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-1a530f84dbf552f98cfd07f445a97baddf7d5edb0f01d69856a5083d992fdb6b | A circle is divided into $n$ equal parts. Marceline assigns whole numbers from $1$ to $n$ to each part such that the distance between two consecutive numbers is always the same. The numbers $887$, $217$, and $1556$ occupy consecutive positions. How many parts was the circumference divided into? | [
"669"
] | [
"A circle is divided into $n$ equal parts. Marceline assigns whole numbers from $1$ to $n$ to each part such that the distance between two consecutive numbers is always the same. The numbers $887$, $217$, and $1556$ occupy consecutive positions. How many parts was the circumference divided into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61f85b08406ebd14ee766c423420855edf160326305fe8d61b43aa745615a296 | A circle is divided into $n$ sectors ($n \geq 3$). Each sector can be filled with either $1$ or $0$. Choose any sector $\mathcal{C}$ occupied by $0$, change it into a $1$ and simultaneously change the symbols $x, y$ in the two adjacent sectors to their complements $1-x$, $1-y$. We repeat this process as long as there e... | [
"\\( n \\equiv 1 \\pmod{3} \\) or \\( n \\equiv 2 \\pmod{3} \\)"
] | [
"A circle is divided into $n$ sectors ($n \\geq 3$). Each sector can be filled with either $1$ or $0$. Choose any sector $\\mathcal{C}$ occupied by $0$, change it into a $1$ and simultaneously change the symbols $x, y$ in the two adjacent sectors to their complements $1-x$, $1-y$. We repeat this process as long as ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 20,
"genselect": 4,
"tir": 13
} | 37 | 0 | 0 |
omr-ws-f463ecf0adbfe8e8c655ad7582501294952c32f967ba85d4463d47a83b892099 | A circle is divided into 100 equal parts, and each point of division is colored either green or yellow. If there are exactly 4 division points between two points $A$ and $B$, and point $A$ is green, then point $B$ must be yellow. Determine whether there are more green points or yellow points. | [
"yellow"
] | [
"A circle is divided into 100 equal parts, and each point of division is colored either green or yellow. If there are exactly 4 division points between two points $A$ and $B$, and point $A$ is green, then point $B$ must be yellow. Determine whether there are more green points or yellow points."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-014b4e16f0df192231d859c0d90bdb4d15b6c8856a6eaf3070d6dc0462cb99d4 | A circle is divided into 13 segments, numbered consecutively from 1 to 13. Five fleas called \(A, B, C, D,\) and \(E\) are sitting in the segments 1, 2, 3, 4, and 5. A flea is allowed to jump to an empty segment five positions away in either direction around the circle. Only one flea jumps at the same time, and two fle... | [
"Cyclic permutations of \\(A, B, C, D, E\\)"
] | [
"A circle is divided into 13 segments, numbered consecutively from 1 to 13. Five fleas called \\(A, B, C, D,\\) and \\(E\\) are sitting in the segments 1, 2, 3, 4, and 5. A flea is allowed to jump to an empty segment five positions away in either direction around the circle. Only one flea jumps at the same time, an... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 7
} | 8 | 0 | 0 |
omr-ws-e22469fb200ba930f55a715037c6de6ba29fb4b7e1e94df14912f1dbacbc9bfd | A circle is divided into 2000 sectors. There are 2001 frogs in these sectors. Every second, two frogs that are in the same sector jump to opposite neighboring sectors. Let $N$ be the maximum number of empty sectors that can occur at any point in time. Find the value of $N$. | [
"1999"
] | [
"A circle is divided into 2000 sectors. There are 2001 frogs in these sectors. Every second, two frogs that are in the same sector jump to opposite neighboring sectors. Let $N$ be the maximum number of empty sectors that can occur at any point in time. Find the value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-c9400258fbf3c1542e89413e3899d3685698e1641e422fdfca98b1377e165f35 | A circle is divided into 2000 sectors. There are 2001 frogs inside these sectors. At every second, two frogs in one sector jump to the two adjacent sectors (in opposite directions). Determine the minimum number of sectors that will be inhabited at some moment. | [
"3"
] | [
"A circle is divided into 2000 sectors. There are 2001 frogs inside these sectors. At every second, two frogs in one sector jump to the two adjacent sectors (in opposite directions). Determine the minimum number of sectors that will be inhabited at some moment."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-96116e5808696c137c772bd32aad3c605824bb823e45cf163213a1c605f247de | A circle is divided into 2017 equal circular sectors numbered $1, 2, 3, \ldots, 2017$ in clockwise order. A token is initially placed on the region with the number $1$. Each time, a fair 6-sided die is rolled: if the number rolled $k$ is even, the token is moved $k$ regions clockwise; if the number rolled $k$ is odd, t... | [
"3"
] | [
"A circle is divided into 2017 equal circular sectors numbered $1, 2, 3, \\ldots, 2017$ in clockwise order. A token is initially placed on the region with the number $1$. Each time, a fair 6-sided die is rolled: if the number rolled $k$ is even, the token is moved $k$ regions clockwise; if the number rolled $k$ is ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-0713619769bd3a69036526d0f9023cbc89e979c93361428194a69425ac5a26de | A circle is divided into \( n \) arcs by \( n \) marked points. After rotating the circle by an angle of \( 2\pi k/n \) (for some positive integer \( k \)), the marked points move to \( n \) new points, dividing the circle into \( n \) new arcs. Let \( f(n) \) be the minimum number of new arcs that lie entirely within ... | [
"0"
] | [
"A circle is divided into \\( n \\) arcs by \\( n \\) marked points. After rotating the circle by an angle of \\( 2\\pi k/n \\) (for some positive integer \\( k \\)), the marked points move to \\( n \\) new points, dividing the circle into \\( n \\) new arcs. Let \\( f(n) \\) be the minimum number of new arcs that ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-792d9bbaba5fe89c211e0dfbc6c6b80350405a39313ae34431d0e20905dc16e5 | A circle is divided into \( n \) sectors. Pawns stand on some of the sectors, with the total number of pawns equaling \( n + 1 \). Any two pawns on the same sector move simultaneously to the neighboring sectors in different directions. Let \( f(n) \) be the minimum number of transformations required to ensure that at l... | [
"2"
] | [
"A circle is divided into \\( n \\) sectors. Pawns stand on some of the sectors, with the total number of pawns equaling \\( n + 1 \\). Any two pawns on the same sector move simultaneously to the neighboring sectors in different directions. Let \\( f(n) \\) be the minimum number of transformations required to ensur... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-405b80775a9e75d9a2b8f3280111f1b68105929465a8dfdd43af1e41d066b432 | A circle is divided into \(2n\) sectors, with \(n\) black sectors and \(n\) white sectors. The white sectors are numbered consecutively clockwise from 1 to \(n\), and the black sectors are numbered consecutively counterclockwise from 1 to \(n\). Determine the number of distinct ways to divide the circle into two halves... | [
"2"
] | [
"A circle is divided into \\(2n\\) sectors, with \\(n\\) black sectors and \\(n\\) white sectors. The white sectors are numbered consecutively clockwise from 1 to \\(n\\), and the black sectors are numbered consecutively counterclockwise from 1 to \\(n\\). Determine the number of distinct ways to divide the circle ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-8452d6624545932e493206b3cdfff3a9c0e3e8167096ac2d2b091164c51321e9 | A circle is divided into \(2n\) sectors, with \(n\) sectors colored blue and \(n\) sectors colored red. Numbers \(1, 2, \ldots, n\) are placed clockwise on the blue sectors and counterclockwise on the red sectors. Determine the sum of all possible values of \(n\) for which there exists a diameter such that all numbers ... | [
"49"
] | [
"A circle is divided into \\(2n\\) sectors, with \\(n\\) sectors colored blue and \\(n\\) sectors colored red. Numbers \\(1, 2, \\ldots, n\\) are placed clockwise on the blue sectors and counterclockwise on the red sectors. Determine the sum of all possible values of \\(n\\) for which there exists a diameter such t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-9042a0cd0378f1db07a482a8f08c3c617a47bcf12f721aa4ba7861145425161c | A circle is divided into four quarter circles, and a point is chosen at random within each quarter circle. Find the probability that the quadrilateral formed by these four points is cyclic. | [
"0"
] | [
