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-6c8283e239d6a816d1f9a4e61ee3e4651bc745844881c74023be3af0d74d51b1 | Triangle $ABC$ has the roots of the polynomial $2x^3 - 10x^2 + 15x - 6$ as its side lengths. What is the perimeter of $ABC$? | [
"5"
] | [
"Triangle $ABC$ has the roots of the polynomial $2x^3 - 10x^2 + 15x - 6$ as its side lengths. What is the perimeter of $ABC$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-423578e07e2ee8c78e9cca60f0881e4207d3a2824935f701f88fd87d2e7299f4 | Triangle $ABC$ has vertices $A(0, 8)$, $B(2, 0)$, $C(8, 0)$. A horizontal line with equation $y=t$ intersects line segment $ \overline{AB} $ at $T$ and line segment $ \overline{AC} $ at $U$, forming $\triangle ATU$ with area 13.5. Compute $t$. | [
"2"
] | [
"Triangle $ABC$ has vertices $A(0, 8)$, $B(2, 0)$, $C(8, 0)$. A horizontal line with equation $y=t$ intersects line segment $ \\overline{AB} $ at $T$ and line segment $ \\overline{AC} $ at $U$, forming $\\triangle ATU$ with area 13.5. Compute $t$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-47d2862e4dc6434e3c11fe1695da60e17573906f3ddd7c80f03483ae06dd7bab | Triangle $ABC$ has vertices $A(0, 8)$, $B(2, 0)$, $C(8, 0)$. A line through $B$ cuts the area of $\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line. | [
"-2"
] | [
"Triangle $ABC$ has vertices $A(0, 8)$, $B(2, 0)$, $C(8, 0)$. A line through $B$ cuts the area of $\\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f26f1f2946b822d1601eb43c4864deccfd2d49e0b4761fdf75b8bc97e1d4048 | Triangle $ABC$ has vertices $A(0,8)$, $B(2,0)$, $C(8,0)$. A vertical line intersects $AC$ at $R$ and $\overline{BC}$ at $S$, forming triangle $RSC$. If the area of $\triangle RSC$ is 12.5, determine the positive difference of the $x$ and $y$ coordinates of point $R$. | [
"2"
] | [
"Triangle $ABC$ has vertices $A(0,8)$, $B(2,0)$, $C(8,0)$. A vertical line intersects $AC$ at $R$ and $\\overline{BC}$ at $S$, forming triangle $RSC$. If the area of $\\triangle RSC$ is 12.5, determine the positive difference of the $x$ and $y$ coordinates of point $R$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0dce674ab1cfc7d75ea9d2d9f1baefc1d1f30c0bc196df2c3467807d31b74542 | Triangle $ABC$ has vertices at $A(5,8)$, $B(3,-2)$, and $C(6,1)$. The point $D$ with coordinates $(m,n)$ is chosen inside the triangle so that the three small triangles $ABD$, $ACD$ and $BCD$ all have equal areas. What is the value of $10m + n$? | [
"49"
] | [
"Triangle $ABC$ has vertices at $A(5,8)$, $B(3,-2)$, and $C(6,1)$. The point $D$ with coordinates $(m,n)$ is chosen inside the triangle so that the three small triangles $ABD$, $ACD$ and $BCD$ all have equal areas. What is the value of $10m + n$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7ceec58560b4e62b8b7b2fe5500cbeb308679f71c3309ad5ed25788ae55363a9 | Triangle $ABC$ has vertices that are lattice points. Two of its sides have lengths from the set $\{\sqrt{17}, \sqrt{1999}, \sqrt{2000}\}$. What is the maximum possible area of triangle $ABC$? | [
"\\(\\frac{\\sqrt{33919}}{2}\\)"
] | [
"Triangle $ABC$ has vertices that are lattice points. Two of its sides have lengths from the set $\\{\\sqrt{17}, \\sqrt{1999}, \\sqrt{2000}\\}$. What is the maximum possible area of triangle $ABC$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cc7880eddb348e91654b2dbff4a9fcb3e3de3f2c9b5850858315923dd60c3e20 | Triangle $ABC$ has vertices with coordinates $A(2,3),$ $B(7,8),$ and $C(-4,6)$. The triangle is reflected about line $L$. The image points are $A'(2,-5),$ $B'(7,-10),$ and $C'(-4,-8)$. What is the equation of line $L$? | [
"y = -1"
] | [
"Triangle $ABC$ has vertices with coordinates $A(2,3),$ $B(7,8),$ and $C(-4,6)$. The triangle is reflected about line $L$. The image points are $A'(2,-5),$ $B'(7,-10),$ and $C'(-4,-8)$. What is the equation of line $L$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fc97697564deb6cc9a6812eaa5a85fcd2405d6918f88883a028b4f28a15f8025 | Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of triangle $CDM$ may be expressed as $\frac {m\sqrt {n}}{p},$ where $m,$ $n,$ and $p$ are positive integers... | [
"578"
] | [
"Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of triangle $CDM$ may be expressed as $\\frac {m\\sqrt {n}}{p},$ where $m,$ $n,$ and $p$ are positive... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 7,
"tir": 6
} | 44 | 0 | 0 |
omr-ws-61eeed800cd1fdd8f33e608a4e2bb826386ec521596b807665d400a95ec0ee59 | Triangle $ABC$ is a right triangle with $AC=7,$ $BC=24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD=BD=15.$ Given that the area of triangle $CDM$ may be expressed as $\frac{m\sqrt{n}}{p},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and ... | [
"578"
] | [
"Triangle $ABC$ is a right triangle with $AC=7,$ $BC=24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD=BD=15.$ Given that the area of triangle $CDM$ may be expressed as $\\frac{m\\sqrt{n}}{p},$ where $m,$ $n,$ and $p$ are positive integers, $... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-bf3f9d4369f105a782409595c92c30ed5d6247117c4ab84b53bed59b3164693e | Triangle $ABC$ is a right triangle with legs $AB$ and $AC$. Points $X$ and $Y$ lie on legs $AB$ and $AC$, respectively, so that $AX:XB = AY:YC = 1:2$. If $BY = 16$ units, and $CX = 28$ units, what is the length of hypotenuse $BC$? Express your answer in simplest radical form. | [
"6\\sqrt{26}"
] | [
"Triangle $ABC$ is a right triangle with legs $AB$ and $AC$. Points $X$ and $Y$ lie on legs $AB$ and $AC$, respectively, so that $AX:XB = AY:YC = 1:2$. If $BY = 16$ units, and $CX = 28$ units, what is the length of hypotenuse $BC$? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-78cc6f3e8542a478e8e254cdb1789d9391c4659e39c44f63ea0a98da36c4b317 | Triangle $ABC$ is a right-angled triangle with the right angle at vertex $C$. Let $P$ be the reflection of $C$ about the hypotenuse $AB$. It is known that $P$ and the midpoints of two sides of $ABC$ lie on a line. Find the angles of the triangle. | [
"\\(30^\\circ\\) and \\(60^\\circ\\)"
] | [
"Triangle $ABC$ is a right-angled triangle with the right angle at vertex $C$. Let $P$ be the reflection of $C$ about the hypotenuse $AB$. It is known that $P$ and the midpoints of two sides of $ABC$ lie on a line. Find the angles of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 15
} | 16 | 0 | 0 |
omr-ws-359f5c6e5ed14e6bbfc128da9d2877cbda84f2e9454c38f7c0ee026128cd5ccc | Triangle $ABC$ is acute. Equilateral triangles $ABC', AB'C, A'BC$ are constructed externally to $ABC$. Let $BB'$ and $CC'$ intersect at $F$. Let $CC'$ intersect $AB$ at $C_1$ and $AA'$ intersect $BC$ at $A_1$, and let $A_1C_1$ intersect $AC$ at $D$. Given $A'F=23$, $CF=13$, and $DF=24$, find the length of $BD$. | [
"26"
] | [
"Triangle $ABC$ is acute. Equilateral triangles $ABC', AB'C, A'BC$ are constructed externally to $ABC$. Let $BB'$ and $CC'$ intersect at $F$. Let $CC'$ intersect $AB$ at $C_1$ and $AA'$ intersect $BC$ at $A_1$, and let $A_1C_1$ intersect $AC$ at $D$. Given $A'F=23$, $CF=13$, and $DF=24$, find the length of $BD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 4
} | 26 | 0 | 0 |
omr-ws-d78991b86dd328dce3e6d5f72d117021b9d56eda451ceb30ae7c1c34116cdb2c | Triangle $ABC$ is an equilateral triangle with $AC = 3$. Points $D$ and $E$ are selected on $AB$ and $BC$, respectively, such that $BD = BE = 1$. Find the perimeter of the quadrilateral $ACED$. | [
"8"
] | [
"Triangle $ABC$ is an equilateral triangle with $AC = 3$. Points $D$ and $E$ are selected on $AB$ and $BC$, respectively, such that $BD = BE = 1$. Find the perimeter of the quadrilateral $ACED$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7abbc8a645d2b7489bec2e71c9943f1bba69eb7cbb7c2a8035d948f2d61be455 | Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\overline{BC}$. Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ can be wri... | [
"12"
] | [
"Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\\overline{BC}$. Points $I$ and $E$ lie on sides $\\overline{AC}$ and $\\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ ca... | [
