\) at \(E\) and \(F\). What are the necessary and sufficient conditions such that there exists another circle which passes through \(E\) and \(F\), and is tangential to \(DA\) and \(... | [
"AB + CD = BC + AD"
] | [
"A circle is tangential to sides \\(AB\\) and \\(AC\\) of convex quadrilateral \\(ABCD\\) at \\(G\\) and \\(H\\) respectively, and cuts diagonal \\(AC\\) at \\(E\\) and \\(F\\). What are the necessary and sufficient conditions such that there exists another circle which passes through \\(E\\) and \\(F\\), and is ta... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-605de4cc75f154b0b9a8ce7bf4a998b1bf87037b9756867c2edb17b4ef73d970 | A circle lies 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 lies 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": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-2bcef64714315e04ec917114ecbc31401a52a776a2113054a5c965dd621f42a5 | A circle of a radius of 4 cm is inscribed in a triangle. One of its sides is divided by the point of tangency into two portions equal to 6 cm and 8 cm. Find the lengths of the other two sides. | [
"$AB=13$ cm; $BC=15$ cm"
] | [
"A circle of a radius of 4 cm is inscribed in a triangle. One of its sides is divided by the point of tangency into two portions equal to 6 cm and 8 cm. Find the lengths of the other two sides."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2aa446655b527106c50ec8f2da248384b374ca0aeecefae1971e22a48aaa72f5 | A circle of area 20 is centered at point $D$. Points $A$, $B$, and $C$ lie on the circumference of the circle, and $\triangle ABC$ has an area of 8. If $\angle ADB = \alpha$, $\angle BDC = \gamma$, and $\angle ADC = \beta$, find the value of $\sin \alpha + \sin \beta + \sin \gamma$. | [
"\\(\\frac{4 \\pi}{5}\\)"
] | [
"A circle of area 20 is centered at point $D$. Points $A$, $B$, and $C$ lie on the circumference of the circle, and $\\triangle ABC$ has an area of 8. If $\\angle ADB = \\alpha$, $\\angle BDC = \\gamma$, and $\\angle ADC = \\beta$, find the value of $\\sin \\alpha + \\sin \\beta + \\sin \\gamma$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-f0c377089206fb3bf8b9e135afceb8b7e77c2c3ce41c60074fce55a75947e0ca | A circle of constant radius passes through a fixed point $A$ and intersects two fixed straight lines $AB$ and $AC$ at points $B$ and $C$, respectively. Let $H$ be the orthocenter of triangle $ABC$. If the coordinates of $A$ are $(0,0)$, the equations of lines $AB$ and $AC$ are $y = x$ and $y = -x$, respectively, and th... | [
"x^2 + y^2 = 25"
] | [
"A circle of constant radius passes through a fixed point $A$ and intersects two fixed straight lines $AB$ and $AC$ at points $B$ and $C$, respectively. Let $H$ be the orthocenter of triangle $ABC$. If the coordinates of $A$ are $(0,0)$, the equations of lines $AB$ and $AC$ are $y = x$ and $y = -x$, respectively, a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-ea5587239fd48ab5afca0254e0e2468718560b4bff73ccea154e51791a981232 | A circle of constant radius passes through a fixed point \( A \) and intersects two fixed straight lines \( AB \) and \( AC \) at points \( B \) and \( C \), respectively. Let \( H \) be the orthocenter of triangle \( ABC \). If the coordinates of \( A \) are \( (0, 0) \), \( B \) are \( (2, 0) \), and \( C \) are \( (... | [
"x^2 + y^2 = 2"
] | [
"A circle of constant radius passes through a fixed point \\( A \\) and intersects two fixed straight lines \\( AB \\) and \\( AC \\) at points \\( B \\) and \\( C \\), respectively. Let \\( H \\) be the orthocenter of triangle \\( ABC \\). If the coordinates of \\( A \\) are \\( (0, 0) \\), \\( B \\) are \\( (2, 0... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-7fc78675ec46f04ca8977cf8d722cfd06edfa65543f58ec9e768fbf1866fa3dc | A circle of diameter 10 is inscribed in a trapezoid ABCD where angles A and B are right angles and AD=8. Find the length of the longest side of this trapezoid. | [
"\\(\\frac{40}{3}\\)"
] | [
"A circle of diameter 10 is inscribed in a trapezoid ABCD where angles A and B are right angles and AD=8. Find the length of the longest side of this trapezoid."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 9
} | 26 | 0 | 0 |
omr-ws-87b809a21cd66a811e2e46c32722dde3450779c5f3df4213cb1875caa99ca044 | A circle of diameter 10 is inscribed in a trapezoid \(ABCD\) where angles \(A\) and \(B\) are right angles and \(AD = 8\). Find the length of the longest side of this trapezoid. Express your answer as an improper fraction reduced to lowest terms. | [
"\\(\\frac{40}{3}\\)"
] | [
"A circle of diameter 10 is inscribed in a trapezoid \\(ABCD\\) where angles \\(A\\) and \\(B\\) are right angles and \\(AD = 8\\). Find the length of the longest side of this trapezoid. Express your answer as an improper fraction reduced to lowest terms."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 6
} | 23 | 0 | 0 |
omr-ws-2bde1d73ec2b0f90da4b084f0c625915ed9bb397eac95f2efd346004a045b531 | A circle of diameter \(d\) is inscribed in a convex quadrilateral \(ABCD\) and touches sides \(BC\) and \(DA\) at points \(K\) and \(L\) respectively. Given that the harmonic mean of the sides \(AB\) and \(CD\) is equal to the segment \(KL\), calculate the value of \(d\) if \(AB = 12\) and \(CD = 18\). | [
"14.4"
] | [
"A circle of diameter \\(d\\) is inscribed in a convex quadrilateral \\(ABCD\\) and touches sides \\(BC\\) and \\(DA\\) at points \\(K\\) and \\(L\\) respectively. Given that the harmonic mean of the sides \\(AB\\) and \\(CD\\) is equal to the segment \\(KL\\), calculate the value of \\(d\\) if \\(AB = 12\\) and \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d13f3975f8cddf56db9bb0a5d323f3e4cc398175d55706d88b94e2924d2f600d | A circle of integer radius $r$ has a chord $PQ$ of length 8. There is a point $X$ on chord $PQ$ such that $\overline{PX} = 2$ and $\overline{XQ} = 6$. Call a chord $AB$ euphonic if it contains $X$ and both $\overline{AX}$ and $\overline{XB}$ are integers. What is the minimal possible integer $r$ such that there exist 6... | [