"A circle is divided into four quarter circles, and a point is chosen at random within each quarter circle. Find the probability that the quadrilateral formed by these four points is cyclic."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d17dd0116c848d0f7ca03d6b6a9e08a4c989e34144fdf8807fe4da0e36d22102 | A circle is divided into three arcs by two secant lines meeting at a point outside the circle. The measures of the arcs outside the secant lines are 80° and 130°. Two lines are drawn between the intersections of the secant lines and the circle, forming an angle \( y \) at the point outside the circle. Find the measure ... | [
"25"
] | [
"A circle is divided into three arcs by two secant lines meeting at a point outside the circle. The measures of the arcs outside the secant lines are 80° and 130°. Two lines are drawn between the intersections of the secant lines and the circle, forming an angle \\( y \\) at the point outside the circle. Find the m... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-893cefb9576e8a86da9f35fd78c74cdb03bbaf0ae9fd44d33243c4083f859dc5 | A circle is drawn inside the kite K so that it touches each of the 4 sides of K. Find the radius of the circle, giving your answer in the form \( p\sqrt{q} - q\sqrt{p} \), where \( p \) and \( q \) are positive integers. | [
"2\\sqrt{3} - 3\\sqrt{2}"
] | [
"A circle is drawn inside the kite K so that it touches each of the 4 sides of K. Find the radius of the circle, giving your answer in the form \\( p\\sqrt{q} - q\\sqrt{p} \\), where \\( p \\) and \\( q \\) are positive integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f6eb011e74ddfa74591bb00ebc4656367558092010428fb1dfeae4fd6720cbde | A circle is drawn on a focal chord of the parabola \( y^2 = 4ax \) using the focal chord as the diameter. Another diameter of this circle intersects the parabola at two points \( t \) and \( t' \) (other than the endpoints of the focal chord). Find the product \( tt' \). | [
"-1"
] | [
"A circle is drawn on a focal chord of the parabola \\( y^2 = 4ax \\) using the focal chord as the diameter. Another diameter of this circle intersects the parabola at two points \\( t \\) and \\( t' \\) (other than the endpoints of the focal chord). Find the product \\( tt' \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-e640634e5748f357ad2db0a678dfbd3270f2a2b447ebb956fbb9f660428b41d6 | A circle is drawn through vertex $B$ of triangle $ABD$ tangent to side $AD$ at its midpoint $E$, and intersecting sides $AB$ and $BD$ at points $K$ and $L$, respectively. On the arc $BL$ that does not contain point $K$, point $C$ is selected so that $\angle CKL = \angle BDA$. Given that triangle $KCD$ is equilateral, f... | [
"\\( 75^\\circ \\)"
] | [
"A circle is drawn through vertex $B$ of triangle $ABD$ tangent to side $AD$ at its midpoint $E$, and intersecting sides $AB$ and $BD$ at points $K$ and $L$, respectively. On the arc $BL$ that does not contain point $K$, point $C$ is selected so that $\\angle CKL = \\angle BDA$. Given that triangle $KCD$ is equilat... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be60c222eae5f073be95044035a5295ae8df19ca0074a2fc87208865a71c866a | A circle is drawn through vertices \( A \) and \( B \) of triangle \( ABC \), intersecting sides \( AC \) and \( BC \) at points \( M \) and \( P \). It is known that the segment \( MP \) contains the center of the circle inscribed in \( ABC \). Find the length of \( MP \) if \( AB = c \), \( BC = a \), and \( CA = b \... | [
"\\(\\frac{2ab}{a+b+c}\\)"
] | [
"A circle is drawn through vertices \\( A \\) and \\( B \\) of triangle \\( ABC \\), intersecting sides \\( AC \\) and \\( BC \\) at points \\( M \\) and \\( P \\). It is known that the segment \\( MP \\) contains the center of the circle inscribed in \\( ABC \\). Find the length of \\( MP \\) if \\( AB = c \\), \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8925b127c551326555ed100356f99b5b72ecb589fac1c17044dafaa708a8aef6 | A circle is given. Each point on the circle is colored either red or blue. Let $X$ be a point on the circle, and consider all isosceles triangles $XYZ$ with $|XY|=|XZ|$ and $Y, Z$ on the circle. Let $N$ be the number of such triangles where $X, Y, Z$ are all the same color. Find the minimum possible value of $N$ over a... | [
"2"
] | [
"A circle is given. Each point on the circle is colored either red or blue. Let $X$ be a point on the circle, and consider all isosceles triangles $XYZ$ with $|XY|=|XZ|$ and $Y, Z$ on the circle. Let $N$ be the number of such triangles where $X, Y, Z$ are all the same color. Find the minimum possible value of $N$ o... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-3721e872071710f2d6f06447c4ad860f8afd12bcab08be2d9049a0a84f1479ef | A circle is in the coordinate plane such that the slope of the line from the origin to its center is $2$ while the slope of one of its tangents through the origin is $3$. What is the slope of the other tangent line through the origin? | [
"\\(\\frac{13}{9}\\)"
] | [
"A circle is in the coordinate plane such that the slope of the line from the origin to its center is $2$ while the slope of one of its tangents through the origin is $3$. What is the slope of the other tangent line through the origin?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-8119ad4543dc28c5edbeae66dfcd4bd170b869c01b6121752b6425f9e94670de | A circle is inscribed in $\triangle ABC$ with sides $| BC | = a$, $| AC | =2a$ and the angle $C=120^\circ$. Another circle is drawn through the points of tangency of that circle with the sides $AC$ and $BC$ and through the vertex $B$. Find the radius of the second circle. | [
"$a\\sqrt {\\frac{13-4\\sqrt 7}{2}}$"
] | [
"A circle is inscribed in $\\triangle ABC$ with sides $| BC | = a$, $| AC | =2a$ and the angle $C=120^\\circ$. Another circle is drawn through the points of tangency of that circle with the sides $AC$ and $BC$ and through the vertex $B$. Find the radius of the second circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5aa38bafc7528b4fa7dfe875151f593ba468b26d6cfb16f2ecfbb6cd0bb26843 | A circle is inscribed in a $3-4-5$ triangle. A segment is drawn from the smaller acute angle to the point of tangency on the opposite side. This segment is divided in the ratio $p : q$ by the segment drawn from the larger acute angle to the point of tangency on its opposite side. If $p > q$, find $p : q$. | [
"9:2"
] | [
"A circle is inscribed in a $3-4-5$ triangle. A segment is drawn from the smaller acute angle to the point of tangency on the opposite side. This segment is divided in the ratio $p : q$ by the segment drawn from the larger acute angle to the point of tangency on its opposite side. If $p > q$, find $p : q$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c06e14d618d14bf62539fec59d029196d69fd372b1a17b286f4426e2e9ee7fd0 | A circle is inscribed in a deltoid \(ABCD\) with \(AB = BC\) and \(AD = DC\). The circle touches the sides \(AB\), \(BC\), and \(AD\) at points \(K\), \(M\), and \(N\), respectively. The diagonal \(AC\) intersects \(MN\) at point \(P\). If the radius of the inscribed circle is \(r\) and the length of \(AC\) is \(d\), f... | [
"\\dfrac{d^2}{4r^2}"
] | [
"A circle is inscribed in a deltoid \\(ABCD\\) with \\(AB = BC\\) and \\(AD = DC\\). The circle touches the sides \\(AB\\), \\(BC\\), and \\(AD\\) at points \\(K\\), \\(M\\), and \\(N\\), respectively. The diagonal \\(AC\\) intersects \\(MN\\) at point \\(P\\). If the radius of the inscribed circle is \\(r\\) and t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-e40a0f0fa9f40308e7a8d4e4de4232be4c2f7314ad669d4fbd64edcde53d1a71 | A circle is inscribed in a quadrilateral \(ABCD\) with center \(O\). Given that the perimeters of triangles \(AOB\), \(BOC\), and \(COD\) are equal, calculate the ratio of the perimeter of quadrilateral \(ABCD\) to the circumference of the inscribed circle. | [
"\\dfrac{4}{\\pi}"
] | [
"A circle is inscribed in a quadrilateral \\(ABCD\\) with center \\(O\\). Given that the perimeters of triangles \\(AOB\\), \\(BOC\\), and \\(COD\\) are equal, calculate the ratio of the perimeter of quadrilateral \\(ABCD\\) to the circumference of the inscribed circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13af73cface6a254193734daefce347fc219a97dd17a456ff2328e667195c014 | A circle is inscribed in a quadrilateral \(ABCD\) with perpendicular diagonals. Given \(AB = 2\) and \(CD = 4\), find the radius of the inscribed circle. | [
"\\dfrac{4}{3}"
] | [
"A circle is inscribed in a quadrilateral \\(ABCD\\) with perpendicular diagonals. Given \\(AB = 2\\) and \\(CD = 4\\), find the radius of the inscribed circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-99b9d2e43ce43cb64c375bdce6f3abf5dd9cb7ec4627c5bef6fdc24f2fc044cf | A circle is inscribed in a quadrilateral \(ABCD\), where \(M\) and \(N\) are midpoints of \(AC\) and \(BD\), respectively. Given \(AB = 5\), \(CD = 9\), and \((AC)^2 + (BD)^2 + 4(MN)^2 = 218\), find the value of \(BC \times DA\). | [
"42"
] | [
