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-7427db7248fe16b10ff0050775ed24450954fcb11673e8c4134b59136e59c60c | Triangle $ABC$ has an area of $92$. $\overline {AD}$ and $\overline {BE}$ intersect at $P$ such that $AP/PD=3$ and $BP/PE=2$, where $D$ is on $BC$ and $E$ is on $AC$. Find the area of triangle $APB$. | [
"$\\frac{115}{3}$"
] | [
"Triangle $ABC$ has an area of $92$. $\\overline {AD}$ and $\\overline {BE}$ intersect at $P$ such that $AP/PD=3$ and $BP/PE=2$, where $D$ is on $BC$ and $E$ is on $AC$. Find the area of triangle $APB$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 18
} | 37 | 0 | 0 |
omr-ws-654c76e37a5cb593cc9fac65a4303ac86a1573715c42146f0177d4dc2050219b | Triangle $ABC$ has an incenter $I$, and its incircle touches the side $BC$ at $T$. The line through $T$ parallel to $IA$ meets the incircle again at $S$. The tangent to the incircle at $S$ meets $AB$ and $AC$ at points $C'$ and $B'$, respectively. If the ratio of the area of triangle $AB'C'$ to the area of triangle $AB... | [
"10"
] | [
"Triangle $ABC$ has an incenter $I$, and its incircle touches the side $BC$ at $T$. The line through $T$ parallel to $IA$ meets the incircle again at $S$. The tangent to the incircle at $S$ meets $AB$ and $AC$ at points $C'$ and $B'$, respectively. If the ratio of the area of triangle $AB'C'$ to the area of triangl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-98ded06376043f6e18860b7e76125b76d2845b3c519c3bf2eefc1f5a97728096 | Triangle $ABC$ has an incenter $I$. Points $X$ and $Y$ are located on line segments $AB$ and $AC$ respectively, such that $BX \cdot AB = IB^2$ and $CY \cdot AC = IC^2$. Given that points $X$, $I$, and $Y$ lie on a straight line, find the possible values of the measure of angle $A$. | [
"\\(60^\\circ\\)"
] | [
"Triangle $ABC$ has an incenter $I$. Points $X$ and $Y$ are located on line segments $AB$ and $AC$ respectively, such that $BX \\cdot AB = IB^2$ and $CY \\cdot AC = IC^2$. Given that points $X$, $I$, and $Y$ lie on a straight line, find the possible values of the measure of angle $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-5ff7e185b2b71868e4f31386a04db0cb6f35caa053582ae5d330595d79afe1ac | Triangle $ABC$ has an incircle centered at $I$. Define $M$ and $N$ as the midpoints of $BC$ and $CA$, respectively. Extend $BI$ and $MN$ to meet at a point $K$. The circumcircle of $\triangle BKC$ intersects the incircle at two points $D$ and $G$, where $D$ is closer to $AB$ than $G$. Line $BK$ intersects the incircle ... | [
"12"
] | [
"Triangle $ABC$ has an incircle centered at $I$. Define $M$ and $N$ as the midpoints of $BC$ and $CA$, respectively. Extend $BI$ and $MN$ to meet at a point $K$. The circumcircle of $\\triangle BKC$ intersects the incircle at two points $D$ and $G$, where $D$ is closer to $AB$ than $G$. Line $BK$ intersects the inc... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-d4a7fa9a05d504093bc454d8426d9e11d69500fc730fa2200dbf3c67cce55f4e | Triangle $ABC$ has an inradius of $5$ and $AB = 12$. Find the minimum possible perimeter of $\triangle ABC$. Express your answer as a mixed number. | [
"78 \\frac{6}{11}"
] | [
"Triangle $ABC$ has an inradius of $5$ and $AB = 12$. Find the minimum possible perimeter of $\\triangle ABC$. Express your answer as a mixed number."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-8428658a05fb48ce0389a49bc4ffd4090f693faffde69e5364c793b938133993 | Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2 \cos B = \cos A + \cos C$, then the area of triangle $ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Comp... | [
"141"
] | [
"Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2 \\cos B = \\cos A + \\cos C$, then the area of triangle $ABC$ can be expressed as $\\frac{a\\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any p... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-ca8dd3c0b8f017d5101bc2864c7abd4fa29ea053176aa28f2a430acc730ac229 | Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\cos{B} = \cos{A} + \cos{C}$, then the area of triangle $ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Comp... | [