"7"
] | [
"A circle of integer radius $r$ has a chord $PQ$ of length 8. There is a point $X$ on chord $PQ$ such that $\\overline{PX} = 2$ and $\\overline{XQ} = 6$. Call a chord $AB$ euphonic if it contains $X$ and both $\\overline{AX}$ and $\\overline{XB}$ are integers. What is the minimal possible integer $r$ such that ther... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-b7c22da3735006b9fd87c3dba955b8cfb21fc900011926b429289edf26c21906 | A circle of length (circumference) $10$ is inscribed in a convex polygon with a perimeter of $15$. What fraction of the area of the polygon is occupied by the circle? | [
"\\(\\frac{2}{3}\\)"
] | [
"A circle of length (circumference) $10$ is inscribed in a convex polygon with a perimeter of $15$. What fraction of the area of the polygon is occupied by the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-eedf131be26900cdd10533af3fba2abef3913fbea040b145ac95b89543453a59 | A circle of maximal area is inscribed in the region bounded by the graph of $y = -x^2 - 7x + 12$ and the $x$-axis. The radius of this circle is $\frac{\sqrt{p} + q}{r}$, where $p$, $q$, and $r$ are integers and $q$ and $r$ are relatively prime. What is $p + q + r$? | [
"98"
] | [
"A circle of maximal area is inscribed in the region bounded by the graph of $y = -x^2 - 7x + 12$ and the $x$-axis. The radius of this circle is $\\frac{\\sqrt{p} + q}{r}$, where $p$, $q$, and $r$ are integers and $q$ and $r$ are relatively prime. What is $p + q + r$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 9
} | 29 | 0 | 0 |
omr-ws-a2bcf75ad1da0eb6dba64d7e4621fd21b28a94998cd73295ce1e70a8f05bd89b | A circle of radius $1$ and center $(0,0)$ is drawn. It is then stretched by a factor of $3$ along the line $x = y$, then stretched by a factor of $\frac{1}{2}$ along the line $y = -x$. The points that lie on the original and transformed circle form a convex polygon with area $\frac{a\sqrt{b}}{c}$ where $a,c$ are relati... | [
"65"
] | [
"A circle of radius $1$ and center $(0,0)$ is drawn. It is then stretched by a factor of $3$ along the line $x = y$, then stretched by a factor of $\\frac{1}{2}$ along the line $y = -x$. The points that lie on the original and transformed circle form a convex polygon with area $\\frac{a\\sqrt{b}}{c}$ where $a,c$ ar... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 3
} | 20 | 0 | 0 |
omr-ws-f766dc9a66148ec76b8d5c69ea94cddaafbe6e8fab3757424d2df4376095912d | A circle of radius $1$ is cut into four equal arcs, which are then arranged to form a specific shape. What is the area of this shape? | [
"2"
] | [
"A circle of radius $1$ is cut into four equal arcs, which are then arranged to form a specific shape. What is the area of this shape?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23e7f3cd66bedfda8d1960800907c978bcc9d93903e728b797868384f5d5921c | A circle of radius $1$ is internally tangent to two circles of radius $2$ at points $A$ and $B$, where $AB$ is a diameter of the smaller circle. What is the area of the region that is outside the smaller circle and inside each of the two larger circles? | [
"\\(\\frac{5\\pi}{3} - 2\\sqrt{3}\\)",
"\\dfrac{5\\pi}{3} - 2\\sqrt{3}"
] | [
"A circle of radius $1$ is internally tangent to two circles of radius $2$ at points $A$ and $B$, where $AB$ is a diameter of the smaller circle. What is the area of the region that is outside the smaller circle and inside each of the two larger circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 27
} | 75 | 0 | 0 |
omr-ws-1306d52d6ad0c6b85440fe40544cb19606e5fc54dcb42fa9ff568c7d72bcbabd | A circle of radius $1$ is surrounded by $4$ circles of radius $r$. What is the value of $r$? | [
"\\( 1 + \\sqrt{2} \\)"
] | [
"A circle of radius $1$ is surrounded by $4$ circles of radius $r$. What is the value of $r$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-7e7aa109939c89fb17b6786ed1d5d4091c29b509a99ba2d8a135defa77538321 | A circle of radius $1$ rolls around a circle of radius $\sqrt{2}$. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made $n$ circuits around the bigger one? | [
"\\( \\lfloor n\\sqrt{2} \\rfloor + 1 \\)"
] | [
"A circle of radius $1$ rolls around a circle of radius $\\sqrt{2}$. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made $n$ circuits around the bigger one?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5ddf58159201c2d94e9f9c6f25aead3d48b45db200abbd3596beefe6e5675f9a | A circle of radius $1$ touches the curve $y = |2x|$ twice. Find the area of the region that lies between the two curves. | [
"\\(\\frac{\\pi}{2} - \\frac{16}{5}\\)"
] | [
"A circle of radius $1$ touches the curve $y = |2x|$ twice. Find the area of the region that lies between the two curves."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b2b60dd0dc96dd476dc23572eb01d3c002831b0fc5a10c06bca9190eb7ff0d56 | A circle of radius $1+\sqrt 2$ is circumscribed about an isosceles right triangle. Find the radius of the circle which touches the legs of the triangle and is internally tangent to the circle circumscribed about it. | [
"$R=2$"
] | [
"A circle of radius $1+\\sqrt 2$ is circumscribed about an isosceles right triangle. Find the radius of the circle which touches the legs of the triangle and is internally tangent to the circle circumscribed about it."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-84b74b99cf57dcf883745fc2c0f33ad0917423a963f251cad19f90fb0d314dbb | A circle of radius $10$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. What is the number of degrees of angle $ADB$? | [