"A circle is inscribed in a quadrilateral \\(ABCD\\), where \\(M\\) and \\(N\\) are midpoints of \\(AC\\) and \\(BD\\), respectively. Given \\(AB = 5\\), \\(CD = 9\\), and \\((AC)^2 + (BD)^2 + 4(MN)^2 = 218\\), find the value of \\(BC \\times DA\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c73c7b7a00325b646f1fb6f5633d5ee4f33d24767c761b0b2f94186eab9cd03e | A circle is inscribed in a quadrilateral with side lengths 5, 7, 8, and \( x \). Find \( x \). | [
"6"
] | [
"A circle is inscribed in a quadrilateral with side lengths 5, 7, 8, and \\( x \\). Find \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-51c62a97af1026689eeed85db1defb3b03d49df261faeea0e65d23978add054b | A circle is inscribed in a quarter circle of radius $1$. What is the radius of the inscribed circle? | [
"\\(\\sqrt{2} - 1\\)"
] | [
"A circle is inscribed in a quarter circle of radius $1$. What is the radius of the inscribed circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-823aa98e2de505db968ee19a363b6a703a65513258ee20c5fe52a8b2123fd066 | A circle is inscribed in a regular hexagon with a radius of 5. What is the area of the hexagon? | [
"259.81"
] | [
"A circle is inscribed in a regular hexagon with a radius of 5. What is the area of the hexagon?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e42b255e22f333a5b35a425a34b53b052b57cc8df9a524adf16e95455fe25483 | A circle is inscribed in a regular triangle with side equal to $a$. Another circle is drawn from the vertex by a radius which is equal to half its side. Find the area of the common part of the circles. | [
"$\\frac{5\\pi -6\\sqrt 3}{72}a^2$"
] | [
"A circle is inscribed in a regular triangle with side equal to $a$. Another circle is drawn from the vertex by a radius which is equal to half its side. Find the area of the common part of the circles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c11dea39f8ac38e5722f5e8d664191d7975f9a0fe919839084a5ee67653ca966 | A circle is inscribed in a rhombus $A BCD$ in which $|AB|=l$ and $\angle BAD=\alpha $. A tangent to the circle cuts the side $AB$ at a point $M$ and the side $AD$ at a point $N$. It is known that $| MN | = 2a$. Find the lengths of the line segments $MB$ and $ND$. | [
"$\\left.\\begin{array}{lll} |MN|\\\\ |ND| \\end{array}\\right\\}=a+l\\sin ^2 \\frac{\\alpha }{2}\\pm \\sqrt {a^2+2al\\sin ^2 \\frac{\\alpha }{2}-l^2\\cos ^2 \\frac{\\alpha }{2} \\sin ^2 \\frac{\\alpha }{2}}$"
] | [
"A circle is inscribed in a rhombus $A BCD$ in which $|AB|=l$ and $\\angle BAD=\\alpha $. A tangent to the circle cuts the side $AB$ at a point $M$ and the side $AD$ at a point $N$. It is known that $| MN | = 2a$. Find the lengths of the line segments $MB$ and $ND$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be1800a4015309a98fa39e6a48dbaab82eb3ab87786d64912ee120c7cd5f9200 | A circle is inscribed in a rhombus with altitude \( h \) and acute angle \( \alpha \). Find the radius of the greatest of two possible circles each of which touches the given circle and two sides of the rhombus. | [
"\\(\\frac{h}{2} \\cdot \\frac{1 - \\sin\\frac{\\alpha}{2}}{1 + \\sin\\frac{\\alpha}{2}}\\)"
] | [
"A circle is inscribed in a rhombus with altitude \\( h \\) and acute angle \\( \\alpha \\). Find the radius of the greatest of two possible circles each of which touches the given circle and two sides of the rhombus."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-f375a35c11164beabac4c9096e0f39f8c535f6a67bd10a0f461f2e3ad5357cff | A circle is inscribed in a rhombus with altitude \( h \) and acute angle \( a \). The radius of the largest of two possible circles that touch the given inscribed circle and two sides of the rhombus is \( r \). Calculate \( r \) in terms of \( h \) and \( a \). | [
"r = \\dfrac{h(1 - \\sin \\frac{a}{2})}{2(1 + \\sin \\frac{a}{2})}"
] | [
"A circle is inscribed in a rhombus with altitude \\( h \\) and acute angle \\( a \\). The radius of the largest of two possible circles that touch the given inscribed circle and two sides of the rhombus is \\( r \\). Calculate \\( r \\) in terms of \\( h \\) and \\( a \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-8fe499c5422ec9b9941cdbcfd839bb6f377a1435c51734b8cb55312fafef3875 | A circle is inscribed in a rhombus with side $a$ and acute angle equal to $60^\circ$. Determine the area of the rectangle, whose vertices lie at the points of tangency of the circle and the sides of the rhombus. | [
"$S=\\frac{3a^2\\sqrt 3}{16}$"
] | [
"A circle is inscribed in a rhombus with side $a$ and acute angle equal to $60^\\circ$. Determine the area of the rectangle, whose vertices lie at the points of tangency of the circle and the sides of the rhombus."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e544b23bdff4d487fc7a51f834de574a25e1b5c56e783638659d25cdeb998fb0 | A circle is inscribed in a right triangle with dimensions \(x\), \(y\), and \(r\). (\(x\) and \(y\) are the distances from the corners to where the circle intersects the legs and \(r\) is the radius of the circle.) Find the area of the triangle in terms of \(x\) and \(y\) only (without \(r\)). | [
"xy"
] | [
"A circle is inscribed in a right triangle with dimensions \\(x\\), \\(y\\), and \\(r\\). (\\(x\\) and \\(y\\) are the distances from the corners to where the circle intersects the legs and \\(r\\) is the radius of the circle.) Find the area of the triangle in terms of \\(x\\) and \\(y\\) only (without \\(r\\))."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94c6e3dd7634edbccb6aec6cbad8a54c199d9f656f180b8353191b7f2323831d | A circle is inscribed in a right triangle. The point of tangency with the circle divides one of the legs of the triangle into segments 6 cm and 10 cm long, reckoning from the vertex of the right angle. Find the area of the triangle. | [
"240 cm$^2$"
] | [
"A circle is inscribed in a right triangle. The point of tangency with the circle divides one of the legs of the triangle into segments 6 cm and 10 cm long, reckoning from the vertex of the right angle. Find the area of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22edbc8c46a9a6998a06802d7dcfcb9435556ade21c942985a51b2fda058e801 | A circle is inscribed in a sector of a larger circle so that the inscribed circle touches the sector in three places: each of the radii and the arc. The ratio of the radius of the sector to the radius of the small circle is $3:1$. The ratio of the area of the sector to the area of the small circle, written in its simpl... | [
"5"
] | [
"A circle is inscribed in a sector of a larger circle so that the inscribed circle touches the sector in three places: each of the radii and the arc. The ratio of the radius of the sector to the radius of the small circle is $3:1$. The ratio of the area of the sector to the area of the small circle, written in its ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c3a15d64c1363a09488a0d203f9606fabbd31025c94eb5719fcb54fa28f90823 | A circle is inscribed in a sector of radius $R$ with a central angle $\alpha $. Determine the radius of the circle. | [
"$x=\\frac{R\\sin\\frac{\\alpha }{2}}{1+\\sin\\frac{\\alpha }{2}} =\\frac{R\\sin\\frac{\\alpha }{2}}{2\\cos^2 \\left(45^\\circ-\\frac{\\alpha }{4}\\right)}$"
] | [
"A circle is inscribed in a sector of radius $R$ with a central angle $\\alpha $. Determine the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6c88c5712fd1255ac68f89311c3b8d70d4798ff46fb117fd0c6bc0eda658a82 | A circle is inscribed in a semicircle such that the diameter of the circle is equal to the radius of the semicircle and perpendicular to the diameter of the semicircle. Two lines are drawn from each endpoint of the diameter of the semicircle to the center of the circle. Four circles are then drawn such that they are th... | [
"\\dfrac{1}{1}"
] | [
"A circle is inscribed in a semicircle such that the diameter of the circle is equal to the radius of the semicircle and perpendicular to the diameter of the semicircle. Two lines are drawn from each endpoint of the diameter of the semicircle to the center of the circle. Four circles are then drawn such that they a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-fa62d669cfe90f998378c7b1e608916edaaf8ebea198cc0a9492b796ca07b887 | A circle is inscribed in a square of side length \( x \). Then, a square is inscribed in that circle, followed by another circle inscribed in that square, and so on. If \( S_n \) denotes the sum of the areas of the first \( n \) circles, find the limit of \( S_n \) as \( n \) approaches infinity. | [
"\\(\\frac{\\pi x^2}{2}\\)"
] | [
"A circle is inscribed in a square of side length \\( x \\). Then, a square is inscribed in that circle, followed by another circle inscribed in that square, and so on. If \\( S_n \\) denotes the sum of the areas of the first \\( n \\) circles, find the limit of \\( S_n \\) as \\( n \\) approaches infinity."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-e9cae29214ad3229700038dd5526f5e53dea11fe9206194a43e678128da6a2be | A circle is inscribed in a square with a side length of 10. In the remaining space of the square, which does not belong to the disk, four smaller circles are inscribed. Find the radius of one of those smaller circles. | [