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 63,
"tir": 6
} | 69 | 0 | 0 |
omr-ws-dbb005df8e32b260911ebbf0cc2c075cd6844a538a68d81182da6b819cbeeabd | Triangle $ABC$ is an isosceles right triangle with the measure of angle $A$ equal to 90 degrees. The length of segment $AC$ is 6 cm. What is the area of triangle $ABC$, in square centimeters? | [
"18"
] | [
"Triangle $ABC$ is an isosceles right triangle with the measure of angle $A$ equal to 90 degrees. The length of segment $AC$ is 6 cm. What is the area of triangle $ABC$, in square centimeters?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e94eefcdf9cf814fb98c386ae11d49df22422dd7e65cef8810b6ca94d6fa50d | Triangle $ABC$ is an isosceles triangle with $AB=BC$. The segments $BE$ and $CD$ are medians drawn from $B$ and $C$ to sides $AC$ and $AB$, respectively. If $BE \perp CD$ and $BE=CD=12$, find the area of triangle $ABC$. | [
"96"
] | [
"Triangle $ABC$ is an isosceles triangle with $AB=BC$. The segments $BE$ and $CD$ are medians drawn from $B$ and $C$ to sides $AC$ and $AB$, respectively. If $BE \\perp CD$ and $BE=CD=12$, find the area of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-c0c971b5006a3e2850657d707799cfb255e34fa2c36c71066a731ce42731c053 | Triangle $ABC$ is an isosceles triangle with side lengths of 25, 25 and 48 centimeters. What is the area of triangle $ABC$, in square centimeters? | [
"168"
] | [
"Triangle $ABC$ is an isosceles triangle with side lengths of 25, 25 and 48 centimeters. What is the area of triangle $ABC$, in square centimeters?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-999acc6ebed956b4938eba2fff83f5c616614e574dbb3cc03323c3b7e92ed42a | Triangle $ABC$ is constructed such that $AB>BC>AC$. Points $E$, $F$, and $G$ are drawn from $A$, $B$, and $C$ to their opposite sides respectively. If $\frac{BE}{EC} = \frac{7}{12}$, the ratio of $\frac{AF}{FC}$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is the s... | [
"19"
] | [
"Triangle $ABC$ is constructed such that $AB>BC>AC$. Points $E$, $F$, and $G$ are drawn from $A$, $B$, and $C$ to their opposite sides respectively. If $\\frac{BE}{EC} = \\frac{7}{12}$, the ratio of $\\frac{AF}{FC}$ can be represented as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-4ae8941ed02eda768dd15da321a32bc4f81af9c89a9f1f42403aec2b03fae5ba | Triangle $ABC$ is equilateral with a side length of $1$, the two circular arcs are centered at $C$ and $F$, and $D$ and $E$ are the midpoints of their respective sides. What is the area of the portion of the triangle outside of the two circular arcs? | [
"\\(\\frac{3\\sqrt{3}}{8} - \\frac{\\pi}{12}\\)"
] | [
"Triangle $ABC$ is equilateral with a side length of $1$, the two circular arcs are centered at $C$ and $F$, and $D$ and $E$ are the midpoints of their respective sides. What is the area of the portion of the triangle outside of the two circular arcs?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 3
} | 7 | 0 | 0 |
omr-ws-f07aa8475dd150d953cb99bdf22058fb2341282f4775fe0ca6e7bacbb76795c4 | Triangle $ABC$ is equilateral with side length $12$. Point $D$ is the midpoint of side $\overline{BC}$. Circles centered at $A$ and $D$ intersect at the midpoints of sides $\overline{AB}$ and $\overline{AC}$. Point $E$ lies on segment $\overline{AD}$ and circle $E$ is tangent to both circles centered at $A$ and $D$. Co... | [
"\\( 6 - 3\\sqrt{3} \\)"
] | [
"Triangle $ABC$ is equilateral with side length $12$. Point $D$ is the midpoint of side $\\overline{BC}$. Circles centered at $A$ and $D$ intersect at the midpoints of sides $\\overline{AB}$ and $\\overline{AC}$. Point $E$ lies on segment $\\overline{AD}$ and circle $E$ is tangent to both circles centered at $A$ an... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-7d8efe33051d9544ac5476c6eb705d0227406ce3ee6d37f9cee49c8a36be2958 | Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$? | [
"\\( 12\\pi \\)"
] | [
"Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-0bc971741677bd48c1a18ec9069f6d0ba19897c644bfae5fe5035b8a29befdc6 | Triangle $ABC$ is equilateral with side length $\sqrt{3}$ and circumcenter at $O$. Point $P$ is in the plane such that $(AP)(BP)(CP) = 7$. Compute the difference between the maximum and minimum possible values of $OP$. | [
"\\( 2 - \\sqrt[3]{6} \\)"
] | [
"Triangle $ABC$ is equilateral with side length $\\sqrt{3}$ and circumcenter at $O$. Point $P$ is in the plane such that $(AP)(BP)(CP) = 7$. Compute the difference between the maximum and minimum possible values of $OP$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-35951c2b2ce222c6b5673dbfd808ec014fcedc225794fd9f859400a40972cc4c | Triangle $ABC$ is equilateral, $D$ is an inner point of triangle $ABC$, and $P$ is a point outside triangle $ABC$ such that $AD = BD$, $AB = BP$, and $BD$ bisects $\angle CBP$. Find $\angle BPD$. | [
"30^\\circ"
] | [
"Triangle $ABC$ is equilateral, $D$ is an inner point of triangle $ABC$, and $P$ is a point outside triangle $ABC$ such that $AD = BD$, $AB = BP$, and $BD$ bisects $\\angle CBP$. Find $\\angle BPD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-eebc2ff7de9a9db93203d0709372cb6187182de6f9cf2be97f08c5930b7b0e1c | Triangle $ABC$ is equilateral, with $AB = 6$. There are points $D$, $E$ on segment $AB$ (in the order $A$, $D$, $E$, $B$), points $F$, $G$ on segment $BC$ (in the order $B$, $F$, $G$, $C$), and points $H$, $I$ on segment $CA$ (in the order $C$, $H$, $I$, $A$) such that $DE = FG = HI = 2$. Considering all such configura... | [
"\\(\\sqrt{3}\\)"
] | [
"Triangle $ABC$ is equilateral, with $AB = 6$. There are points $D$, $E$ on segment $AB$ (in the order $A$, $D$, $E$, $B$), points $F$, $G$ on segment $BC$ (in the order $B$, $F$, $G$, $C$), and points $H$, $I$ on segment $CA$ (in the order $C$, $H$, $I$, $A$) such that $DE = FG = HI = 2$. Considering all such conf... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9ff5fe366ba7343c13ce2ede6f461774062adc25782aad040e9e0f690b8cdfa1 | Triangle $ABC$ is given. Circle $\omega_1$ with center on segment $AB$ passes through point $A$ and intersects segments $AB$ and $AC$ for the second time at points $A_1$ and $A_2$, respectively. Circle $\omega_2$ with center on segment $BC$ passes through point $C$ and intersects segments $BC$ and $AC$ for the second t... | [
"\\angle BAC"
] | [
"Triangle $ABC$ is given. Circle $\\omega_1$ with center on segment $AB$ passes through point $A$ and intersects segments $AB$ and $AC$ for the second time at points $A_1$ and $A_2$, respectively. Circle $\\omega_2$ with center on segment $BC$ passes through point $C$ and intersects segments $BC$ and $AC$ for the s... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-e856e2ebc1f2974333d847e473f764fa26ada115fd8192cfb32413e31b63b594 | Triangle $ABC$ is given. Let $O$ be the center of its circumscribed circle, $B_1$ and $C_1$ be the midpoints of the sides $AC$ and $AB$, respectively. Among the circles that contain the vertex $A$ and the point $O$, but do not pass through the points $B_1$ and $C_1$, choose a circle. Let this circle intersect the lines... | [
"1"
] | [
"Triangle $ABC$ is given. Let $O$ be the center of its circumscribed circle, $B_1$ and $C_1$ be the midpoints of the sides $AC$ and $AB$, respectively. Among the circles that contain the vertex $A$ and the point $O$, but do not pass through the points $B_1$ and $C_1$, choose a circle. Let this circle intersect the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-5eaffd70b49182cb833eb230ce3ca8480182547945fcceb497ce6ccc7f93ce68 | Triangle $ABC$ is given. The line perpendicular to $AC$ and passing through $B$ intersects the circle with diameter $AC$ at points $X$ and $K$, where $X$ is closer to $B$ than $K$. Similarly, the line perpendicular to $AB$ and passing through $C$ intersects the circle with diameter $AB$ at points $Y$ and $L$, where $Y$... | [
"\\dfrac{1}{2}"
] | [