"30"
] | [
"A circle of radius $10$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. What is the number of degrees of angle $ADB$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 30,
"tir": 2
} | 32 | 0 | 0 |
omr-ws-fba18b3ca1d1b651a0203bf09740cc73b3941c3dc675ac4c7197f571158a7a46 | A circle of radius $12\text{ cm}$ touches all four sides of a quadrilateral $ABCD$ with $AB$ parallel to $DC$. If $BC = 25\text{ cm}$ and the area of $ABCD$ is $648\text{ cm}^2$, determine the length of $DA$. | [
"29"
] | [
"A circle of radius $12\\text{ cm}$ touches all four sides of a quadrilateral $ABCD$ with $AB$ parallel to $DC$. If $BC = 25\\text{ cm}$ and the area of $ABCD$ is $648\\text{ cm}^2$, determine the length of $DA$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-3c7788f9d3616fc4e557a2d1f107217eabd03c3b2d0e5a256972d40230cca471 | A circle of radius $2$ has center at $(2,0)$. A circle of radius $1$ has center at $(5,0)$. A line is tangent to the two circles at points in the first quadrant. What is the $y$-intercept of the line? | [
"2\\sqrt{2}"
] | [
"A circle of radius $2$ has center at $(2,0)$. A circle of radius $1$ has center at $(5,0)$. A line is tangent to the two circles at points in the first quadrant. What is the $y$-intercept of the line?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-57f95568bc268dcacfaaa6a8255fc7bbd98f4ea028e235fa74f44cbfbd87ed7e | A circle of radius $2$ is centered at $A$. An equilateral triangle with side $4$ has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle? | [
"\\( 4(\\pi - \\sqrt{3}) \\)"
] | [
"A circle of radius $2$ is centered at $A$. An equilateral triangle with side $4$ has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-1b8dc22ae1af8d6a94988c0e08ce905ce7eedf2db19ddf4575f4219faa8ccdc8 | A circle of radius $2$ is inscribed in an isosceles trapezoid with an area of $28$. Find the length of the side of the trapezoid. | [
"\\(7\\)"
] | [
"A circle of radius $2$ is inscribed in an isosceles trapezoid with an area of $28$. Find the length of the side of the trapezoid."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-41853fdcb906bb4f65ed71633aefb9cfffc0b1671d7fe79bf399ac7a2e137577 | A circle of radius $2$ touches a circle of radius $4$ at point $B$. A straight line passing through point $B$ intersects the circle of smaller radius at point $A$, and the circle of larger radius at point $C$. Find $BC$ if $AC = 3\sqrt{2}$. | [
"\\(2\\sqrt{2}\\)"
] | [
"A circle of radius $2$ touches a circle of radius $4$ at point $B$. A straight line passing through point $B$ intersects the circle of smaller radius at point $A$, and the circle of larger radius at point $C$. Find $BC$ if $AC = 3\\sqrt{2}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-3f29962dace6ff45896485fd4bed8982ffb978b61a0b643c782f57ba428d9a52 | A circle of radius $2$ touches a circle of radius $4$ at point $B$. A straight line passing through point $B$ intersects the smaller circle at point $A$ and the larger circle at point $C$. Find the length of $BC$ if $AC = 3\sqrt{2}$. | [
"\\(2\\sqrt{2}\\)"
] | [
"A circle of radius $2$ touches a circle of radius $4$ at point $B$. A straight line passing through point $B$ intersects the smaller circle at point $A$ and the larger circle at point $C$. Find the length of $BC$ if $AC = 3\\sqrt{2}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-0d93a36d7c105c0ec6339e9857b9461696906b596424ed51bc931d35e781b299 | A circle of radius $4$ is inscribed in a triangle $ABC$. The touchpoint between the circle and side $BC$ is $D$. Given $CD = 8$ and $DB = 10$, find the lengths of sides $AB$ and $AC$. | [
"\\( AB = \\frac{29}{2} \\), \\( AC = \\frac{25}{2} \\)"
] | [
"A circle of radius $4$ is inscribed in a triangle $ABC$. The touchpoint between the circle and side $BC$ is $D$. Given $CD = 8$ and $DB = 10$, find the lengths of sides $AB$ and $AC$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 21
} | 21 | 0 | 0 |
omr-ws-a3efb815a4cceac68fde10ad1356792f25b8dfddd0bb82d3964cf362cabc4b74 | A circle of radius $4\sqrt{5}$ has its center on the y-axis and is tangent to the parabola $y = \frac{1}{8}x^2$. Point $(p,q)$ is their common point in the first quadrant. Find $p+q$. | [
"\\(4\\sqrt{4 + \\sqrt{2}} + 8 + 2\\sqrt{2}\\)"
] | [
"A circle of radius $4\\sqrt{5}$ has its center on the y-axis and is tangent to the parabola $y = \\frac{1}{8}x^2$. Point $(p,q)$ is their common point in the first quadrant. Find $p+q$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-74db1ce5bdac0271817880556f71c48c34fae68c549a862815113c126cc2e936 | A circle of radius $6\sqrt{7}$ is circumscribed about an acute-angled triangle $ABC$. A straight line is drawn through vertex $B$ and the center of the circle, intersecting $AC$ at point $D$. Perpendiculars $DK$ and $DN$ are dropped from point $D$ to sides $AB$ and $BC$, respectively. Given that $\frac{AD}{DC}=\frac{5}... | [
"20"
] | [
"A circle of radius $6\\sqrt{7}$ is circumscribed about an acute-angled triangle $ABC$. A straight line is drawn through vertex $B$ and the center of the circle, intersecting $AC$ at point $D$. Perpendiculars $DK$ and $DN$ are dropped from point $D$ to sides $AB$ and $BC$, respectively. Given that $\\frac{AD}{DC}=\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-fadffaf7a60b262e65ad55987e58e42a60c2405f67cd779694cfea3112e48eb2 | A circle of radius $R$ is divided into 8 equal parts, with the points of division denoted successively by $A, B, C, D, E, F, G,$ and $H$. Find the area of the square formed by drawing the chords $AF$, $BE$, $CH$, and $DG$. | [