"\\( 15 - 10\\sqrt{2} \\)"
] | [
"A circle is inscribed in a square with a side length of 10. In the remaining space of the square, which does not belong to the disk, four smaller circles are inscribed. Find the radius of one of those smaller circles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-053457a73146417599e87c492614bfed3a306b2b00b51b8c1db86d096e0a9897 | A circle is inscribed in a square, and a rectangle is placed inside the square but outside the circle. Two sides of the rectangle lie along the sides of the square, and one vertex of the rectangle lies on the circle. The rectangle is twice as high as it is wide. What is the ratio of the area of the square to the area o... | [
"50"
] | [
"A circle is inscribed in a square, and a rectangle is placed inside the square but outside the circle. Two sides of the rectangle lie along the sides of the square, and one vertex of the rectangle lies on the circle. The rectangle is twice as high as it is wide. What is the ratio of the area of the square to the a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-3bcc240568a97bc3558dbddac8b60fb99554871ffbf8003efe44acbd19442ef6 | A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square? | [
"\\(\\frac{\\pi}{8}\\)",
"\\frac{\\pi}{8}"
] | [
"A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?",
"A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a... | [
"MATH_training_set",
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 31
} | 62 | 0 | 0 |
omr-ws-45f3392cd8d7bf26e71f194a4384dc96f328c2f1ce90fe2db0086a4882b88f26 | A circle is inscribed in a square. A rectangle of length $8$ and height $4$ is placed in the bottom right-hand-corner of the square so that its top left-hand-corner touches the circle. Find the radius of the circle. | [
"20"
] | [
"A circle is inscribed in a square. A rectangle of length $8$ and height $4$ is placed in the bottom right-hand-corner of the square so that its top left-hand-corner touches the circle. Find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-bb8a291b57aa0cca110ff8bee2227a5082b92a0d1427613bff6a5929455350b9 | A circle is inscribed in a square. A rectangle with length 12 and breadth 6 is inscribed in one corner of the square such that one of its vertices touches the circle. Find the radius of the circle. | [
"30"
] | [
"A circle is inscribed in a square. A rectangle with length 12 and breadth 6 is inscribed in one corner of the square such that one of its vertices touches the circle. Find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 5
} | 22 | 0 | 0 |
omr-ws-5101dd9a5ddbebb13223de92670a814f1da32296f0b816ec4f24b0e560d1abc2 | A circle is inscribed in a square. A second circle, inside the square, is externally tangent to the first circle and is also tangent to two sides of the square. Compute the ratio of the radius of the smaller circle to the radius of the first circle. | [
"\\( 3 - 2\\sqrt{2} \\)"
] | [
"A circle is inscribed in a square. A second circle, inside the square, is externally tangent to the first circle and is also tangent to two sides of the square. Compute the ratio of the radius of the smaller circle to the radius of the first circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-a85eaa40d5a874b6f70e7cfd1d0cabd6e76707402ed2663a9b065ae762f74357 | A circle is inscribed in a square. The circumference of the circle is increasing at a constant rate of 6 inches per second. At the instant when the area of the circle is \(25\pi\) square inches, find the rate of increase of the area enclosed between the circle and the square. | [
"\\(\\frac{30(4 - \\pi)}{\\pi}\\)"
] | [
"A circle is inscribed in a square. The circumference of the circle is increasing at a constant rate of 6 inches per second. At the instant when the area of the circle is \\(25\\pi\\) square inches, find the rate of increase of the area enclosed between the circle and the square."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-9edd6743348b957fb2995487de072f183b9d928bff73ed1c8a96d2c362c85cb4 | A circle is inscribed in a square. The circumference of the circle is increasing at a constant rate of 6 inches per second. Find the rate at which the perimeter of the square is increasing. | [
"\\(\\frac{24}{\\pi}\\)"
] | [
"A circle is inscribed in a square. The circumference of the circle is increasing at a constant rate of 6 inches per second. Find the rate at which the perimeter of the square is increasing."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-1bc9a2d565ba545db05ed97e2a25b44ebbdd79c08077ba84e133cdff8a2565df | A circle is inscribed in a trapezoid $ABCD$ with the base $| AD | = 40$,\\ the vertex angles $A$ and $D$ equal to $60^\circ$ and the nonparallel sides\linebreak $| AB| = | CD | = 10$ so that it touches both bases $AD$ and $BC$ and the side $AB$. A tangent is drawn to the circle through the point $M$ of the base $AD$ wh... | [
"$10:11$"
] | [
"A circle is inscribed in a trapezoid $ABCD$ with the base $| AD | = 40$,\\\\ the vertex angles $A$ and $D$ equal to $60^\\circ$ and the nonparallel sides\\linebreak $| AB| = | CD | = 10$ so that it touches both bases $AD$ and $BC$ and the side $AB$. A tangent is drawn to the circle through the point $M$ of the b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a44f6fa1a236f8221abe6958488ac686a7c80033df028393a02f51baaca75c4b | A circle is inscribed in a triangle \(ABC\) with sides \(a\), \(b\), and \(c\). Tangents to the circle parallel to the sides of the triangle are constructed, each cutting off a smaller triangle from \(\triangle ABC\). In each of these smaller triangles, a circle is inscribed. Find the sum of the areas of all four inscr... | [
"\\(\\pi \\cdot \\frac{(a+b-c)(b+c-a)(c+a-b)(a^2+b^2+c^2)}{(a+b+c)^3}\\)"
] | [
"A circle is inscribed in a triangle \\(ABC\\) with sides \\(a\\), \\(b\\), and \\(c\\). Tangents to the circle parallel to the sides of the triangle are constructed, each cutting off a smaller triangle from \\(\\triangle ABC\\). In each of these smaller triangles, a circle is inscribed. Find the sum of the areas o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 3
} | 20 | 0 | 0 |
omr-ws-8d3b89a0dffc797d03fd1c72a9e9491749d9e73fe03fea1848db6b4f755ffa50 | A circle is inscribed in a triangle whose sides are 10, 10, and 12 units in measure. A second, smaller circle is inscribed tangent to the first circle and to the two equal sides of the triangle. What is the measure of the radius of the second, smaller circle? | [
"\\dfrac{3}{4}"
] | [
"A circle is inscribed in a triangle whose sides are 10, 10, and 12 units in measure. A second, smaller circle is inscribed tangent to the first circle and to the two equal sides of the triangle. What is the measure of the radius of the second, smaller circle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-da1e6a5b2cd932fab86d680568610b6be1747d1a29d93de674f4a2c579a13c87 | A circle is inscribed in a triangle with side lengths $8$, $13$, and $17$. Let the segments of the side of length $8$, made by a point of tangency, be $r$ and $s$, with $r<s$. What is the ratio $r:s$? | [
"1:3"
] | [
"A circle is inscribed in a triangle with side lengths $8$, $13$, and $17$. Let the segments of the side of length $8$, made by a point of tangency, be $r$ and $s$, with $r<s$. What is the ratio $r:s$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-03f8d9dc1d0fab62fa5fc71fe0a4af1e87de5885278af62fb18b2db9abddd9ca | A circle is inscribed in a triangle with side lengths 6, 10, and 12. Find the radius of the circle. | [
"2.13"
] | [
"A circle is inscribed in a triangle with side lengths 6, 10, and 12. Find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-25563ea0eb3e9e2bf877963d758c3eabf8e4a5822d179be141d237d178c6ceab | A circle is inscribed in a triangle with sides 10, 10, and 12. A second, smaller circle is inscribed tangent to the first circle and to the equal sides of the triangle. Compute the radius of the smaller circle. | [
"\\dfrac{3}{4}"
] | [
"A circle is inscribed in a triangle with sides 10, 10, and 12. A second, smaller circle is inscribed tangent to the first circle and to the equal sides of the triangle. Compute the radius of the smaller circle."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-77347cfed3f6872f4180a4ef3ca67866b973f02ebe810044b0f47178ee1f8b04 | A circle is inscribed in a triangle with sides of lengths 3, 4, and 5. A second circle, interior to the triangle, is tangent to the first circle and to both sides of the larger acute angle of the triangle. Find the radius of the second circle. | [
"\\(\\frac{3 - \\sqrt{5}}{2}\\)"
] | [
"A circle is inscribed in a triangle with sides of lengths 3, 4, and 5. A second circle, interior to the triangle, is tangent to the first circle and to both sides of the larger acute angle of the triangle. Find the radius of the second circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 43
} | 74 | 0 | 0 |