"Triangle $ABC$ is given. The line perpendicular to $AC$ and passing through $B$ intersects the circle with diameter $AC$ at points $X$ and $K$, where $X$ is closer to $B$ than $K$. Similarly, the line perpendicular to $AB$ and passing through $C$ intersects the circle with diameter $AB$ at points $Y$ and $L$, wher... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0c75bfd1ed585a44d2a0e03b738ef6daab36864469df607d06adb7375accc0bc | Triangle $ABC$ is good if it satisfies the following conditions: $AB = 7$, the other two sides are integers, and $\cos A = \frac{2}{7}$. Find the sides of a good isosceles triangle. | [
"\\( AC = 4, AB = BC = 7 \\)"
] | [
"Triangle $ABC$ is good if it satisfies the following conditions: $AB = 7$, the other two sides are integers, and $\\cos A = \\frac{2}{7}$. Find the sides of a good isosceles triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-b3146f605fad6446d5004c4ac6751aa772f51c34932bd76651f16a60a72a67f4 | Triangle $ABC$ is good if it satisfies the following conditions: $AB = 7$, the other two sides are integers, and $\cos A = \frac{2}{7}$. Find the sides of a good scalene triangle. | [
"\\( AB = 7 \\), \\( AC = 8 \\), \\( BC = 9 \\)"
] | [
"Triangle $ABC$ is good if it satisfies the following conditions: $AB = 7$, the other two sides are integers, and $\\cos A = \\frac{2}{7}$. Find the sides of a good scalene triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 23
} | 25 | 0 | 0 |
omr-ws-03fb0f824245fdef2bae77893fc34fd835387ab3c79a187ed997b523bddf6ba8 | Triangle $ABC$ is inscribed in $\odot O$, and $AD \perp BC$ at $D$. Point $K$ is on $AO$ such that $\angle BKC = 90^\circ$. Extend $BK$ and $CK$ to meet $AC$ and $AB$ at $E$ and $F$, respectively. If $DK$ intersects $EF$ at point $P$, and $DP = 5$, find the length of $KP$. | [
"\\dfrac{5}{4}"
] | [
"Triangle $ABC$ is inscribed in $\\odot O$, and $AD \\perp BC$ at $D$. Point $K$ is on $AO$ such that $\\angle BKC = 90^\\circ$. Extend $BK$ and $CK$ to meet $AC$ and $AB$ at $E$ and $F$, respectively. If $DK$ intersects $EF$ at point $P$, and $DP = 5$, find the length of $KP$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-395c60e24123a4a30a3cb30a762938cf00672555be4c9b5e56e38863be69f744 | Triangle $ABC$ is inscribed in a circle $(O)$. A circle $(K)$ passes through points $B$ and $C$ and intersects $AC$ and $AB$ at points $E$ and $F$, respectively. The lines $BE$ and $CF$ intersect at point $H$, and the line $HK$ intersects $EF$ at point $I$. The line through $I$ perpendicular to $BC$ intersects the perp... | [
"16"
] | [
"Triangle $ABC$ is inscribed in a circle $(O)$. A circle $(K)$ passes through points $B$ and $C$ and intersects $AC$ and $AB$ at points $E$ and $F$, respectively. The lines $BE$ and $CF$ intersect at point $H$, and the line $HK$ intersects $EF$ at point $I$. The line through $I$ perpendicular to $BC$ intersects the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-399504919f57d88b6de48af8dc75967c9eb8d8cf9e9e39572dd449e8ab86ca91 | Triangle $ABC$ is inscribed in a circle $C$ with center $O$ and radius $1$. For any point $M \in C \setminus \{A, B, C\}$, let $s(M) = OH_1^2 + OH_2^2 + OH_3^2$, where $H_1, H_2, H_3$ are the orthocenters of triangles $MAB$, $MBC$, and $MCA$, respectively. Determine the smallest non-zero number $k$ such that if points ... | [
"3"
] | [
"Triangle $ABC$ is inscribed in a circle $C$ with center $O$ and radius $1$. For any point $M \\in C \\setminus \\{A, B, C\\}$, let $s(M) = OH_1^2 + OH_2^2 + OH_3^2$, where $H_1, H_2, H_3$ are the orthocenters of triangles $MAB$, $MBC$, and $MCA$, respectively. Determine the smallest non-zero number $k$ such that i... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-19aef24f8a7d63e83f36258dac62f3f202316fcae245efd9e08adae5c44553bf | Triangle $ABC$ is inscribed in a circle of radius $2$ with $\angle ABC \geq 90^\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC,b=CA,c=AB$. Find all possible values of $x$. | [
"-\\frac{\\sqrt{6}+\\sqrt{2}}{2}, -\\frac{\\sqrt{6}-\\sqrt{2}}{2}"
] | [
"Triangle $ABC$ is inscribed in a circle of radius $2$ with $\\angle ABC \\geq 90^\\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC,b=CA,c=AB$. Find all possible values of $x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab6ab6716125e1ec48d4a1f6d824cb9260d919ccab8dbfee6dd1ef2f0657b1f1 | Triangle $ABC$ is inscribed in a circle of radius 2 with $\angle ABC \geq 90^\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC$, $b=CA$, $c=AB$. Find all possible values of $x$. | [
"\\( x = \\frac{-\\sqrt{6} \\pm \\sqrt{2}}{2} \\)"
] | [
"Triangle $ABC$ is inscribed in a circle of radius 2 with $\\angle ABC \\geq 90^\\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC$, $b=CA$, $c=AB$. Find all possible values of $x$."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-887816deb56d29a93a04905efdbd4e78eb1075579a2e757cbff2526f904d11f5 | Triangle $ABC$ is inscribed in a circle with $AB=20$ and $AC=12$. If $\angle{ACB}=\angle{ABC}+90^\circ$, determine the area of the circle. | [
"\\( 136 \\pi \\)"
] | [
"Triangle $ABC$ is inscribed in a circle with $AB=20$ and $AC=12$. If $\\angle{ACB}=\\angle{ABC}+90^\\circ$, determine the area of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-6be7322a8a0a4f9aaf1e0ed31ad7bc85497a90a3c46d4170acfd1a229f6393f7 | Triangle $ABC$ is inscribed in a circle with $AC=12$, $BC=16$, and $AB=7$. The angle bisector of $\angle ACB$ meets the circle at $E$ and $AB$ at $D$. If $CD=p$, find the length of $EB$ in terms of $p$. | [
"\\(\\frac{48}{p}\\)"
] | [
"Triangle $ABC$ is inscribed in a circle with $AC=12$, $BC=16$, and $AB=7$. The angle bisector of $\\angle ACB$ meets the circle at $E$ and $AB$ at $D$. If $CD=p$, find the length of $EB$ in terms of $p$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80425290d31f6d398343aa7a63522273d7bf2c6ec9a327ed2a5ae18340a08ee3 | Triangle $ABC$ is inscribed in a circle with center $O$. $H$ is the orthocenter, and $E$ is the foot of the perpendicular from $C$ to $AB$. Let $M$ be a point on $AC$ such that $ME \perp OE$. If $\angle BAC = 30^\circ$, find the measure of $\angle MHE$ in degrees. | [
"30"
] | [
"Triangle $ABC$ is inscribed in a circle with center $O$. $H$ is the orthocenter, and $E$ is the foot of the perpendicular from $C$ to $AB$. Let $M$ be a point on $AC$ such that $ME \\perp OE$. If $\\angle BAC = 30^\\circ$, find the measure of $\\angle MHE$ in degrees."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-69506f93915355145e3f1527fb5bea6bcfaa9d290c2121119ae48e6c6064e189 | Triangle $ABC$ is inscribed in a circle with center $O$. Let $q$ be the circle passing through $A$, $O$, and $B$. The lines $CA$ and $CB$ intersect $q$ at points $D$ and $E$ (different from $A$ and $B$). If the coordinates of $C$ are $(0,0)$, $O$ are $(1,0)$, and $A$ and $B$ are on the unit circle such that $A = (\cos ... | [
"2 - \\sqrt{3}"
] | [
"Triangle $ABC$ is inscribed in a circle with center $O$. Let $q$ be the circle passing through $A$, $O$, and $B$. The lines $CA$ and $CB$ intersect $q$ at points $D$ and $E$ (different from $A$ and $B$). If the coordinates of $C$ are $(0,0)$, $O$ are $(1,0)$, and $A$ and $B$ are on the unit circle such that $A = (... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-52cc608db2c8aa999bdd30a67d261e60958e0432a1fb7bd635c3206a441c4ee6 | Triangle $ABC$ is inscribed in a circle with radius $\sqrt{3}-1$. Given that $\angle{BAC}=60^\circ$ and the radius of the A-excircle is $1$, find the angles $\angle{ABC}$ and $\angle{ACB}$. | [
"\\(\\angle ABC = 30^\\circ\\), \\(\\angle ACB = 90^\\circ\\) or \\(\\angle ABC = 90^\\circ\\), \\(\\angle ACB = 30^\\circ\\)"
] | [
"Triangle $ABC$ is inscribed in a circle with radius $\\sqrt{3}-1$. Given that $\\angle{BAC}=60^\\circ$ and the radius of the A-excircle is $1$, find the angles $\\angle{ABC}$ and $\\angle{ACB}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 8
} | 9 | 0 | 0 |