"\\( R^2 (2 - \\sqrt{2}) \\)"
] | [
"A circle of radius $R$ is divided into 8 equal parts, with the points of division denoted successively by $A, B, C, D, E, F, G,$ and $H$. Find the area of the square formed by drawing the chords $AF$, $BE$, $CH$, and $DG$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 30,
"tir": 10
} | 40 | 0 | 0 |
omr-ws-948f8db6b00a08997adfbff644284622a8bbb0563203976949f8bca92cb65c12 | A circle of radius $R$ is inscribed in the triangle $ABC$ and touches the side $AC$ at a point $D$, the side $AB$ at a point $E$ and the side $BC$ at a point $F$. The length of the segment $AD$ is equal to $R$, and that of the segment $DC$ is equal to $a$. Find the area of $\triangle BEF$. | [
"$S=\\frac{1}{2}\\cdot \\frac{R^2(a+R)^3}{2(a-R)(a^2+R^2)}$"
] | [
"A circle of radius $R$ is inscribed in the triangle $ABC$ and touches the side $AC$ at a point $D$, the side $AB$ at a point $E$ and the side $BC$ at a point $F$. The length of the segment $AD$ is equal to $R$, and that of the segment $DC$ is equal to $a$. Find the area of $\\triangle BEF$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf5aa1d32f267ec992bed1f467e8324ce7830d61d75a24c6549c9edb6fc620df | A circle of radius $R$ passes through the vertices $A$ and $B$ of $\triangle ABC$ and touches the straight line $AC$ and a point $A$. Find the area of $\triangle ABC$ knowing that $\angle ABC=\beta $, $\angle ACB=\alpha $. | [
"$2R^2\\frac{\\sin ^3(\\alpha +\\beta )\\sin \\beta }{\\sin \\alpha }$"
] | [
"A circle of radius $R$ passes through the vertices $A$ and $B$ of $\\triangle ABC$ and touches the straight line $AC$ and a point $A$. Find the area of $\\triangle ABC$ knowing that $\\angle ABC=\\beta $, $\\angle ACB=\\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-986fe861801924c094c8141f04c192ecf87f1ef10052b483cae6f11a5309444d | A circle of radius $\frac{1}{2}$ is inside a square with sides of length $1$. In the upper right-hand corner, there are a series of circles which are tangent to two sides of the square and the bordering circle. Find the sum of the areas of these circles. | [
"\\(\\frac{\\pi (4 + 3\\sqrt{2})}{32}\\)"
] | [
"A circle of radius $\\frac{1}{2}$ is inside a square with sides of length $1$. In the upper right-hand corner, there are a series of circles which are tangent to two sides of the square and the bordering circle. Find the sum of the areas of these circles."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19d534b99365e60d1973b0a4dbff30661e935562f3f21592b17b2af22c22eb68 | A circle of radius $a$ is centered at the origin $O$. A rectangle $PQRS$ lies in the minor sector $OMN$ of this circle where $M$ is $(a, 0)$ and $N$ is $(a \cos \beta, a \sin \beta)$, and $\beta$ is a constant with $0 < \beta < \frac{\pi}{2}$. Vertex $P$ lies on the positive $x$-axis at $(x, 0)$; vertex $Q$ lies on $ON... | [
"\\dfrac{a^2 (\\sqrt{15} - \\sqrt{3})}{16}"
] | [
"A circle of radius $a$ is centered at the origin $O$. A rectangle $PQRS$ lies in the minor sector $OMN$ of this circle where $M$ is $(a, 0)$ and $N$ is $(a \\cos \\beta, a \\sin \\beta)$, and $\\beta$ is a constant with $0 < \\beta < \\frac{\\pi}{2}$. Vertex $P$ lies on the positive $x$-axis at $(x, 0)$; vertex $Q... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-38b9524d8f3cf466c8a8ba6177de080e8f0f11bfb5510048670604e75c268f24 | A circle of radius $a$ is revolved through $180^{\circ}$ about a line in its plane, which is a distance $b$ from the center of the circle, where $b > a$. For what value of the ratio $\frac{b}{a}$ does the center of gravity of the solid thus generated lie on the surface of the solid? | [
"2"
] | [
"A circle of radius $a$ is revolved through $180^{\\circ}$ about a line in its plane, which is a distance $b$ from the center of the circle, where $b > a$. For what value of the ratio $\\frac{b}{a}$ does the center of gravity of the solid thus generated lie on the surface of the solid?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40d6fc4b4c9f25c99c733178930b09958218fb0efb9792104dc5ad06367be278 | A circle of radius $r$ and a tangent line $l$ to the circle through a given point $P$ on the circle are given. $R$ is on the circumference of the circle, and $Q$ is the foot of the perpendicular from $R$ to $l$. Determine the maximum area of $\triangle PQR$. | [
"\\dfrac{3\\sqrt{3}}{8} r^2"
] | [
"A circle of radius $r$ and a tangent line $l$ to the circle through a given point $P$ on the circle are given. $R$ is on the circumference of the circle, and $Q$ is the foot of the perpendicular from $R$ to $l$. Determine the maximum area of $\\triangle PQR$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b928d763c77a8f587662bf7b8fec7b1798658ea3210e25f446fd353cd5d62d20 | A circle of radius $r$ contains two smaller circles, one concentric with the big circle and one tangent to the edge of the big circle. Both smaller circles have a radius of 1. The concentric circle chases the tangent circle, with both moving at an equal, constant speed. What is the smallest value of $r$, if any, that w... | [
"3"
] | [
"A circle of radius $r$ contains two smaller circles, one concentric with the big circle and one tangent to the edge of the big circle. Both smaller circles have a radius of 1. The concentric circle chases the tangent circle, with both moving at an equal, constant speed. What is the smallest value of $r$, if any, t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-f62bca88a7a6f1f1934f97411f60a2f8be78a722383ba61e7ad2576badc691ad | A circle of radius $r$ is inscribed in a rhombus $CDEF$ in which $\angle DCF = \gamma $. A tangent to the circle cuts the side $CD$ at a point $A$ and the side $CF$ at a point $B$. It is known that $|AD|=m$. Find the area of the triangle $ABC$. | [