omr-ws-12e790da06598f138762be4b859b784dc2ef987fc543308719f76c5d72e9ba97 | A circle is inscribed in a triangle with sides of lengths 8, 13, and 17. Let the segments of the side of length 8 made by a point of tangency be \( r \) and \( s \), with \( r < s \). Find the ratio \( r:s \). | [
"\\(\\frac{1}{3}\\)"
] | [
"A circle is inscribed in a triangle with sides of lengths 8, 13, and 17. Let the segments of the side of length 8 made by a point of tangency be \\( r \\) and \\( s \\), with \\( r < s \\). Find the ratio \\( r:s \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-507885dbbad945dcafd6963c46b1b57ec877c414341dad2b9771664a59e20734 | A circle is inscribed in a triangle with sides of lengths 8, 13, and 17. Let the segments of the side of length 8 made by a point of tangency be r and s, with r < s. Find the ratio of r:s. | [
"1:3"
] | [
"A circle is inscribed in a triangle with sides of lengths 8, 13, and 17. Let the segments of the side of length 8 made by a point of tangency be r and s, with r < s. Find the ratio of r:s."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-98330ba9ec277ec1165dd6fd708b5af20722fedef18a7952b9fa6df32ddd4394 | A circle is inscribed in a triangle with sides of lengths 8, 13, and 17. Let the segments of the side of length 8, made by the point of tangency, be r and s with r < s. What is the ratio of r to s? | [
"\\dfrac{1}{3}"
] | [
"A circle is inscribed in a triangle with sides of lengths 8, 13, and 17. Let the segments of the side of length 8, made by the point of tangency, be r and s with r < s. What is the ratio of r to s?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-604b7b1a812b66839b57ee6f213ae4a693027dca5e533c852e3298bb8ea2a2d7 | A circle is inscribed in a unit square, and a diagonal of the square is drawn. Find the total length of the segments of the diagonal not contained within the circle. | [
"\\(\\sqrt{2} - 1\\)",
"\\sqrt{2} - 1"
] | [
"A circle is inscribed in a unit square, and a diagonal of the square is drawn. Find the total length of the segments of the diagonal not contained within the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 31
} | 63 | 0 | 0 |
omr-ws-50b6dddae74f4a6f0e47b5d1172be0be7eb1f546e663f95ed933002fab59d70f | A circle is inscribed in an angle with vertex $O$, touching its sides at points $A$ and $B$. A straight line is drawn from point $A$ parallel to $OB$ and intersects the circle again at point $C$. The segment $OC$ intersects the circle again at point $E$. Let line $AE$ intersect the segment $OB$ at point $K$. If the len... | [
"10"
] | [
"A circle is inscribed in an angle with vertex $O$, touching its sides at points $A$ and $B$. A straight line is drawn from point $A$ parallel to $OB$ and intersects the circle again at point $C$. The segment $OC$ intersects the circle again at point $E$. Let line $AE$ intersect the segment $OB$ at point $K$. If th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2cc1d9bcd834b550a54d35f6bd9826a2bf3612813f50308bc5e7f8028eb4d164 | A circle is inscribed in an equilateral triangle \(ABC\). Let \(M\) be the point where the circle touches side \(AB\), and let \(N\) be the second intersection of segment \(CM\) and the circle. Compute the ratio \(\dfrac{MN}{CN}\). | [
"2"
] | [
"A circle is inscribed in an equilateral triangle \\(ABC\\). Let \\(M\\) be the point where the circle touches side \\(AB\\), and let \\(N\\) be the second intersection of segment \\(CM\\) and the circle. Compute the ratio \\(\\dfrac{MN}{CN}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b530583efb707575627a2f86ec77155cdab29dc77f98295a3d1366567c41ae2a | A circle is inscribed in an equilateral triangle with side $a$. Then three more circles are inscribed in the same triangle so that they are tangent to the first one and to the sides of the triangle, and then another three circles tangent to the above three circles and to the sides of the triangle, and so forth. Find th... | [
"The required area is equal to $\\displaystyle\\frac{11}{8}S=\\displaystyle\\frac{11}{96}\\pi a^2$"
] | [
"A circle is inscribed in an equilateral triangle with side $a$. Then three more circles are inscribed in the same triangle so that they are tangent to the first one and to the sides of the triangle, and then another three circles tangent to the above three circles and to the sides of the triangle, and so forth. Fi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88cc50bcc9a5a2a8673e519925a5460c3c81dde9359a3361bb3f02b59aff4417 | A circle is inscribed in an equilateral triangle. A square is inscribed in the circle, a smaller circle is inscribed in the square, and an equilateral triangle is inscribed in the smaller circle. If the ratio of the area of the smaller equilateral triangle to the larger equilateral triangle can be expressed as \(\frac{... | [
"9"
] | [
"A circle is inscribed in an equilateral triangle. A square is inscribed in the circle, a smaller circle is inscribed in the square, and an equilateral triangle is inscribed in the smaller circle. If the ratio of the area of the smaller equilateral triangle to the larger equilateral triangle can be expressed as \\(... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a90ad9fd7948f554506fbbc4b42669071bda71bf325d9656268b1170be79f3a7 | A circle is inscribed in an isosceles trapezium \(ABCD\) where \(AB\) is parallel to \(DC\), \(AB = 10 \, \text{cm}\), and \(DC = 30 \, \text{cm}\). Find the radius and the area of the circle. | [
"Radius = \\(5\\sqrt{3}\\) cm, Area = \\(75\\pi\\) square cm"
] | [
"A circle is inscribed in an isosceles trapezium \\(ABCD\\) where \\(AB\\) is parallel to \\(DC\\), \\(AB = 10 \\, \\text{cm}\\), and \\(DC = 30 \\, \\text{cm}\\). Find the radius and the area of the circle."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-e76b306d99d41b4bfd8a30af6cf2f65994404f51f5e07daeab21e70fe30fd6b2 | A circle is inscribed in an isosceles trapezium \(ABCD\) where \(AB\) is parallel to \(DC\), \(AB = 10\) cm, and \(DC = 30\) cm. Find the radius and the area of the circle. | [
"75\\pi"
] | [
"A circle is inscribed in an isosceles trapezium \\(ABCD\\) where \\(AB\\) is parallel to \\(DC\\), \\(AB = 10\\) cm, and \\(DC = 30\\) cm. Find the radius and the area of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-88bc539d170e2279a04eaf03e48f0f06ef640630b061dc828e379f20b9b2bd11 | A circle is inscribed in an isosceles trapezium with parallel sides measuring $8$ cm and $18$ cm. Is it possible to determine the area of the circle? | [
"\\(36\\pi\\)"
] | [
"A circle is inscribed in an isosceles trapezium with parallel sides measuring $8$ cm and $18$ cm. Is it possible to determine the area of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-2c9d57c364199cc5f2c37e4c8ef97ebf45e45665ebe5079a8177e321d79b34ca | A circle is inscribed in an isosceles trapezoid $ABCD$ $(AB=CD)$. Let $M$ be the tangency point of the circle with side $CD$, $K$ the intersection point of the circle with segment $AM$, and $L$ the intersection point of the circle with segment $BM$. Find the quantity $\frac{AM}{AK}+\frac{BM}{BL}$. | [
"2"
] | [
"A circle is inscribed in an isosceles trapezoid $ABCD$ $(AB=CD)$. Let $M$ be the tangency point of the circle with side $CD$, $K$ the intersection point of the circle with segment $AM$, and $L$ the intersection point of the circle with segment $BM$. Find the quantity $\\frac{AM}{AK}+\\frac{BM}{BL}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e3f76668306955d767ffba29d02d2d76299f792eda216b3da58f8fd5f74d5a1 | A circle is inscribed in an isosceles trapezoid \(ABCD\). Diagonal \(AC\) intersects the circle at points \(K\) and \(L\), with \(K\) between \(A\) and \(L\). Find the value of \(\frac{AL \cdot KC}{AK \cdot LC}\). | [
"17 + 12\\sqrt{2}"
] | [
"A circle is inscribed in an isosceles trapezoid \\(ABCD\\). Diagonal \\(AC\\) intersects the circle at points \\(K\\) and \\(L\\), with \\(K\\) between \\(A\\) and \\(L\\). Find the value of \\(\\frac{AL \\cdot KC}{AK \\cdot LC}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-f7212b5e84537ad3c5edd619625e997991fb3f8f17e9d4ac0fb083d83c7e071f | A circle is inscribed in an isosceles trapezoid \(ABCD\). The circle intersects diagonal \(AC\) at points \(K\) and \(L\) (with \(K\) between \(A\) and \(L\)). Find the value of \(\frac{AL \cdot KC}{AK \cdot LC}\). | [
"\\((\\sqrt{2} + 1)^4\\)"
] | [
"A circle is inscribed in an isosceles trapezoid \\(ABCD\\). The circle intersects diagonal \\(AC\\) at points \\(K\\) and \\(L\\) (with \\(K\\) between \\(A\\) and \\(L\\)). Find the value of \\(\\frac{AL \\cdot KC}{AK \\cdot LC}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 6,
"tir": 5
} | 16 | 0 | 0 |
omr-ws-0b7ac895ae5ceb6d782dfcb5373a66e89e313b8a31de446491a238fd127dc8db | A circle is inscribed in an isosceles trapezoid \(ABCD\). The diagonal \(AC\) intersects the circle at points \(K\) and \(L\) in the order \(A, K, L, C\). Find the value of \(\sqrt[4]{\frac{AL \cdot KC}{AK \cdot LC}}\). | [
"\\(\\sqrt{2} + 1\\)."