omr-ws-b1af28de6d57077255d4e9577ed86b248092f65f50ffc508d22a366c5e82f803 | Triangle $ABC$ is inscribed in a circle, and $BP$ bisects angle $ABC$. Given $AB = 6$, $BC = 8$, and $AC = 7$, compute the length of $BP$. | [
"8"
] | [
"Triangle $ABC$ is inscribed in a circle, and $BP$ bisects angle $ABC$. Given $AB = 6$, $BC = 8$, and $AC = 7$, compute the length of $BP$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b112bf00db71b1a735bd089f8c5105c1a49bc19f41785c4104707d5e584ad21 | Triangle $ABC$ is inscribed in a circle. Through points $A$ and $B$ tangents to this circle are drawn, which intersect at point $P$. Points $X$ and $Y$ are the orthogonal projections of point $P$ onto lines $AC$ and $BC$. Let $M$ be the midpoint of $AB$, and let $CM$ intersect $XY$ at point $Z$. If $CM = 10$ and $AB = ... | [
"\\dfrac{250}{9}"
] | [
"Triangle $ABC$ is inscribed in a circle. Through points $A$ and $B$ tangents to this circle are drawn, which intersect at point $P$. Points $X$ and $Y$ are the orthogonal projections of point $P$ onto lines $AC$ and $BC$. Let $M$ be the midpoint of $AB$, and let $CM$ intersect $XY$ at point $Z$. If $CM = 10$ and $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-dc784ddc7438e78684cf3f1bfca823f3cdd72c931c34c9e12ec231ff1f90cc1a | Triangle $ABC$ is inscribed in a unit circle in the complex plane with circumcenter $O$ at $0$. The affixes of points $A$, $B$, and $C$ are $a^2$, $b^2$, and $c^2$ respectively. Let $D$ be the point where the $A$-excircle of $\triangle ABC$ touches side $BC$. Define points $E$ and $F$ similarly for sides $CA$ and $AB$.... | [
"\\( D = \\frac{b^2 + c^2 - a^2}{2} \\), \\( E = \\frac{c^2 + a^2 - b^2}{2} \\), \\( F = \\frac{a^2 + b^2 - c^2}{2} \\)"
] | [
"Triangle $ABC$ is inscribed in a unit circle in the complex plane with circumcenter $O$ at $0$. The affixes of points $A$, $B$, and $C$ are $a^2$, $b^2$, and $c^2$ respectively. Let $D$ be the point where the $A$-excircle of $\\triangle ABC$ touches side $BC$. Define points $E$ and $F$ similarly for sides $CA$ and... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93e03834938c5a77c8b0730bb9091a87bdad86b9322d7834547cbacf7d0d25c8 | Triangle $ABC$ is inscribed in circle $(O)$ with $BP$ and $CQ$ being diameters of $(O)$. $H$ is the orthocenter of triangle $ABC$. Let $M$ and $N$ be the points such that $CM$ is perpendicular to $CA$, $PM$ is perpendicular to $PH$, $BN$ is perpendicular to $BA$, and $QN$ is perpendicular to $QH$. If $QM$, $NP$, and $A... | [
"\\dfrac{4}{5}"
] | [
"Triangle $ABC$ is inscribed in circle $(O)$ with $BP$ and $CQ$ being diameters of $(O)$. $H$ is the orthocenter of triangle $ABC$. Let $M$ and $N$ be the points such that $CM$ is perpendicular to $CA$, $PM$ is perpendicular to $PH$, $BN$ is perpendicular to $BA$, and $QN$ is perpendicular to $QH$. If $QM$, $NP$, a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-5783ce51ed553e63329cb67616197d74d955cc41bc0403a52f722c44dfd2367a | Triangle $ABC$ is inscribed in circle $(O)$ with incircle $(I)$. The circle with diameter $IA$ intersects $(O)$ at $A_1$ (distinct from $A$). Points $B_1, C_1$ are defined similarly. Line $B_1C_1$ intersects $BC$ at $A_2$, and points $B_2, C_2$ are defined similarly. If $O$ is the orthocenter of triangle $A_2B_2C_2$, f... | [
"1"
] | [
"Triangle $ABC$ is inscribed in circle $(O)$ with incircle $(I)$. The circle with diameter $IA$ intersects $(O)$ at $A_1$ (distinct from $A$). Points $B_1, C_1$ are defined similarly. Line $B_1C_1$ intersects $BC$ at $A_2$, and points $B_2, C_2$ are defined similarly. If $O$ is the orthocenter of triangle $A_2B_2C_... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-4f79cb80d8f9d3f39f903e2e213bf229bdbf44fc128e8523915c746f29bb4fb1 | Triangle $ABC$ is inscribed in circle $(O)$, and $AD$ is the internal angle bisector of triangle $ABC$. Let $H$ and $K$ be the circumcenters of triangles $ABD$ and $ACD$, respectively. If the radius of circle $(O)$ is $r$ and $OH = OK = x$, find the value of $x^2$ in terms of $r$. | [
"\\dfrac{r^2}{2}"
] | [
"Triangle $ABC$ is inscribed in circle $(O)$, and $AD$ is the internal angle bisector of triangle $ABC$. Let $H$ and $K$ be the circumcenters of triangles $ABD$ and $ACD$, respectively. If the radius of circle $(O)$ is $r$ and $OH = OK = x$, find the value of $x^2$ in terms of $r$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5bcec09aee4ee86ed9aa2c2c93dc0d5e9a8243f88b56d7071d42e624c9444b14 | Triangle $ABC$ is inscribed in circle $(O)$. $A$ varies on $(O)$ such that $AB > BC$. $M$ is the midpoint of $AC$. The circle with diameter $BM$ intersects $(O)$ at $R$. $RM$ intersects $(O)$ at $Q$ and intersects $BC$ at $P$. The circle with diameter $BP$ intersects $AB$ and $BO$ at $K$ and $S$, respectively. Let $N$ ... | [
"\\dfrac{1}{2}"
] | [
"Triangle $ABC$ is inscribed in circle $(O)$. $A$ varies on $(O)$ such that $AB > BC$. $M$ is the midpoint of $AC$. The circle with diameter $BM$ intersects $(O)$ at $R$. $RM$ intersects $(O)$ at $Q$ and intersects $BC$ at $P$. The circle with diameter $BP$ intersects $AB$ and $BO$ at $K$ and $S$, respectively. Let... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ec6b0bd9b6340a1249d4a5c4f53d3433ef7843e8931a7984f49cbc76fd09e33a | Triangle $ABC$ is inscribed in circle $(O)$. $A$ varies on $(O)$ such that $AB > BC$. $M$ is the midpoint of $AC$. The circle with diameter $BM$ intersects $(O)$ at $R$. $RM$ intersects $(O)$ at $Q$ and intersects $BC$ at $P$. The circle with diameter $BP$ intersects $AB$ and $BO$ at $K$ and $S$, respectively. Let $T$ ... | [
"(0,1)"
] | [
"Triangle $ABC$ is inscribed in circle $(O)$. $A$ varies on $(O)$ such that $AB > BC$. $M$ is the midpoint of $AC$. The circle with diameter $BM$ intersects $(O)$ at $R$. $RM$ intersects $(O)$ at $Q$ and intersects $BC$ at $P$. The circle with diameter $BP$ intersects $AB$ and $BO$ at $K$ and $S$, respectively. Let... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d738ae5d48a9c1024b0b41902bd57f8957e1eadee1d2da2970df22b1bd9b7e55 | Triangle $ABC$ is inscribed in circle $(O)$. Point $P$ is on arc $BC$ (not containing $A$) such that $AP$ is the symmedian of triangle $ABC$. Points $E$ and $F$ are the reflections of $P$ with respect to $CA$ and $AB$, respectively. Point $K$ is the reflection of $A$ with respect to $EF$. Point $L$ is the projection of... | [
"10"
] | [
"Triangle $ABC$ is inscribed in circle $(O)$. Point $P$ is on arc $BC$ (not containing $A$) such that $AP$ is the symmedian of triangle $ABC$. Points $E$ and $F$ are the reflections of $P$ with respect to $CA$ and $AB$, respectively. Point $K$ is the reflection of $A$ with respect to $EF$. Point $L$ is the projecti... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c30dfa3e80d87b00263f0e5d780d6ea1d9538da4778ddc3210583ce138c5470 | Triangle $ABC$ is inscribed in circle $(O)$. The tangent lines to $(O)$ at $B$ and $C$ meet at $T$. Point $S$ lies on ray $CB$ such that $AS \perp AT$. Points $B_{1}$ and $C_{1}$ lie on ray $ST$ (with $T$ between $B_{1}$ and $C_{1}$, and $B_{1}$ between $S$ and $T$) such that $B_{1}T = BT = C_{1}T$. If the ratio of the... | [
"\\dfrac{1}{4}"
] | [
"Triangle $ABC$ is inscribed in circle $(O)$. The tangent lines to $(O)$ at $B$ and $C$ meet at $T$. Point $S$ lies on ray $CB$ such that $AS \\perp AT$. Points $B_{1}$ and $C_{1}$ lie on ray $ST$ (with $T$ between $B_{1}$ and $C_{1}$, and $B_{1}$ between $S$ and $T$) such that $B_{1}T = BT = C_{1}T$. If the ratio ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-b7e2317aa7d9ea727d03749f190608664a4af2d9c06982365779222ed345c2e5 | Triangle $ABC$ is inscribed in circle $C$. The bisectors of the angles $A, B,$ and $C$ meet the circle $C$ again at the points $A', B', C'$. Let $I$ be the incenter of $ABC$. Calculate the minimum value of the expression $\frac{IA'}{IA} + \frac{IB'}{IB} + \frac{IC'}{IC}$. | [
"3"
] | [