"$\\left(\\frac{2r}{\\sin \\gamma }-m\\right)\\left(r-\\frac{r^2}{m}\\tan\\frac{\\gamma }{2}\\right)$"
] | [
"A circle of radius $r$ is inscribed in a rhombus $CDEF$ in which $\\angle DCF = \\gamma $. A tangent to the circle cuts the side $CD$ at a point $A$ and the side $CF$ at a point $B$. It is known that $|AD|=m$. Find the area of the triangle $ABC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7870232573165660c285ecce3ab24e8d52704113cda0faa52f236faa4775459f | A circle of radius $r$ is inscribed in a triangle. Tangent lines to this circle, parallel to the sides of the triangle, cut out three smaller triangles. The radii of the circles inscribed in these smaller triangles are $1$, $2$, and $3$. Find $r$. | [
"6"
] | [
"A circle of radius $r$ is inscribed in a triangle. Tangent lines to this circle, parallel to the sides of the triangle, cut out three smaller triangles. The radii of the circles inscribed in these smaller triangles are $1$, $2$, and $3$. Find $r$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb6f25be98471f8573a382649602890040d389fafa5cbade6ac706c2adfc1770 | A circle of radius $r$ is inscribed in an isosceles triangle whose perimeter is $kr$. Find all possible values of $k$ for which this situation is possible. | [
"[6\\sqrt{3}, \\infty)"
] | [
"A circle of radius $r$ is inscribed in an isosceles triangle whose perimeter is $kr$. Find all possible values of $k$ for which this situation is possible."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94513e479eb35eb92445c513546dd1b0016fe9e1738211cca1d909b1c0329356 | A circle of radius $t$ is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius $r = 1 + \sin \frac{\pi}{8}$. Find the product $rt$. | [
"1"
] | [
"A circle of radius $t$ is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius $r = 1 + \\sin \\frac{\\pi}{8}$. Find the product $rt$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-470b9ae5920da66b0616dae5ce0bcfe5bf8cfa0f86147af9f70fc1e1088e11ea | A circle of radius $t$ is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius $r=1+\sin\frac{\pi}{8}$. Compute the product $rt$. | [
"\\(\\frac{2+\\sqrt{2}}{4}\\)"
] | [
"A circle of radius $t$ is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius $r=1+\\sin\\frac{\\pi}{8}$. Compute the product $rt$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 5,
"tir": 7
} | 14 | 0 | 0 |
omr-ws-d16a837cf537b0b934038e849c6656c78a4f26a7af2c659980bbd3009c77cac9 | A circle of radius 'r' goes through two neighboring vertices of a square and is tangent to the side of the square opposite to these vertices. What is the area of the square? | [
"\\( \\frac{64}{25} r^2 \\)"
] | [
"A circle of radius 'r' goes through two neighboring vertices of a square and is tangent to the side of the square opposite to these vertices. What is the area of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-b59fc63a48a89f9fd4248ee82f0f9f7822f605d8d064a4503b233ff4bbef1db3 | A circle of radius 1 is drawn on a chessboard such that it contains an entire white cell whose sides are equal to 1 unit. Let $L$ be the total length of the circumference passing through the white cells. Calculate the maximum possible value of $\frac{L}{2\pi}$. | [
"\\dfrac{1}{2}"
] | [
"A circle of radius 1 is drawn on a chessboard such that it contains an entire white cell whose sides are equal to 1 unit. Let $L$ be the total length of the circumference passing through the white cells. Calculate the maximum possible value of $\\frac{L}{2\\pi}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-379b487097e8576f3c4f46029a4df68be2d97a1dc788949c81b0fcb49f542704 | A circle of radius 1 is inscribed in a 30-60-90 triangle. Find the perimeter of the triangle. | [
"\\(6 + 4\\sqrt{3}\\)"
] | [
"A circle of radius 1 is inscribed in a 30-60-90 triangle. Find the perimeter of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-dab29a04750497feac15c1cfad3908f289df110731faedfb77ec8436858fcca6 | A circle of radius 1 is inscribed in a triangle ABC, where \(\cos B = 0.8\). The circle touches the midline of the triangle ABC parallel to the side AC. Find the length of AC. | [
"\\(10.67\\)"
] | [
"A circle of radius 1 is inscribed in a triangle ABC, where \\(\\cos B = 0.8\\). The circle touches the midline of the triangle ABC parallel to the side AC. Find the length of AC."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1bf7bfa991ca13186105b3fed137aad7cecb546a0bf03defdcc810d96c05a8a0 | A circle of radius 1 is inscribed in the parabola \( y = x^2 \). What is the \( x \)-coordinate of the intersection in the first quadrant? | [
"\\(\\frac{\\sqrt{3}}{2}\\)"
] | [
"A circle of radius 1 is inscribed in the parabola \\( y = x^2 \\). What is the \\( x \\)-coordinate of the intersection in the first quadrant?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-f700a32b95107062bb45bec56f4c3860d5158e0f5f695900813c284fb766fd07 | A circle of radius 1 is placed in a corner of a room such that it touches the horizontal floor and two vertical walls perpendicular to each other. Find the locus of the center of the circle for all of its possible positions. | [
"(1, 1)"
] | [
"A circle of radius 1 is placed in a corner of a room such that it touches the horizontal floor and two vertical walls perpendicular to each other. Find the locus of the center of the circle for all of its possible positions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 17,
"genselect": 4,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-a8e85f5717826d4da78570b52c2cbcefe38f421ba68b86937b520236b013913c | A circle of radius 1 is randomly placed in a $15 \times 35$ rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m + n$. | [