"141"
] | [
"Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\\cos{B} = \\cos{A} + \\cos{C}$, then the area of triangle $ABC$ can be expressed as $\\frac{a\\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any p... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6dd194a38d9d2b04749ebabc0e31355f9b6425c490410eb466a7a1a042d50bfc | Triangle $ABC$ has area $1$. Points $E$, $F$, and $G$ lie, respectively, on sides $BC$, $CA$, and $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of triangle $RST$. | [
"$\\frac{7 - 3\\sqrt{5}}{4}$"
] | [
"Triangle $ABC$ has area $1$. Points $E$, $F$, and $G$ lie, respectively, on sides $BC$, $CA$, and $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of triangle $RST$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-28df88d9f93137fe9844e8c1f4140f96a2aa120c9d87d31d5f7187a17db69038 | Triangle $ABC$ has area $5$ and $AB^2 + AC^2 = 17 + BC^2$. If the largest possible value of $\tan A$ can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $a + b$. | [
"37"
] | [
"Triangle $ABC$ has area $5$ and $AB^2 + AC^2 = 17 + BC^2$. If the largest possible value of $\\tan A$ can be expressed in the form $\\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $a + b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-517b1e233e78d5b7e8c605fe230232f0a8dcf37473cbf67944fbb8ae6848e327 | Triangle $ABC$ has base $AC$ equal to 218 and altitude 100. Squares $s_1, s_2, s_3, \ldots$ are drawn such that $s_1$ has a side on $AC$ and touches $AB$ and $BC$ at one point each, and square $s_k$ has a side on square $s_{k-1}$ and also touches $AB$ and $BC$ exactly once each. What is the sum of the areas of these sq... | [
"\\dfrac{1090000}{209}"
] | [
"Triangle $ABC$ has base $AC$ equal to 218 and altitude 100. Squares $s_1, s_2, s_3, \\ldots$ are drawn such that $s_1$ has a side on $AC$ and touches $AB$ and $BC$ at one point each, and square $s_k$ has a side on square $s_{k-1}$ and also touches $AB$ and $BC$ exactly once each. What is the sum of the areas of th... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-f8f59c6e09e0047e561e6747934f3ab77dc949544d5bdbb7519f656e32f74e1c | Triangle $ABC$ has centroid $Z(0,0)$ and $AB=AC$. Point $B$ lies on the line $y=-2$ and point $C$ lies on the line $2x+y+2=0$. The distance of point $A$ to the line $x+y-6=0$ equals $\sqrt{2}$. Determine all possible triangles. | [
"The possible triangles are given by the coordinates:\n1. \\(A(8+4\\sqrt{2}, -4\\sqrt{2})\\), \\(B(-6+2\\sqrt{2}, -2)\\), \\(C(-2+2\\sqrt{2}, -2-4\\sqrt{2})\\)\n2. \\(A(8-4\\sqrt{2}, 4\\sqrt{2})\\), \\(B(-6-2\\sqrt{2}, -2)\\), \\(C(-2-2\\sqrt{2}, -2+4\\sqrt{2})\\)"
] | [
"Triangle $ABC$ has centroid $Z(0,0)$ and $AB=AC$. Point $B$ lies on the line $y=-2$ and point $C$ lies on the line $2x+y+2=0$. The distance of point $A$ to the line $x+y-6=0$ equals $\\sqrt{2}$. Determine all possible triangles."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6593cbc746c96b4bcf5b751bf101d704a41b8e1d122e773bfdb6cf675745aae7 | Triangle $ABC$ has circumcenter $O$ and orthocenter $H$. Let $M$ and $N$ be the midpoints of $AH$ and $BH$, respectively. If $M$, $N$, $O$, and $H$ all lie on a circle $\omega$, find the ratio of the radius of $\omega$ to the radius of the circumcircle of $ABC$. | [
"\\dfrac{1}{2}"
] | [
"Triangle $ABC$ has circumcenter $O$ and orthocenter $H$. Let $M$ and $N$ be the midpoints of $AH$ and $BH$, respectively. If $M$, $N$, $O$, and $H$ all lie on a circle $\\omega$, find the ratio of the radius of $\\omega$ to the radius of the circumcircle of $ABC$."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-10b82b9e4f8fb10fbc87cc32c5aee28d95bac91df4460e3a7b76338d36994bd6 | Triangle $ABC$ has circumcenter $O$. $M$ is the midpoint of arc $BC$ not containing $A$. $S$ is a point on the circumcircle of $\triangle ABC$ such that $AS$ and $BC$ intersect on the line passing through $O$ and perpendicular to $AM$. $D$ is a point such that $ABDC$ is a parallelogram. If $SM$ intersects $AD$ at point... | [