"Triangle $ABC$ is inscribed in circle $C$. The bisectors of the angles $A, B,$ and $C$ meet the circle $C$ again at the points $A', B', C'$. Let $I$ be the incenter of $ABC$. Calculate the minimum value of the expression $\\frac{IA'}{IA} + \\frac{IB'}{IB} + \\frac{IC'}{IC}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c4be699572168712817ed3487d43f5dc8a5237d38630b12e1cf2ef044dd2aa2d | Triangle $ABC$ is inscribed in circle $O$. Tangent $PD$ is drawn from $A$, with $D$ on ray $BC$ and $P$ on ray $DA$. Line $PU$ ($U \in BD$) intersects circle $O$ at $Q$ and $T$, and intersects $AB$ and $AC$ at $R$ and $S$ respectively. If $QR=ST$, and $PQ=10$, find the length of $UT$. | [
"10"
] | [
"Triangle $ABC$ is inscribed in circle $O$. Tangent $PD$ is drawn from $A$, with $D$ on ray $BC$ and $P$ on ray $DA$. Line $PU$ ($U \\in BD$) intersects circle $O$ at $Q$ and $T$, and intersects $AB$ and $AC$ at $R$ and $S$ respectively. If $QR=ST$, and $PQ=10$, find the length of $UT$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-ab846f476a9e73ce2de7521fa5dd6b0886d1968eb7921ec29161a7fd80487745 | Triangle $ABC$ is inscribed in circle $\Omega$ with center $O$. The incircle of $ABC$ is tangent to $BC$, $AC$, $AB$ at $D$, $E$, $F$ respectively, and its center is $I$. The reflection of the tangent line to $\Omega$ at $A$ with respect to $EF$ is denoted $\ell_A$. Similarly, define $\ell_B$ and $\ell_C$. Let $H$ be t... | [
"20"
] | [
"Triangle $ABC$ is inscribed in circle $\\Omega$ with center $O$. The incircle of $ABC$ is tangent to $BC$, $AC$, $AB$ at $D$, $E$, $F$ respectively, and its center is $I$. The reflection of the tangent line to $\\Omega$ at $A$ with respect to $EF$ is denoted $\\ell_A$. Similarly, define $\\ell_B$ and $\\ell_C$. Le... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-6d828dcd4b310c2f6b9df7a198c1b67543f23ea415fad4279f3a026cd56e43af | Triangle $ABC$ is inscribed in circle $\Omega$. The interior angle bisector of angle $A$ intersects side $BC$ and $\Omega$ at $D$ and $L$ (other than $A$), respectively. Let $M$ be the midpoint of side $BC$. The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respect... | [
"25"
] | [
"Triangle $ABC$ is inscribed in circle $\\Omega$. The interior angle bisector of angle $A$ intersects side $BC$ and $\\Omega$ at $D$ and $L$ (other than $A$), respectively. Let $M$ be the midpoint of side $BC$. The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-12e95292890e7b248e78ae47c4b19351a09157297363a353df3fc680cf595625 | Triangle $ABC$ is inscribed in circle $\odot(O)$, with $H$ as the orthocenter. Let $P$ be any point on $\odot(O)$, and $Q$ be the intersection of line $PH$ with $\odot(O)$. The lines through $P$ that are perpendicular to $PB$ and $PC$ intersect $AC$ and $AB$ at points $E$ and $F$, respectively. If the radius of $\odot(... | [
"20"
] | [
"Triangle $ABC$ is inscribed in circle $\\odot(O)$, with $H$ as the orthocenter. Let $P$ be any point on $\\odot(O)$, and $Q$ be the intersection of line $PH$ with $\\odot(O)$. The lines through $P$ that are perpendicular to $PB$ and $PC$ intersect $AC$ and $AB$ at points $E$ and $F$, respectively. If the radius of... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-abb7bb4f1c334833c46ce7085bb0f7a8e0180df69e33ed87c89376ae83ce6291 | Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The altitude from $A$ to $BC$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2=\frac{a}{b}$, ... | [
"919"
] | [
"Triangle $ABC$ is inscribed in circle $\\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The altitude from $A$ to $BC$ meets side $BC$ at $D$ and circle $\\omega$ at a second point $E$. Let $\\gamma$ be the circle with diameter $DE$. Circles $\\omega$ and $\\gamma$ meet at $E$ and a second point $F$. Then $AF^2=\\fr... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-d8b869c919522a3a0fe41580986e61e36428bbd7c3194f4b318e2d3caa4d14c7 | Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The altitude from $A$ to $BC$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2=\frac{a}{b}$, ... | [
"-654"
] | [
"Triangle $ABC$ is inscribed in circle $\\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The altitude from $A$ to $BC$ meets side $BC$ at $D$ and circle $\\omega$ at a second point $E$. Let $\\gamma$ be the circle with diameter $DE$. Circles $\\omega$ and $\\gamma$ meet at $E$ and a second point $F$. Then $AF^2=\\fr... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-11d963aaaa87339c0ed8e9d7c71f507e0a9ca6dd68b7f1fd5ae10712b4835664 | Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac{m}{n}$, wh... | [
"919"
] | [
"Triangle $ABC$ is inscribed in circle $\\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\\omega$ at a second point $E$. Let $\\gamma$ be the circle with diameter $DE$. Circles $\\omega$ and $\\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \\frac... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 12
} | 33 | 0 | 0 |
omr-ws-12d59c96366c63b5b67b2ce9de067c512fea000714b79764b56a22be1f722ea5 | Triangle $ABC$ is inscribed in circle $\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $\overline{DE}$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = ... | [
"919"
] | [
"Triangle $ABC$ is inscribed in circle $\\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\\overline{BC}$ at $D$ and circle $\\omega$ at a second point $E$. Let $\\gamma$ be the circle with diameter $\\overline{DE}$. Circles $\\omega$ and $\\gamma$ meet at $E$ and a second point $F$. T... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 5,
"tir": 11
} | 47 | 0 | 0 |
omr-ws-87c68730ef7508ac9c45baf4118a8cfcbff0a672fa532208ca8529652abd9cd9 | Triangle $ABC$ is inscribed in circle $\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$ and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$. Gi... | [
"1"
] | [
"Triangle $ABC$ is inscribed in circle $\\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$ and rays $LR$ and $LS$ intersect $\\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\\angle BDE$ meets line $ER$ at... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-33fe668b8d19736a0a1558214e7b19dc54387a34bca3f497769cc99dd5613363 | Triangle $ABC$ is inscribed in circle $\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$. G... | [
"2"
] | [
"Triangle $ABC$ is inscribed in circle $\\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\\angle BDE$ meets line $ER$ a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-3a92ad62737bb25d9722e447658e4ffa3485b5f9f4c666a6584fe6196a666f0d | Triangle $ABC$ is inscribed in circle $\omega$. A variable line $l$ parallel to $BC$ intersects $AB$ at $D$, $AC$ at $E$, and $\omega$ at $K$ and $L$ (with $D$ between $K$ and $E$). Circle $\gamma_1$ is tangent to segments $KD$, $BD$, and $\omega$, while circle $\gamma_2$ is tangent to segments $LE$, $CE$, and $\omega$... | [
"\\text{The A-symmedian of triangle } ABC"
] | [
"Triangle $ABC$ is inscribed in circle $\\omega$. A variable line $l$ parallel to $BC$ intersects $AB$ at $D$, $AC$ at $E$, and $\\omega$ at $K$ and $L$ (with $D$ between $K$ and $E$). Circle $\\gamma_1$ is tangent to segments $KD$, $BD$, and $\\omega$, while circle $\\gamma_2$ is tangent to segments $LE$, $CE$, an... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-9303bc2a61cce521688717abccac746d8fbeb8c4da73c121f8ffdf70b79ed60d | Triangle $ABC$ is inscribed in circle $\omega$. Circle $\gamma$ is tangent to $AB$ and $AC$ at points $P$ and $Q$ respectively, and is also tangent to circle $\omega$ at point $S$. Let the intersection of $AS$ and $PQ$ be $T$. If $\angle BTP = \alpha$ and $\angle CTQ = \beta$, find the value of $\alpha - \beta$. | [
"0"
] | [
"Triangle $ABC$ is inscribed in circle $\\omega$. Circle $\\gamma$ is tangent to $AB$ and $AC$ at points $P$ and $Q$ respectively, and is also tangent to circle $\\omega$ at point $S$. Let the intersection of $AS$ and $PQ$ be $T$. If $\\angle BTP = \\alpha$ and $\\angle CTQ = \\beta$, find the value of $\\alpha - \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-62df13df57083f7ea34ae5df3ccdd203bcd06e68893cd234649fbab8461c026e | Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies inside the triangle. Rays $AP$, $BP$, and $CP$ meet $\omega$ again at $A_1$, $B_1$, and $C_1$, respectively. Let $A_2$, $B_2$, and $C_2$ be the reflections of $A_1$, $B_1$, and $C_1$ across the midpoints of sides $BC$, $CA$, and $AB$, respectively. If the c... | [