"1765"
] | [
"A circle of radius 1 is randomly placed in a $15 \\times 35$ rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m + n$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d12fcfad708145a90491c5205ef60df70f0954046920b31511a2c918f010dfa | A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ Contents 1 Problem 2 Solution 2.1 Solu... | [
"817"
] | [
"A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ \nContents 1 Problem ... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 2,
"tir": 2
} | 41 | 0 | 0 |
omr-ws-1bd5890fda1612b2f453a2e87a4dd45e053fb3f9ed4ae5c5da035939f409257f | A circle of radius 1 is randomly placed in a 15-by-36 rectangle \(ABCD\) so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal \(AC\) is \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\). | [
"817"
] | [
"A circle of radius 1 is randomly placed in a 15-by-36 rectangle \\(ABCD\\) so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal \\(AC\\) is \\(\\frac{m}{n}\\), where \\(m\\) and \\(n\\) are relatively prime positive integers. Find \\(m + n\\)."... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 4
} | 21 | 0 | 0 |
omr-ws-39741c522458c03d71ddae3b89a7b0420c60825f14e3da74909cb7c848c843a6 | A circle of radius 1 is randomly placed inside a 15-by-36 rectangle such that the circle lies completely inside the rectangle. Find the probability that the circle will not touch the diagonal AC of the rectangle. | [
"\\dfrac{375}{442}"
] | [
"A circle of radius 1 is randomly placed inside a 15-by-36 rectangle such that the circle lies completely inside the rectangle. Find the probability that the circle will not touch the diagonal AC of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-f2bc0184570576a5914a02687605f3075daeb2955ee7c20aa8029e7d685a1d00 | A circle of radius 1 is tangent to a circle of radius 2. The sides of $\triangle ABC$ are tangent to the circles, and the sides $\overline{AB}$ and $\overline{AC}$ are congruent. What is the area of $\triangle ABC$? | [
"\\(16\\sqrt{2}\\)"
] | [
"A circle of radius 1 is tangent to a circle of radius 2. The sides of $\\triangle ABC$ are tangent to the circles, and the sides $\\overline{AB}$ and $\\overline{AC}$ are congruent. What is the area of $\\triangle ABC$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 15
} | 32 | 0 | 0 |
omr-ws-a8b55df20bdcce2d34f57d9420493f13f007bd50994e88eeef3cb05ec6ffe3ab | A circle of radius 1 is tangent to a circle of radius 2. The sides of \( \triangle ABC \) are tangent to the circles, and the sides \( \overline{AB} \) and \( \overline{AC} \) are congruent. What is the area of \( \triangle ABC \)? | [
"\\( 16\\sqrt{2} \\)"
] | [
"A circle of radius 1 is tangent to a circle of radius 2. The sides of \\( \\triangle ABC \\) are tangent to the circles, and the sides \\( \\overline{AB} \\) and \\( \\overline{AC} \\) are congruent. What is the area of \\( \\triangle ABC \\)?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 13,
"tir": 18
} | 67 | 0 | 0 |
omr-ws-860ddbf1884851754f3a115360e4a8eba95a4fd3b070bc6f9355236921faf89f | A circle of radius 1 rolls around the inside of a 6-8-10 triangle. How far does the center of the circle travel? | [
"12"
] | [
"A circle of radius 1 rolls around the inside of a 6-8-10 triangle. How far does the center of the circle travel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 1
} | 6 | 0 | 0 |
omr-ws-8ecd67f40957af774c1109c432439b1ebe162c32aec42fda50e5276df4b75ec1 | A circle of radius 1 rolls toward the origin from the positive x-axis and is stopped by the line \( y = x \). What is the x-coordinate of the center of the circle when it is stopped by the line? | [
"\\( 1 + \\sqrt{2} \\)"
] | [
"A circle of radius 1 rolls toward the origin from the positive x-axis and is stopped by the line \\( y = x \\). What is the x-coordinate of the center of the circle when it is stopped by the line?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 8,
"tir": 25
} | 78 | 0 | 0 |
omr-ws-2d1676618b89950545451eb8a0b06e93f898ba772a99faed7395f69258ff5440 | A circle of radius 12 has its center at the origin. A smaller circle of radius 8 is initially tangent to the larger circle at the point (0, 12). The smaller circle rolls inside the larger circle without slipping. What are the coordinates of point A on the smaller circle when the circles are tangent at the point (0, -12... | [
"(8, -4)"
] | [
"A circle of radius 12 has its center at the origin. A smaller circle of radius 8 is initially tangent to the larger circle at the point (0, 12). The smaller circle rolls inside the larger circle without slipping. What are the coordinates of point A on the smaller circle when the circles are tangent at the point (0... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-4d4f6acdba1b64fe6d7d7e15fe531999b788f695dcb08fa774a04676df3ab397 | A circle of radius 144 has three smaller congruent circles inside it. Each smaller circle is tangent to the other two and to the large circle. Find the radius of one of the smaller circles. | [
"\\( 72\\sqrt{3} \\)"
] | [
"A circle of radius 144 has three smaller congruent circles inside it. Each smaller circle is tangent to the other two and to the large circle. Find the radius of one of the smaller circles."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aa5a8629c3fa32a0829ce7a837bebe44a16ca6b9af96ebcb19ac2a4273a6527a | A circle of radius 2 cm is inscribed in an equilateral triangle and is tangent at three points. Three smaller circles are inscribed so that they are each tangent to two sides of the triangle and to the larger circle. The process is continued with circles of ever decreasing diameters. What is the sum of the circumferenc... | [
"\\( 10\\pi \\) cm."