] | [
"A circle is inscribed in an isosceles trapezoid \\(ABCD\\). The diagonal \\(AC\\) intersects the circle at points \\(K\\) and \\(L\\) in the order \\(A, K, L, C\\). Find the value of \\(\\sqrt[4]{\\frac{AL \\cdot KC}{AK \\cdot LC}}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 39,
"tir": 5
} | 44 | 0 | 0 |
omr-ws-2b24a6a06d161134ad9848215546a4ef967d774347aced8d9b9d783a0f8bee8a | A circle is inscribed in an isosceles trapezoid such that all four sides of the trapezoid are tangent to the circle. If the radius of the circle is \(1\), and the upper base of the trapezoid is \(1\), what is the area of the trapezoid? | [
"5"
] | [
"A circle is inscribed in an isosceles trapezoid such that all four sides of the trapezoid are tangent to the circle. If the radius of the circle is \\(1\\), and the upper base of the trapezoid is \\(1\\), what is the area of the trapezoid?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ef6691c552044b6c27e421a3368b1430c6bfdc88bf913326fb73ae7cdfe86666 | A circle is inscribed in an isosceles trapezoid with bases of lengths 4 and 8. What is the diameter of the circle? | [
"\\( 4\\sqrt{2} \\)"
] | [
"A circle is inscribed in an isosceles trapezoid with bases of lengths 4 and 8. What is the diameter of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-04f634363b173849f8fe9a5823d901043fe67992e83149fc296d7e009b8c7516 | A circle is inscribed in an isosceles trapezoid with lengths of its parallel sides as 75 and 108. What is the length of the diameter of the circle? | [
"90"
] | [
"A circle is inscribed in an isosceles trapezoid with lengths of its parallel sides as 75 and 108. What is the length of the diameter of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-f02057ef772be008df717785f90fc24296ef9ae5c83c2804524c5da05a191da2 | A circle is inscribed in an isosceles trapezoid. The distance between the center of the circle and the point of intersection of the diagonals of the trapezoid is related to the radius as $3 : 5$. Find the ratio of the perimeter of the trapezoid to the length of the inscribed circle. | [
"$\\frac{5}{\\pi }$"
] | [
"A circle is inscribed in an isosceles trapezoid. The distance between the center of the circle and the point of intersection of the diagonals of the trapezoid is related to the radius as $3 : 5$. Find the ratio of the perimeter of the trapezoid to the length of the inscribed circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a5b37c626d7a5656797c64ea16e0fa0ef044f1f1b7db087325764a72ea5c2f98 | A circle is inscribed in an isosceles trapezoid. The longer base of the trapezoid is 20, and the legs are each 12. Find the diameter of the inscribed circle. | [
"\\(4\\sqrt{5}\\)"
] | [
"A circle is inscribed in an isosceles trapezoid. The longer base of the trapezoid is 20, and the legs are each 12. Find the diameter of the inscribed circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-0bbb23a2502646601e0c29a3616f1f76bc37192c2b06631236c35085ffd6364e | A circle is inscribed in an isosceles triangle with base $a$ and the base angle $\alpha $. In addition, a second circle is constructed which touches the base, one of the lateral sides of the triangle and the first circle inscribed in it. Find the radius of the second circle. | [
"$\\frac{a}{2}\\cdot \\frac{1-\\sin (\\alpha /2)}{1+\\sin (\\alpha /2)}\\tan\\frac{\\alpha }{2}$"
] | [
"A circle is inscribed in an isosceles triangle with base $a$ and the base angle $\\alpha $. In addition, a second circle is constructed which touches the base, one of the lateral sides of the triangle and the first circle inscribed in it. Find the radius of the second circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0827b6a43cdb459179175e7805304a696d6228f7d1fc8d7b24dadde6e241e319 | A circle is inscribed in an isosceles triangle with sides 10, 10, and 12 units. A second, smaller circle is inscribed tangent to the first circle and to the two equal sides of the triangle. Find the radius of the second, smaller circle. | [
"\\(\\frac{3}{4}\\)"
] | [
"A circle is inscribed in an isosceles triangle with sides 10, 10, and 12 units. A second, smaller circle is inscribed tangent to the first circle and to the two equal sides of the triangle. Find the radius of the second, smaller circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-81075b30e6b3531013a1fc99c07a34d545feecf88a336f62642cd17be397c811 | A circle is inscribed in an isosceles triangle. The tangents drawn to this circle cut off three triangles from the given triangle, the sum of the perimeters of which is \(48\). Find the lateral side of this triangle if its base is \(12\). | [
"10"
] | [
"A circle is inscribed in an isosceles triangle. The tangents drawn to this circle cut off three triangles from the given triangle, the sum of the perimeters of which is \\(48\\). Find the lateral side of this triangle if its base is \\(12\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-3f94c782224bc2623970bda26b96b9e149e1c21af23d7597b84abbe4cc0dfda0 | A circle is inscribed in one regular hexagon and circumscribed about another. If the circle has a radius of 6, find the ratio of the area of the smaller hexagon to the area of the larger hexagon. | [
"\\(\\frac{3}{4}\\)"
] | [
"A circle is inscribed in one regular hexagon and circumscribed about another. If the circle has a radius of 6, find the ratio of the area of the smaller hexagon to the area of the larger hexagon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-b17219971858ec277e6c795cb1a3b801622722654ad2147a349c3c195c50a692 | A circle is inscribed in pentagon \(ABCDE\). Given that \(AB = BC\), \(CD = DE\), \(AE = 6\), \(AC = 8\), and \(CE = 7\), find the radius of the inscribed circle. | [
"\\(\\frac{11}{\\sqrt{15}}\\)"
] | [
"A circle is inscribed in pentagon \\(ABCDE\\). Given that \\(AB = BC\\), \\(CD = DE\\), \\(AE = 6\\), \\(AC = 8\\), and \\(CE = 7\\), find the radius of the inscribed circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c3436dbfdff1138f2e239ea04a28e4454c75afc968fc3fb00c39076b7d9db291 | A circle is inscribed in quadrilateral $ABCD$, tangent to $\overline{AB}$ at $P$ and to $\overline{CD}$ at $Q$. Given that $AP=19$, $PB=26$, $CQ=37$, and $QD=23$, find the square of the radius of the circle. | [
"361",
"647"
] | [
"A circle is inscribed in quadrilateral $ABCD$, tangent to $\\overline{AB}$ at $P$ and to $\\overline{CD}$ at $Q$. Given that $AP=19$, $PB=26$, $CQ=37$, and $QD=23$, find the square of the radius of the circle.",
"A circle is inscribed in quadrilateral $ABCD$, tangent to $\\overline{AB}$ at $P$ and to $\\o... | [
"MATH_training_set",
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 7,
"tir": 11
} | 25 | 0 | 0 |
omr-ws-b99f34a9be3628a1c1b69264a0f80b283e09550a7e97b2267cd24c07b1c83bb0 | A circle is inscribed in quadrilateral \(ABCD\) and touches sides \(AD\), \(DC\), and \(CB\) at points \(K\), \(L\), and \(M\) respectively. Let \(N\) be the intersection of the line through \(L\) parallel to \(AD\) and the line \(KM\). Let \(P\) be the intersection of \(LN\) and \(KC\). If \(|PL| = |PN| = x\), find th... | [