"5"
] | [
"Triangle $ABC$ is inscribed in circle $\\omega$. Point $P$ lies inside the triangle. Rays $AP$, $BP$, and $CP$ meet $\\omega$ again at $A_1$, $B_1$, and $C_1$, respectively. Let $A_2$, $B_2$, and $C_2$ be the reflections of $A_1$, $B_1$, and $C_1$ across the midpoints of sides $BC$, $CA$, and $AB$, respectively. I... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-f17b3f593b131147085c1e005329b6a18b5254a0fb447989dbd1b89430ce1f90 | Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies inside triangle $ABC$. Lines $AP$, $BP$, and $CP$ intersect $\omega$ again at points $A_1$, $B_1$, and $C_1$ (other than $A$, $B$, $C$), respectively. The tangent lines to $\omega$ at $A_1$ and $B_1$ intersect at $C_2$. The tangent lines to $\omega$ at $B_1... | [
"20"
] | [
"Triangle $ABC$ is inscribed in circle $\\omega$. Point $P$ lies inside triangle $ABC$. Lines $AP$, $BP$, and $CP$ intersect $\\omega$ again at points $A_1$, $B_1$, and $C_1$ (other than $A$, $B$, $C$), respectively. The tangent lines to $\\omega$ at $A_1$ and $B_1$ intersect at $C_2$. The tangent lines to $\\omega... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-f654d9d5b19dd86f755fdd0858a8f4d214b9d05049bea3150afcea3875627d4e | Triangle $ABC$ is inscribed in circle $\omega$. Points $P$ and $Q$ are on side $\overline{AB}$ with $AP < AQ$. Rays $CP$ and $CQ$ meet $\omega$ again at $S$ and $T$ (other than $C$), respectively. Given $AP = 4$, $PQ = 3$, $QB = 6$, $BT = 5$, and $AS = 7$, find the length of $ST$ in the form $\frac{m}{n}$, where $m$ an... | [
"43"
] | [
"Triangle $ABC$ is inscribed in circle $\\omega$. Points $P$ and $Q$ are on side $\\overline{AB}$ with $AP < AQ$. Rays $CP$ and $CQ$ meet $\\omega$ again at $S$ and $T$ (other than $C$), respectively. Given $AP = 4$, $PQ = 3$, $QB = 6$, $BT = 5$, and $AS = 7$, find the length of $ST$ in the form $\\frac{m}{n}$, whe... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 5,
"tir": 8
} | 14 | 0 | 0 |
omr-ws-ed22bc451ba4031ab81060170d5ec444cee2b9bc74adf340c4bb4ddc1ebd107c | Triangle $ABC$ is inscribed in circle $\omega$. Points $P$ and $Q$ are on side $\overline{AB}$ with $AP<AQ$. Rays $CP$ and $CQ$ meet $\omega$ again at $S$ and $T$ (other than $C$ ), respectively. If $AP=4,PQ=3,QB=6,BT=5,$ and $AS=7$, then $ST=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find ... | [
"043"
] | [
"Triangle $ABC$ is inscribed in circle $\\omega$. Points $P$ and $Q$ are on side $\\overline{AB}$ with $AP<AQ$. Rays $CP$ and $CQ$ meet $\\omega$ again at $S$ and $T$ (other than $C$ ), respectively. If $AP=4,PQ=3,QB=6,BT=5,$ and $AS=7$, then $ST=\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive intege... | [
"MATH_training_set"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-2f7ae9021fd343cfaf530fb647bc4706ac6e77023512eefbccc4a8529f128445 | Triangle $ABC$ is inscribed in circle $\omega$. The tangents to $\omega$ from $B$ and $C$ meet at $D$, and segments $AD$ and $BC$ intersect at $E$. If $\angle BAC = 60^\circ$ and the area of $\triangle BDE$ is twice the area of $\triangle CDE$, what is $\frac{AB}{AC}$? | [
"\\(\\sqrt{2}\\)"
] | [
"Triangle $ABC$ is inscribed in circle $\\omega$. The tangents to $\\omega$ from $B$ and $C$ meet at $D$, and segments $AD$ and $BC$ intersect at $E$. If $\\angle BAC = 60^\\circ$ and the area of $\\triangle BDE$ is twice the area of $\\triangle CDE$, what is $\\frac{AB}{AC}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 13
} | 24 | 0 | 0 |
omr-ws-91d8a3d68b4abd812e9b2437dde4e421b35c4f4a615d0cf8e868e50abcaa55d4 | Triangle $ABC$ is inscribed in triangle $ADE$ with right angles at $B$ and $D$. Given that $CE = 1$, $CB = 3$, and $AB = 4$, if the ratio $\frac{BD}{AD}$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $(10m + n)$. | [
"29"
] | [
"Triangle $ABC$ is inscribed in triangle $ADE$ with right angles at $B$ and $D$. Given that $CE = 1$, $CB = 3$, and $AB = 4$, if the ratio $\\frac{BD}{AD}$ can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $(10m + n)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-0e20af1eb11e0d1f57287a2c1a6f314f47d787b01b3278faac35323f4f407f66 | Triangle $ABC$ is inscribed with a circle centered at point $I$, which touches sides $AB$ and $AC$ at points $M$ and $N$ respectively. The point $Q$ is chosen on the smaller arc $MN$ of the inscribed circle. Point $F$ on segment $AB$ is such that $FN$ is perpendicular to $IQ$. The line passing through the point $F$ and... | [
"no_valid_answer_found"
] | [
"Triangle $ABC$ is inscribed with a circle centered at point $I$, which touches sides $AB$ and $AC$ at points $M$ and $N$ respectively. The point $Q$ is chosen on the smaller arc $MN$ of the inscribed circle. Point $F$ on segment $AB$ is such that $FN$ is perpendicular to $IQ$. The line passing through the point $F... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-862c7e17f8ee923a30021020d888dc4ae501fea0c7934e5b6946b848ffcd487d | Triangle $ABC$ is isosceles with $AB = BC = 1$ and $\angle ABC = 120^\circ$. A circle is tangent to the line $AB$ at $A$ and to the line $BC$ at $C$. What is the radius of the circle? | [
"\\(\\sqrt{3}\\)"
] | [
"Triangle $ABC$ is isosceles with $AB = BC = 1$ and $\\angle ABC = 120^\\circ$. A circle is tangent to the line $AB$ at $A$ and to the line $BC$ at $C$. What is the radius of the circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 26
} | 74 | 0 | 0 |
omr-ws-17bda34f486557b400e3b56712332349820b957638fc24f478e376f6717d14e8 | Triangle $ABC$ is isosceles with $AB=AC$. Point $D$ lies on $AB$ such that the inradius of $\triangle ADC$ and the inradius of $\triangle BDC$ both equal $\frac{3-\sqrt{3}}{2}$. The inradius of $\triangle ABC$ equals $1$. What is the length of $BD$? | [
"\\(\\sqrt{3}\\)"
] | [
"Triangle $ABC$ is isosceles with $AB=AC$. Point $D$ lies on $AB$ such that the inradius of $\\triangle ADC$ and the inradius of $\\triangle BDC$ both equal $\\frac{3-\\sqrt{3}}{2}$. The inradius of $\\triangle ABC$ equals $1$. What is the length of $BD$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-02ebb6848a59c2454f4a93bdbd89be57d0bfd1cd60ef1f7704f5dbcbca4341e7 | Triangle $ABC$ is isosceles with $AC = BC$ and $\angle ACB = 106^\circ.$ Point $M$ is in the interior of the triangle so that $\angle MAC = 7^\circ$ and $\angle MCA = 23^\circ.$ Find the number of degrees in $\angle CMB.$ [asy] pointpen = black; pathpen = black+linewidth(0.7); size(220); /* We will WLOG AB = 2 to draw ... | [
"083"
] | [
"Triangle $ABC$ is isosceles with $AC = BC$ and $\\angle ACB = 106^\\circ.$ Point $M$ is in the interior of the triangle so that $\\angle MAC = 7^\\circ$ and $\\angle MCA = 23^\\circ.$ Find the number of degrees in $\\angle CMB.$ \n[asy] pointpen = black; pathpen = black+linewidth(0.7); size(220); /* We will W... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-b8469ffdb0c6fd033756fde041dcbc5924943ab81b5071bd32dd6ddb1316d81a | Triangle $ABC$ is isosceles with $AC = BC$. The angle bisector at $A$ intersects side $BC$ at point $D$ and the angle bisector at $C$ intersects side $AB$ at point $E$. If $AD = 2CE$, find the measure of the angles of triangle $ABC$. | [
"\\(\\angle ABC = 36^\\circ\\), \\(\\angle BAC = 36^\\circ\\), \\(\\angle ACB = 108^\\circ\\)"
] | [
"Triangle $ABC$ is isosceles with $AC = BC$. The angle bisector at $A$ intersects side $BC$ at point $D$ and the angle bisector at $C$ intersects side $AB$ at point $E$. If $AD = 2CE$, find the measure of the angles of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-d478257112bf63803abd7f1ecda1e40f7641fc7861531550b60688ca47a66690 | Triangle $ABC$ is isosceles with $AC=BC$ and $\angle C=120^\circ$. Points $D$ and $E$ are chosen on segment $AB$ so that $AD=DE=EB$. Find $\angle CDE$ (in degrees). | [