] | [
"A circle of radius 2 cm is inscribed in an equilateral triangle and is tangent at three points. Three smaller circles are inscribed so that they are each tangent to two sides of the triangle and to the larger circle. The process is continued with circles of ever decreasing diameters. What is the sum of the circumf... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-43a077659e88b25bca5a4bd79d4fda8d72e8cfd9ba1015d6ed02b516c6c8d30f | A circle of radius 2 has its center at (2,0), and a circle of radius 1 has its center at (5,0). A line is tangent to both circles at points in the first quadrant. What is the y-intercept of this tangent line? | [
"\\(2\\sqrt{2}\\)"
] | [
"A circle of radius 2 has its center at (2,0), and a circle of radius 1 has its center at (5,0). A line is tangent to both circles at points in the first quadrant. What is the y-intercept of this tangent line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-a9953c8b1b0ed874d99d608a6989c866b21fffb19dd5cd77b3f91f19dae39e62 | A circle of radius 2 is centered at O. Square OABC has side length 1. Sides AB and CB are extended past B to meet the circle at D and E, respectively. What is the area of the shaded region in the figure, which is bounded by BD, BE, and the minor arc connecting D and E? | [
"\\(\\frac{\\pi}{3} + 1 - \\sqrt{3}\\)",
"\\(\\frac{\\pi}{3}+1-\\sqrt{3}\\)"
] | [
"A circle of radius 2 is centered at O. Square OABC has side length 1. Sides AB and CB are extended past B to meet the circle at D and E, respectively. What is the area of the shaded region in the figure, which is bounded by BD, BE, and the minor arc connecting D and E?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 63,
"tir": 55
} | 118 | 0 | 0 |
omr-ws-153c131552719976ba657b764f00497dfbb38d3e70871f138d3171edccc6c575 | A circle of radius 2 is centered at \( O \). Square \( OABC \) has side length 1. Sides \( \overline{AB} \) and \( \overline{CB} \) are extended past \( B \) to meet the circle at \( D \) and \( E \), respectively. What is the area of the shaded region in the figure, which is bounded by \( \overline{BD} \), \( \overlin... | [
"\\(\\frac{\\pi}{3} + 1 - \\sqrt{3}\\)"
] | [
"A circle of radius 2 is centered at \\( O \\). Square \\( OABC \\) has side length 1. Sides \\( \\overline{AB} \\) and \\( \\overline{CB} \\) are extended past \\( B \\) to meet the circle at \\( D \\) and \\( E \\), respectively. What is the area of the shaded region in the figure, which is bounded by \\( \\overl... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 23
} | 37 | 0 | 0 |
omr-ws-3f14c8b934d2b6a97ee3cf6ce465ef17d2486f76cca8c9b5da992f2d02ccaa38 | A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form a star figure. What is the ratio of the area of the star figure to the area of the original circle? | [
"\\(\\dfrac{4-\\pi}{\\pi}\\)"
] | [
"A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form a star figure. What is the ratio of the area of the star figure to the area of the original circle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-8cd2cd4bd8ac44aedf2c3c077d84204c229188ccf6c6648bbd5e0dc3309179fe | A circle of radius 3 is tangent at the midpoint of one side of an equilateral triangle and passes through the vertex opposite this side. What is the area of the region that is outside of the circle and inside of the equilateral triangle? | [
"\\( \\frac{15\\sqrt{3}}{2} - 3\\pi \\)"
] | [
"A circle of radius 3 is tangent at the midpoint of one side of an equilateral triangle and passes through the vertex opposite this side. What is the area of the region that is outside of the circle and inside of the equilateral triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 8
} | 14 | 0 | 0 |
omr-ws-4958a55e3a97b9aa3016fff0a1ca17e1cd8f64f4efbf39dcf66210c11f562c98 | A circle of radius 3.8 inches is drawn on an 8 by 8 inch chess board with its center at the center of the board. Through how many of the 64 squares does the circle pass? | [
"28"
] | [
"A circle of radius 3.8 inches is drawn on an 8 by 8 inch chess board with its center at the center of the board. Through how many of the 64 squares does the circle pass?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 8
} | 18 | 0 | 0 |
omr-ws-14e830195800111fd4a1ed85451cffa3b46c08554e4f5e06e5312bf8da38aefd | A circle of radius 320 is tangent to a circle of radius 1000. The smaller circle is tangent to a diameter of the larger circle at a point \( P \). How far is the point \( P \) from the outside of the larger circle? | [
"400"
] | [
"A circle of radius 320 is tangent to a circle of radius 1000. The smaller circle is tangent to a diameter of the larger circle at a point \\( P \\). How far is the point \\( P \\) from the outside of the larger circle?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 64,
"tir": 25
} | 89 | 0 | 0 |
omr-ws-e52c86d4d726ede724a518d6135cd5d57ba8298f5cbd61aa095673ba117f4f73 | A circle of radius 6 is inscribed in a triangle. One of the sides of the triangle is divided by the point of tangency into segments of lengths 6 and 15. Compute the length of the shortest side of the triangle. | [
"20"
] | [
"A circle of radius 6 is inscribed in a triangle. One of the sides of the triangle is divided by the point of tangency into segments of lengths 6 and 15. Compute the length of the shortest side of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6b73450282abbb97afcd951b209e654b8b3312283b1f8ccbb4cb306709af54e3 | A circle of radius 8 and a circle of radius 24 are externally tangent. A rubber band is slipped around the two circles. What is the length of the stretched rubber band? | [
"32\\sqrt{3} + \\dfrac{64}{3}\\pi"
] | [