"no_valid_answer_found"
] | [
"A circle is inscribed in quadrilateral \\(ABCD\\) and touches sides \\(AD\\), \\(DC\\), and \\(CB\\) at points \\(K\\), \\(L\\), and \\(M\\) respectively. Let \\(N\\) be the intersection of the line through \\(L\\) parallel to \\(AD\\) and the line \\(KM\\). Let \\(P\\) be the intersection of \\(LN\\) and \\(KC\\)... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8f39e3479676d98b0f37a671a208b5c16c18436437d060e2ead10543ad11644 | A circle is inscribed in quadrilateral \(ABCD\) with \(AB \parallel CD\) and \(BC = AD\). The diagonals \(AC\) and \(BD\) intersect at \(E\). The circles inscribed in \(\triangle ABE\), \(\triangle BCE\), \(\triangle CDE\), and \(\triangle DAE\) have radii \(r_1\), \(r_2\), \(r_3\), and \(r_4\) respectively. Calculate ... | [
"0"
] | [
"A circle is inscribed in quadrilateral \\(ABCD\\) with \\(AB \\parallel CD\\) and \\(BC = AD\\). The diagonals \\(AC\\) and \\(BD\\) intersect at \\(E\\). The circles inscribed in \\(\\triangle ABE\\), \\(\\triangle BCE\\), \\(\\triangle CDE\\), and \\(\\triangle DAE\\) have radii \\(r_1\\), \\(r_2\\), \\(r_3\\), ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-835dfd5570e0c5b9b9a0ef36940ee7708c3767383a3e1d3816b1bdba35b0deb2 | A circle is inscribed in quadrilateral \(ABCD\), tangent to \(AB\) at \(P\) and to \(CD\) at \(Q\). Given that \(AP=19\), \(PB=26\), \(CQ=37\), and \(QD=23\), find the square of the radius of the circle. | [
"647"
] | [
"A circle is inscribed in quadrilateral \\(ABCD\\), tangent to \\(AB\\) at \\(P\\) and to \\(CD\\) at \\(Q\\). Given that \\(AP=19\\), \\(PB=26\\), \\(CQ=37\\), and \\(QD=23\\), find the square of the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 6,
"tir": 12
} | 21 | 0 | 0 |
omr-ws-0e02627167bc25311fdcfe97bfa28607e40492e08f8c98350e82e9f5e837ac92 | A circle is inscribed in quadrilateral \(ABCD\), tangent to \(\overline{AB}\) at \(P\) and to \(\overline{CD}\) at \(Q\). Given that \(AP = 19\), \(PB = 26\), \(CQ = 37\), and \(QD = 23\), find the square of the radius of the circle. | [
"647"
] | [
"A circle is inscribed in quadrilateral \\(ABCD\\), tangent to \\(\\overline{AB}\\) at \\(P\\) and to \\(\\overline{CD}\\) at \\(Q\\). Given that \\(AP = 19\\), \\(PB = 26\\), \\(CQ = 37\\), and \\(QD = 23\\), find the square of the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 2,
"tir": 13
} | 22 | 0 | 0 |
omr-ws-2d9859a022b088090ee3672c44b9c2b49742c60795e2f8b0947d15def03058b1 | A circle is inscribed in quadrilateral \(ABCD\), tangent to \(\overline{AB}\) at \(P\) and to \(\overline{CD}\) at \(Q\). Given that \(AP=19\), \(PB=26\), \(CQ=37\), and \(QD=23\), find the square of the radius of the circle. | [
"647"
] | [
"A circle is inscribed in quadrilateral \\(ABCD\\), tangent to \\(\\overline{AB}\\) at \\(P\\) and to \\(\\overline{CD}\\) at \\(Q\\). Given that \\(AP=19\\), \\(PB=26\\), \\(CQ=37\\), and \\(QD=23\\), find the square of the radius of the circle."
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 11,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-b4c8b78a5ccc70e4064d0836af3e0d273c0863360d19bbb48f0eebf19907e54f | A circle is inscribed in rectangle $ABCD$ such that it touches sides $AB$, $AD$, and $CD$. Diagonal $AC$ intersects the circle at points $P$ and $Q$. Given $AB = 8$ and $AD = 4$, determine the length of $PQ$. | [
"\\(\\frac{8}{\\sqrt{5}}\\)"
] | [
"A circle is inscribed in rectangle $ABCD$ such that it touches sides $AB$, $AD$, and $CD$. Diagonal $AC$ intersects the circle at points $P$ and $Q$. Given $AB = 8$ and $AD = 4$, determine the length of $PQ$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 25
} | 48 | 0 | 0 |
omr-ws-7eb6cefd25937bc6e5a4080ae34a5784908055448ab42305bfdf188b488b0873 | A circle is inscribed in square \(ABCD\) and is tangent to sides \(AB\) and \(CD\) at points \(M\) and \(K\) respectively. Line \(BK\) intersects the circle at point \(L\), and \(X\) is the midpoint of \(KL\). Find the measure of angle \(\angle MXK\). | [
"\\(135^\\circ\\)"
] | [
"A circle is inscribed in square \\(ABCD\\) and is tangent to sides \\(AB\\) and \\(CD\\) at points \\(M\\) and \\(K\\) respectively. Line \\(BK\\) intersects the circle at point \\(L\\), and \\(X\\) is the midpoint of \\(KL\\). Find the measure of angle \\(\\angle MXK\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 25
} | 73 | 0 | 0 |
omr-ws-28d58a2fc6663b36a7355d0625baa9db11b534e201cd20ea0fa5a9078eca22b3 | A circle is inscribed in the angle \(ABC\), which is equal to \(60^\circ\). The tangent to this circle, drawn so that the center of the circle and the vertex of the angle are located on opposite sides of it, intersects the sides of the angle at points \(M\) and \(N\). The perimeter of the triangle \(MBN\) is \(18\). Fi... | [
"3\\sqrt{3}"
] | [
"A circle is inscribed in the angle \\(ABC\\), which is equal to \\(60^\\circ\\). The tangent to this circle, drawn so that the center of the circle and the vertex of the angle are located on opposite sides of it, intersects the sides of the angle at points \\(M\\) and \\(N\\). The perimeter of the triangle \\(MBN\... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-c3c7d5df8ffe948d35f31bf33ea8b0422ff0ced3052ce7e70e9c14113d178852 | A circle is inscribed in the angle \(AOB\) with tangency at points \(A\) and \(B\). Point \(C\) is on the circle and inside the triangle \(AOC\). The distances from point \(C\) to the lines \(OA\) and \(OB\) are 2 and 8, respectively. Find the distance from point \(C\) to the chord \(AB\). | [
"4"
] | [
"A circle is inscribed in the angle \\(AOB\\) with tangency at points \\(A\\) and \\(B\\). Point \\(C\\) is on the circle and inside the triangle \\(AOC\\). The distances from point \\(C\\) to the lines \\(OA\\) and \\(OB\\) are 2 and 8, respectively. Find the distance from point \\(C\\) to the chord \\(AB\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-ac541eb0beeca650b058e1c35ce87b7798d3a2dbb420b4b3272aab4b4a808f17 | A circle is inscribed in the isosceles trapezoid $ABCD$. Let $K$ and $L$ be the intersection points of the circle with $AC$ (with $K$ between $A$ and $L$). Find the value of $$\frac{AL \cdot KC}{AK \cdot LC}.$$ | [
"1"
] | [
"A circle is inscribed in the isosceles trapezoid $ABCD$. Let $K$ and $L$ be the intersection points of the circle with $AC$ (with $K$ between $A$ and $L$). Find the value of $$\\frac{AL \\cdot KC}{AK \\cdot LC}.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-70ac1c08f90b45db88c14e9c7438030392d200655eaa0106cb9dd9b985f2a1bb | A circle is inscribed in the quadrilateral \(ABCD\) with tangent points \(E \in AB\), \(F \in BC\), \(G \in CD\), and \(H \in AD\). Suppose that \(EH \cap FG = O\). If \(BD\) intersects \(EH\) at \(P\) and \(FG\) at \(Q\), find the ratio \(\frac{OP}{OQ}\). | [