"60"
] | [
"Triangle $ABC$ is isosceles with $AC=BC$ and $\\angle C=120^\\circ$. Points $D$ and $E$ are chosen on segment $AB$ so that $AD=DE=EB$. Find $\\angle CDE$ (in degrees)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-21f3dce1c958c3e2724e85bea8a1acbac3f2be3267bd4d57599d737720e83514 | Triangle $ABC$ is isosceles with $CA = CB$. Point $D$ lies on $AB$ such that $AD = AC$ and $DB = DC$. What is the size in degrees of angle $BCA$? | [
"108^\\circ"
] | [
"Triangle $ABC$ is isosceles with $CA = CB$. Point $D$ lies on $AB$ such that $AD = AC$ and $DB = DC$. What is the size in degrees of angle $BCA$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-917dbd157ea0a27be2769015445dd952ddeb2aa476a10d0f7911f74a796bc4f9 | Triangle $ABC$ is isosceles with $|AB| = |AC|$. Points $D, E, F$ are drawn such that $|AE| = |ED| = |DF| = |FC| = |BC|$. What is the measure of angle $BAC$? | [
"100^\\circ"
] | [
"Triangle $ABC$ is isosceles with $|AB| = |AC|$. Points $D, E, F$ are drawn such that $|AE| = |ED| = |DF| = |FC| = |BC|$. What is the measure of angle $BAC$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-ea6066388949324cb5c97811c3e2024afee7a165540b2137cba202b752ac9428 | Triangle $ABC$ is isosceles with angle $A$ congruent to angle $B$. The measure of angle $C$ is 30 degrees more than the measure of angle $A$. What is the number of degrees in the measure of angle $C$? | [
"80"
] | [
"Triangle $ABC$ is isosceles with angle $A$ congruent to angle $B$. The measure of angle $C$ is 30 degrees more than the measure of angle $A$. What is the number of degrees in the measure of angle $C$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fa5021a5ac570a70321b211e0099e69bbe4e8288000fb2101ebf004cdf86762d | Triangle $ABC$ is isosceles with angle $B$ congruent to angle $C$. The measure of angle $C$ is four times the measure of angle $A$. What is the number of degrees in the measure of angle $B$? | [
"80"
] | [
"Triangle $ABC$ is isosceles with angle $B$ congruent to angle $C$. The measure of angle $C$ is four times the measure of angle $A$. What is the number of degrees in the measure of angle $B$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f838691b0fa06dd125293aa5851913bcf79c9f00124da23e647233a7baf4612d | Triangle $ABC$ is isosceles with base $AC$. Points $P$ and $Q$ are respectively on $CB$ and $AB$ such that $AC=AP=PQ=QB$. Determine the measure of angle $B$ in degrees. | [
"\\(25 \\frac{5}{7}\\)",
"\\(25\\frac{5}{7}\\)"
] | [
"Triangle $ABC$ is isosceles with base $AC$. Points $P$ and $Q$ are respectively on $CB$ and $AB$ such that $AC=AP=PQ=QB$. Determine the measure of angle $B$ in degrees."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 13
} | 14 | 0 | 0 |
omr-ws-37998557396000e63a42f035ea489d98793bb9f03c8085929c94c41a80283a45 | Triangle $ABC$ is isosceles with base $AC$. Points $P$ and $Q$ are respectively on $CB$ and $AB$ such that $AC=AP=PQ=QB$. Find $\angle B$. | [
"\\(\\frac{180}{7}\\)"
] | [
"Triangle $ABC$ is isosceles with base $AC$. Points $P$ and $Q$ are respectively on $CB$ and $AB$ such that $AC=AP=PQ=QB$. Find $\\angle B$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-6e63f96a1c6da408a6e3d36aa6d2b72f14bd78fd1abd405e9575fa9fec37aba7 | Triangle $ABC$ is isosceles with vertices $A(2,7)$, $B(-2,0)$, and $C(3,-1)$. Triangle $ABC$ is reflected over line $BC$ to form triangle $A'BC$, which is then reflected over line $BC$ again to form triangle $A'B'C'$. Triangle $A'B'C'$ is then reflected over line $B'C'$ to form triangle $A''B'C'$. What is the slope of ... | [
"5"
] | [
"Triangle $ABC$ is isosceles with vertices $A(2,7)$, $B(-2,0)$, and $C(3,-1)$. Triangle $ABC$ is reflected over line $BC$ to form triangle $A'BC$, which is then reflected over line $BC$ again to form triangle $A'B'C'$. Triangle $A'B'C'$ is then reflected over line $B'C'$ to form triangle $A''B'C'$. What is the slop... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6c01c26606211cc3faa2acde5413cdfaa8d11a247a4ed76fd3e5d90e7dfab7c | Triangle $ABC$ is isosceles, with $AB = BC = 18$ and has circumcircle $\omega$. Tangents to $\omega$ at $A$ and $B$ intersect at point $D$. If $AD = 27$, what is the length of $AC$? | [
"12"
] | [
"Triangle $ABC$ is isosceles, with $AB = BC = 18$ and has circumcircle $\\omega$. Tangents to $\\omega$ at $A$ and $B$ intersect at point $D$. If $AD = 27$, what is the length of $AC$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-15bc97e065edb881d529f4158de12edad5a03bbcb4cda0119b9061012677a0e2 | Triangle $ABC$ is regular. The point $M$ lies inside $\angle ABC$, and $\angle AMB=30^\circ$. Let lines $AC$ and $BM$ intersect at a point $K$. Find the angles $\angle MAB$ and $\angle MCB$ if $\triangle MKC$ is known to be similar to $\triangle MCB$. | [
"15^\\circ"
] | [
"Triangle $ABC$ is regular. The point $M$ lies inside $\\angle ABC$, and $\\angle AMB=30^\\circ$. Let lines $AC$ and $BM$ intersect at a point $K$. Find the angles $\\angle MAB$ and $\\angle MCB$ if $\\triangle MKC$ is known to be similar to $\\triangle MCB$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-07430cf44afc9d6a8449d77e73389e590ffa9595bed744c4561a6b78d145ab83 | Triangle $ABC$ is right at $C$. Let $E_1$, $E_2$, and $E_3$ be the ellipses with foci at $(A, B)$, $(A, C)$, and $(B, C)$ respectively, and each passing through the third vertex of the triangle. If these ellipses intersect at a common point $P$, find the coordinates of $P$ in terms of the coordinates of $A$, $B$, and $... | [
"(A_x, B_y)"
] | [
"Triangle $ABC$ is right at $C$. Let $E_1$, $E_2$, and $E_3$ be the ellipses with foci at $(A, B)$, $(A, C)$, and $(B, C)$ respectively, and each passing through the third vertex of the triangle. If these ellipses intersect at a common point $P$, find the coordinates of $P$ in terms of the coordinates of $A$, $B$, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-c5f0ec3f84ba2fcd02541463023207de96f65cc8ad2c29d1c7bafba99f1e6827 | Triangle $ABC$ is right isosceles. The square $ADEF$ has an area of $2250$. Determine the area of the square $GHIJ$ inscribed in the triangle in the other basic way. | [
"2000"
] | [
"Triangle $ABC$ is right isosceles. The square $ADEF$ has an area of $2250$. Determine the area of the square $GHIJ$ inscribed in the triangle in the other basic way."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 4,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-0b3194fcd4ab74d76c36bb70e44dde9d56c7819dff97aea8b71976c8d9b190b5 | Triangle $ABC$ is right-angled and isosceles with a right angle at vertex $C$. Points $F$ and $G$ are selected on rays $CB$ and $BA$ respectively such that $AG = BF$. The ray $GD$ is drawn so that it intersects ray $AC$ at point $D$ with $\angle FGD = 45^\circ$. Find $\angle FDG$. | [
"\\(45^\\circ\\)"
] | [
"Triangle $ABC$ is right-angled and isosceles with a right angle at vertex $C$. Points $F$ and $G$ are selected on rays $CB$ and $BA$ respectively such that $AG = BF$. The ray $GD$ is drawn so that it intersects ray $AC$ at point $D$ with $\\angle FGD = 45^\\circ$. Find $\\angle FDG$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-dca2222ab101c01f97ea74f9da71f223575c4ed274ddafeaee9b49e908a0f4b9 | Triangle $ABC$ is right-angled at $A$. Let $D$ be a point on $BC$. $E$ and $F$ are the reflections of $D$ on $AC$ and $AB$, respectively. If the area of triangle $ABC$ is equal to the area of triangle $DEF$, calculate the ratio $\frac{BD}{DC}$. | [
"1"
] | [
"Triangle $ABC$ is right-angled at $A$. Let $D$ be a point on $BC$. $E$ and $F$ are the reflections of $D$ on $AC$ and $AB$, respectively. If the area of triangle $ABC$ is equal to the area of triangle $DEF$, calculate the ratio $\\frac{BD}{DC}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a3955846a71703948eb8c13a360bd6ff4693a6ae4c3d5d10b60631d85ec97562 | Triangle $ABC$ is right-angled at $A$. The circle with center $A$ and radius $AB$ intersects $BC$ and $AC$ internally at points $D$ and $E$ respectively. Given that $BD = 20$ and $DC = 16$, determine $AC^2$. | [