"A circle of radius 8 and a circle of radius 24 are externally tangent. A rubber band is slipped around the two circles. What is the length of the stretched rubber band?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-8e624c41fe24f7558a0e3b124de04158abff383c33ce3e5a6523e874884aff17 | A circle of radius 9 cm is inscribed in an equilateral triangle and is tangent at three points. Three smaller circles are inscribed so that each is tangent to two sides of the triangle and to the larger circle. This process is continued with circles of ever-decreasing diameters. What is the sum of the circumferences of... | [
"\\( 27\\pi \\)"
] | [
"A circle of radius 9 cm is inscribed in an equilateral triangle and is tangent at three points. Three smaller circles are inscribed so that each is tangent to two sides of the triangle and to the larger circle. This process is continued with circles of ever-decreasing diameters. What is the sum of the circumferenc... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-de2c359996ce88170afd1b5be6feaa66f6b81cacc4da249cbe813e5b058bd8f7 | A circle of radius 9 cm is inscribed in an equilateral triangle and is tangent at three points. Three smaller circles are inscribed so that they are each tangent to two sides of the triangle and to the larger circle. The process is continued with circles of ever decreasing diameters. What is the number of centimeters i... | [
"45"
] | [
"A circle of radius 9 cm is inscribed in an equilateral triangle and is tangent at three points. Three smaller circles are inscribed so that they are each tangent to two sides of the triangle and to the larger circle. The process is continued with circles of ever decreasing diameters. What is the number of centimet... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 7
} | 30 | 0 | 0 |
omr-ws-010ead6f3888813c92a6fe9387a47e42e13d8015d56d8a06975265fd43db0176 | A circle of radius \( e \) passes through the center of a square with side length \( \pi \). Find the difference in the areas of the non-overlapping regions between the circle and the square. | [
"\\( \\pi (e^2 - \\pi) \\)"
] | [
"A circle of radius \\( e \\) passes through the center of a square with side length \\( \\pi \\). Find the difference in the areas of the non-overlapping regions between the circle and the square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 18
} | 38 | 0 | 0 |
omr-ws-166ef255bd4be9305629291b5e38f8f231ff819f5af8bc3badf76d73699b4cd6 | A circle of radius \( r \) has chord \( AB \) of length 10 and chord \( CD \) of length 7. When \( AB \) and \( CD \) are extended through \( B \) and \( C \), respectively, they intersect at point \( P \) outside the circle. Given that \( \angle APD = 60^\circ \) and \( BP = 8 \), find \( r^2 \). | [
"73"
] | [
"A circle of radius \\( r \\) has chord \\( AB \\) of length 10 and chord \\( CD \\) of length 7. When \\( AB \\) and \\( CD \\) are extended through \\( B \\) and \\( C \\), respectively, they intersect at point \\( P \\) outside the circle. Given that \\( \\angle APD = 60^\\circ \\) and \\( BP = 8 \\), find \\( r... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d3f7691b00319f53febbc30ed188daf4bdc2ee267162f5373e257abdf3a5354d | A circle of radius \( r \) has chords \( \overline{AB} \) of length 10 and \( \overline{CD} \) of length 7. When \( \overline{AB} \) and \( \overline{CD} \) are extended through \( B \) and \( C \), respectively, they intersect at \( P \), which is outside the circle. If \( \angle APD = 60^\circ \) and \( BP = 8 \), th... | [
"73"
] | [
"A circle of radius \\( r \\) has chords \\( \\overline{AB} \\) of length 10 and \\( \\overline{CD} \\) of length 7. When \\( \\overline{AB} \\) and \\( \\overline{CD} \\) are extended through \\( B \\) and \\( C \\), respectively, they intersect at \\( P \\), which is outside the circle. If \\( \\angle APD = 60^\\... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-8f7ce7422c9f24c02a3e4be6576256062ff32b418b363e17b7bee9c79e918f2e | A circle of radius \( r \) is concentric with and outside a regular hexagon of side length 2. The probability that 3 entire sides of the hexagon are visible from a random point on the circle is \( \frac{1}{2} \). What is \( r \)? | [
"\\( 2\\sqrt{3} \\)"
] | [
"A circle of radius \\( r \\) is concentric with and outside a regular hexagon of side length 2. The probability that 3 entire sides of the hexagon are visible from a random point on the circle is \\( \\frac{1}{2} \\). What is \\( r \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-7164f1ccd1fac85b5edab9c751786e94b08bd92c0f67814a255e41bbc50183c5 | A circle of radius \( r \) is concentric with and outside a regular hexagon of side length 2. The probability that three entire sides of the hexagon are visible from a randomly chosen point on the circle is \( \frac{1}{2} \). What is \( r \)? | [
"2\\sqrt{3}"
] | [
"A circle of radius \\( r \\) is concentric with and outside a regular hexagon of side length 2. The probability that three entire sides of the hexagon are visible from a randomly chosen point on the circle is \\( \\frac{1}{2} \\). What is \\( r \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-1352c20c7f7e331a507b22b4d107363b9e451077ad8326220b48b6405f9d6705 | A circle of radius \( r \) is inscribed in a quadrilateral \( ABCD \). It touches \( AB \) at \( P \) and \( CD \) at \( Q \). Given \( AP = 19 \), \( PB = 26 \), \( CQ = 37 \), and \( QD = 23 \), find \( r \). | [
"\\(\\sqrt{647}\\)"
] | [
"A circle of radius \\( r \\) is inscribed in a quadrilateral \\( ABCD \\). It touches \\( AB \\) at \\( P \\) and \\( CD \\) at \\( Q \\). Given \\( AP = 19 \\), \\( PB = 26 \\), \\( CQ = 37 \\), and \\( QD = 23 \\), find \\( r \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.