"1"
] | [
"A circle is inscribed in the quadrilateral \\(ABCD\\) with tangent points \\(E \\in AB\\), \\(F \\in BC\\), \\(G \\in CD\\), and \\(H \\in AD\\). Suppose that \\(EH \\cap FG = O\\). If \\(BD\\) intersects \\(EH\\) at \\(P\\) and \\(FG\\) at \\(Q\\), find the ratio \\(\\frac{OP}{OQ}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-dace5136a85018963bfe928a6f0b5e2f4574c6beece560224e05bf91c4f4f60e | A circle is inscribed in the trapezoid \(ABCD\). Let \(K, L, M, N\) be the points of tangency of this circle with the diagonals \(AC\) and \(BD\), respectively (\(K\) is between \(A\) and \(L\), and \(M\) is between \(B\) and \(N\)). Given that \(AK \cdot LC = 16\) and \(BM \cdot ND = \frac{9}{4}\), find the radius of ... | [
"6"
] | [
"A circle is inscribed in the trapezoid \\(ABCD\\). Let \\(K, L, M, N\\) be the points of tangency of this circle with the diagonals \\(AC\\) and \\(BD\\), respectively (\\(K\\) is between \\(A\\) and \\(L\\), and \\(M\\) is between \\(B\\) and \\(N\\)). Given that \\(AK \\cdot LC = 16\\) and \\(BM \\cdot ND = \\fr... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-4989916922ffb82e8553c2f52b9f80c8cbba84ba94a9c7ccefeb68bd7e7a8a36 | A circle is inscribed in the triangle $A BC$ with sides $| AB | =10$, $| BC | = 20$ and the angle $C$ equal to $30^\circ$. A tangent to the circle is drawn through the point $M$ of the side $AC$ which is at the distance of 10 from the vertex $A$. Assume that $K$ is a point of intersection of the tangent and the straigh... | [
"$\\frac{50}{3}\\left(6-\\sqrt 3\\right)$"
] | [
"A circle is inscribed in the triangle $A BC$ with sides $| AB | =10$, $| BC | = 20$ and the angle $C$ equal to $30^\\circ$. A tangent to the circle is drawn through the point $M$ of the side $AC$ which is at the distance of 10 from the vertex $A$. Assume that $K$ is a point of intersection of the tangent and the s... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff4b6ebec2ada95c2055f85af43444c6e4a98a51541cfa0d4bc703c15bf06b40 | A circle is inscribed in triangle \(ABC\) and is tangent to \(BC\), \(CA\), and \(AB\) at points \(D\), \(E\), and \(F\) respectively. Points \(M\), \(N\), and \(J\) are the midpoints of the arcs \(EF\), \(FD\), and \(DE\) of the incircle. If the coordinates of \(F\) are \((1, 0)\) and the line \(MN\) is given by the e... | [
"no_valid_answer_found"
] | [
"A circle is inscribed in triangle \\(ABC\\) and is tangent to \\(BC\\), \\(CA\\), and \\(AB\\) at points \\(D\\), \\(E\\), and \\(F\\) respectively. Points \\(M\\), \\(N\\), and \\(J\\) are the midpoints of the arcs \\(EF\\), \\(FD\\), and \\(DE\\) of the incircle. If the coordinates of \\(F\\) are \\((1, 0)\\) an... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-62aa463a43231326a9e0ee9574a029225d383e6e52bc8ecd940b7f430915c2de | A circle is inscribed in triangle \(ABC\) and touches side \(BC\) at point \(E\). A section \(CD\) is perpendicular to the plane \(BC\) and has the same length as \(CA\). Given that \(CE = 1\) and the length of \(BD\) is 2 more than the length of \(BA\), find the radius of the circle inscribed in triangle \(BCD\). | [
"1"
] | [
"A circle is inscribed in triangle \\(ABC\\) and touches side \\(BC\\) at point \\(E\\). A section \\(CD\\) is perpendicular to the plane \\(BC\\) and has the same length as \\(CA\\). Given that \\(CE = 1\\) and the length of \\(BD\\) is 2 more than the length of \\(BA\\), find the radius of the circle inscribed in... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1bc9bf8b1cb7046e8f78b8cbcc88decd95497093ec854c627677d70ac6f94617 | A circle is inscribed in triangle \(ABC\) and touches side \(BC\) at point \(E\). Segment \(CD\) is perpendicular to \(BC\) and has the same length as \(CA\). Given that \(CE = 1\) cm and the length of \(BD\) is 2 cm shorter than the length of \(BA\), find the radius of the circle inscribed in triangle \(BCD\). | [
"2"
] | [
"A circle is inscribed in triangle \\(ABC\\) and touches side \\(BC\\) at point \\(E\\). Segment \\(CD\\) is perpendicular to \\(BC\\) and has the same length as \\(CA\\). Given that \\(CE = 1\\) cm and the length of \\(BD\\) is 2 cm shorter than the length of \\(BA\\), find the radius of the circle inscribed in tr... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8c4d319dbbccd3b7cc70f7ed30cb93eea1da08b3624fe1001e0a66dac702a4de | A circle is inscribed in triangle \(ABC\). It touches side \(AB\) at point \(D\). Given \(AD = 5 \, \text{cm}\), \(DB = 3 \, \text{cm}\), and \(\angle A = 60^\circ\), find the length of side \(BC\). | [
"13"
] | [
"A circle is inscribed in triangle \\(ABC\\). It touches side \\(AB\\) at point \\(D\\). Given \\(AD = 5 \\, \\text{cm}\\), \\(DB = 3 \\, \\text{cm}\\), and \\(\\angle A = 60^\\circ\\), find the length of side \\(BC\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-74d5c17cc7e6ab349c930fcdc6dac2b853eb8080a2d0999cd81f8d2cdffc5a62 | A circle is inscribed in triangle \(ABC\). Points \(M\) and \(N\) are the points of tangency with sides \(BC\) and \(CA\), respectively. Segment \(AM\) intersects \(BN\) at point \(P\) and the inscribed circle at point \(Q\). Given that \(MP = a\) and \(PQ = b\), find the length of \(AQ\). | [
"\\(a + b\\)"
] | [
"A circle is inscribed in triangle \\(ABC\\). Points \\(M\\) and \\(N\\) are the points of tangency with sides \\(BC\\) and \\(CA\\), respectively. Segment \\(AM\\) intersects \\(BN\\) at point \\(P\\) and the inscribed circle at point \\(Q\\). Given that \\(MP = a\\) and \\(PQ = b\\), find the length of \\(AQ\\)."... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5cb6783568aab8465b7e8a66ff33674123a273dbe6df19519ee884acd0d02be1 | A circle is inscribed in triangle \(ABC\). The sides \(BC\), \(CA\), and \(AB\) are tangent to the circle at points \(A_1\), \(B_1\), and \(C_1\) respectively. The segment \(AA_1\) intersects the inscribed circle again at point \(Q\). Line \(\ell\) passes through point \(A\) and is parallel to line \(BC\). Lines \(A_1B... | [
"\\theta"
] | [
"A circle is inscribed in triangle \\(ABC\\). The sides \\(BC\\), \\(CA\\), and \\(AB\\) are tangent to the circle at points \\(A_1\\), \\(B_1\\), and \\(C_1\\) respectively. The segment \\(AA_1\\) intersects the inscribed circle again at point \\(Q\\). Line \\(\\ell\\) passes through point \\(A\\) and is parallel ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-4f012f9078d83fc98034e077e155a5707a4ef54743790d65c97d7707426dfa63 | A circle is inscribed inside a square with a side length of 10. Two additional arcs are quarter circles. Calculate the area of the shaded region without using calculus. | [
"50\\pi - 100"
] | [
"A circle is inscribed inside a square with a side length of 10. Two additional arcs are quarter circles. Calculate the area of the shaded region without using calculus."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
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