"936"
] | [
"Triangle $ABC$ is right-angled at $A$. The circle with center $A$ and radius $AB$ intersects $BC$ and $AC$ internally at points $D$ and $E$ respectively. Given that $BD = 20$ and $DC = 16$, determine $AC^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-51cce6025c0e965a4f32a23b05d8b8e7426ef587a8cdbf36896d35fa0cbca631 | Triangle $ABC$ is right-angled at $B$ and has incentre $I$. Points $D$, $E$, and $F$ are the points where the incircle of the triangle touches the sides $BC$, $AC$, and $AB$ respectively. Lines $CI$ and $EF$ intersect at point $P$. Lines $DP$ and $AB$ intersect at point $Q$. If $AB = 12$ and $BC = 16$, find the length ... | [
"4"
] | [
"Triangle $ABC$ is right-angled at $B$ and has incentre $I$. Points $D$, $E$, and $F$ are the points where the incircle of the triangle touches the sides $BC$, $AC$, and $AB$ respectively. Lines $CI$ and $EF$ intersect at point $P$. Lines $DP$ and $AB$ intersect at point $Q$. If $AB = 12$ and $BC = 16$, find the le... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3ad6c46723287f2353fae7026c319870636d7f09e5b3fe1ffdec4f634a80ae81 | Triangle $ABC$ is right-angled at $B$ and has integer side lengths. A second triangle, $PQR$, is located inside $\bigtriangleup ABC$ such that its sides are parallel to the sides of $\bigtriangleup ABC$ and the distance between parallel lines is 2. Determine the side lengths of all possible triangles $ABC$ such that th... | [
"$(9, 15, 12)$, $(12, 15, 9)$, $(7, 25, 24)$, $(24, 25, 7)$, $(8, 17, 15)$, $(15, 17, 8)$"
] | [
"Triangle $ABC$ is right-angled at $B$ and has integer side lengths. A second triangle, $PQR$, is located inside $\\bigtriangleup ABC$ such that its sides are parallel to the sides of $\\bigtriangleup ABC$ and the distance between parallel lines is 2. Determine the side lengths of all possible triangles $ABC$ such ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4c31b16ac23e9228163ee3891cc8f0fe63e8e0ecdc4d912f8c154e8f7c9f15e9 | Triangle $ABC$ is right-angled at $B$. Let $I$ be the center of the inscribed circle in $ABC$ and $O$ be the midpoint of the side $AC$. If $\angle AOI = 45^\circ$, what is the angle $\angle ACB$? | [
"30^\\circ"
] | [
"Triangle $ABC$ is right-angled at $B$. Let $I$ be the center of the inscribed circle in $ABC$ and $O$ be the midpoint of the side $AC$. If $\\angle AOI = 45^\\circ$, what is the angle $\\angle ACB$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-c2168a4ea2c7650e4437b37da8c9e0ad874ec78eb4b23eb75f7bd76d3cb871d6 | Triangle $ABC$ is right-angled at $B$. The bisector of $\angle BAC$ meets $BC$ at $D$. Let $G$ denote the centroid of $\triangle ABC$. Suppose that $GD$ is parallel to $AB$. Find $\angle C$. | [
"30"
] | [
"Triangle $ABC$ is right-angled at $B$. The bisector of $\\angle BAC$ meets $BC$ at $D$. Let $G$ denote the centroid of $\\triangle ABC$. Suppose that $GD$ is parallel to $AB$. Find $\\angle C$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-9daaf647bbffbecea614997d22ba17d21cf475d928f818517f0ce0d13d7970b6 | Triangle $ABC$ is right-angled at $B$. The internal and external bisectors of angle $\angle BAC$ intersect the line $BC$ at $D$ and $E$, respectively. Given that $AD = 360$ and $AE = 480$, calculate $AB$. | [
"288"
] | [
"Triangle $ABC$ is right-angled at $B$. The internal and external bisectors of angle $\\angle BAC$ intersect the line $BC$ at $D$ and $E$, respectively. Given that $AD = 360$ and $AE = 480$, calculate $AB$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-6bbbfe871fe404d43e46f0043ad2462121f9e782501681c9d3afeef7c26967ab | Triangle $ABC$ is right-angled at $C$ with $\angle ACB = 90^\circ$ and $\frac{AC}{BC} = 2$. A line parallel to $AC$ intersects $AB$ and $BC$ at points $M$ and $N$ respectively, such that $\frac{CN}{BN} = 2$. Let $O$ be the intersection point of lines $CM$ and $AN$. On segment $ON$, point $K$ is such that $OM + OK = KN$... | [
"$90^\\circ$"
] | [
"Triangle $ABC$ is right-angled at $C$ with $\\angle ACB = 90^\\circ$ and $\\frac{AC}{BC} = 2$. A line parallel to $AC$ intersects $AB$ and $BC$ at points $M$ and $N$ respectively, such that $\\frac{CN}{BN} = 2$. Let $O$ be the intersection point of lines $CM$ and $AN$. On segment $ON$, point $K$ is such that $OM +... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-2f90122247c7724bcba148e819351a68a1567cf8e8d616313057e4f23013f1e0 | Triangle $ABC$ is right-angled at point $B$ and has incentre $I$. Points $D$, $E$, and $F$ are the points where the incircle of the triangle touches the sides $BC$, $CA$, and $AB$ respectively. Lines $CI$ and $EF$ intersect at point $M$. Lines $DM$ and $AB$ intersect at point $N$. If $AB = 12$ and $BC = 16$, find the l... | [
"4"
] | [
"Triangle $ABC$ is right-angled at point $B$ and has incentre $I$. Points $D$, $E$, and $F$ are the points where the incircle of the triangle touches the sides $BC$, $CA$, and $AB$ respectively. Lines $CI$ and $EF$ intersect at point $M$. Lines $DM$ and $AB$ intersect at point $N$. If $AB = 12$ and $BC = 16$, find ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93dc2fab373af0f74207b00f5861a37eb23926ca2ddd3610448bcec51ff3fc92 | Triangle $ABC$ is rotated around point $A$ by $30$ degrees, resulting in points $A'$, $B'$, and $C'$. Given that point $C$ lies on the side $B'C'$ and $AB = 8$, find the area of triangle $ABC$. | [
"16"
] | [
"Triangle $ABC$ is rotated around point $A$ by $30$ degrees, resulting in points $A'$, $B'$, and $C'$. Given that point $C$ lies on the side $B'C'$ and $AB = 8$, find the area of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-4f62c7c7ebd202ab1a9c9f5093e8d2c64960e06e4b90fa4c551296d66029c18e | Triangle $ABC$ is situated within an ellipse whose major and minor axes have lengths $10$ and $8$. Point $A$ is located at one focus, point $B$ is located at the endpoint of the minor axis, and point $C$ is located on the ellipse so that the other focus lies on $BC$. Determine the inradius of triangle $ABC$. | [
"\\dfrac{30}{17}"
] | [
"Triangle $ABC$ is situated within an ellipse whose major and minor axes have lengths $10$ and $8$. Point $A$ is located at one focus, point $B$ is located at the endpoint of the minor axis, and point $C$ is located on the ellipse so that the other focus lies on $BC$. Determine the inradius of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d3cd3957b1c3728cee7128c5b3f209dec8a5cf44771b6c03c7bb95804e7c7e71 | Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. If $X$ is the intersection of lines $HM$ and $AN$... | [
"0"
] | [
"Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. If $X$ is the intersection of lines $HM$ and... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-678065a4055dda603550d01ba485d2b782ecc4f9cdd229c88c574070febe7cfa | Triangle $ABC$ lies in circle $O$. Tangents from $A$ and $C$ intersect at $P$. Line $BP$ intersects $O$ at $K$. If $AK = 12$, $BC = 15$, and $CK = 9$, find the length of $AB$. | [
"20"
] | [
"Triangle $ABC$ lies in circle $O$. Tangents from $A$ and $C$ intersect at $P$. Line $BP$ intersects $O$ at $K$. If $AK = 12$, $BC = 15$, and $CK = 9$, find the length of $AB$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-2ba602b3ecee03607aa8b5023c03348a7cbf680c65ce0232061b84ead18fc561 | Triangle $ABC$ lies in the Cartesian Plane with vertices $B(12,19)$ and $C(23,20)$. The coordinates of $A$ are $(p,q)$, and the area of the triangle is 70. The line containing the median to side $BC$ has a slope of $-5$. Find the largest possible value of $p+q$. | [
"47"
] | [
"Triangle $ABC$ lies in the Cartesian Plane with vertices $B(12,19)$ and $C(23,20)$. The coordinates of $A$ are $(p,q)$, and the area of the triangle is 70. The line containing the median to side $BC$ has a slope of $-5$. Find the largest possible value of $p+q$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
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