"1"
] | [
"Triangle $ABC$ has circumcenter $O$. $M$ is the midpoint of arc $BC$ not containing $A$. $S$ is a point on the circumcircle of $\\triangle ABC$ such that $AS$ and $BC$ intersect on the line passing through $O$ and perpendicular to $AM$. $D$ is a point such that $ABDC$ is a parallelogram. If $SM$ intersects $AD$ at... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-6ecee07fce0823f793b62c4f0a95a87986e90c6fcda08509f2a924576925b0f3 | Triangle $ABC$ has circumcenter $O$. Point $D$ is on $BC$, and $AD$ intersects the circumcircle of $\triangle ABC$ at $E$. Point $S$ is on the circumcircle of $\triangle ABC$ such that $D, O, E, S$ are collinear. $AS$ intersects $BC$ at $P$. Point $Q$ is on $BC$ such that $D, O, A, Q$ are concyclic. If the radius of th... | [
"2"
] | [
"Triangle $ABC$ has circumcenter $O$. Point $D$ is on $BC$, and $AD$ intersects the circumcircle of $\\triangle ABC$ at $E$. Point $S$ is on the circumcircle of $\\triangle ABC$ such that $D, O, E, S$ are collinear. $AS$ intersects $BC$ at $P$. Point $Q$ is on $BC$ such that $D, O, A, Q$ are concyclic. If the radiu... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-aa6a7cacac8b75299bb6c5f72c02989e47c38f97be4b30c13c9a70a7b410925b | Triangle $ABC$ has circumcentre $O$ and centroid $M$. The lines $OM$ and $AM$ are perpendicular. Let $AM$ meet the circumcircle of $ABC$ again at $K$. Lines $CK$ and $AB$ intersect at $D$ and $BK$ and $AC$ intersect at $E$. If the circumcentre of triangle $ADE$ is $P$, calculate the value of $\frac{OP}{OA}$, where $OA$... | [
"1"
] | [
"Triangle $ABC$ has circumcentre $O$ and centroid $M$. The lines $OM$ and $AM$ are perpendicular. Let $AM$ meet the circumcircle of $ABC$ again at $K$. Lines $CK$ and $AB$ intersect at $D$ and $BK$ and $AC$ intersect at $E$. If the circumcentre of triangle $ADE$ is $P$, calculate the value of $\\frac{OP}{OA}$, wher... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-69320d60afaa7a3ed1aeaec93530355171bc84feea35e708452aeaff0846e692 | Triangle $ABC$ has circumcircle $\Gamma$ and $M$ is the midpoint of $BC$. Suppose $AM$ meets $\Gamma$ again in $X$ and $P$ is a point such that $BXCP$ is a parallelogram. The tangent to $\Gamma$ at $A$ intersects the line $BC$ in $K$. Calculate the ratio $\frac{KA}{KP}$ if $KA = 10$. | [
"1"
] | [
"Triangle $ABC$ has circumcircle $\\Gamma$ and $M$ is the midpoint of $BC$. Suppose $AM$ meets $\\Gamma$ again in $X$ and $P$ is a point such that $BXCP$ is a parallelogram. The tangent to $\\Gamma$ at $A$ intersects the line $BC$ in $K$. Calculate the ratio $\\frac{KA}{KP}$ if $KA = 10$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-e46a06cfd98e5a37dfaec2c3b64565a6f0ee1638e5e9eac6375aaa7dbaf50189 | Triangle $ABC$ has circumcircle $\Omega$ and incircle $\omega$, where $\omega$ is tangent to $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. $T$ is an arbitrary point on $\omega$. $EF$ meets $BC$ at $K$, $AT$ meets $\Omega$ again at $P$, $PK$ meets $\Omega$ again at $S$. $X$ is a point on $\Omega$ such that $S... | [
"1"
] | [
"Triangle $ABC$ has circumcircle $\\Omega$ and incircle $\\omega$, where $\\omega$ is tangent to $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. $T$ is an arbitrary point on $\\omega$. $EF$ meets $BC$ at $K$, $AT$ meets $\\Omega$ again at $P$, $PK$ meets $\\Omega$ again at $S$. $X$ is a point on $\\Omega$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-503e2a14588b4e77deda9e29a630224f906cc66e768f27ef414d48d229ed46d2 | Triangle $ABC$ has circumcircle $\Omega$. Let the angle bisectors of $\angle CAB$ and $\angle CBA$ meet $\Omega$ again at $D$ and $E$. Given that $AB = 4$, $BC = 5$, and $AC = 6$, find the product of the side lengths of pentagon $BAECD$. | [
"512"
] | [
"Triangle $ABC$ has circumcircle $\\Omega$. Let the angle bisectors of $\\angle CAB$ and $\\angle CBA$ meet $\\Omega$ again at $D$ and $E$. Given that $AB = 4$, $BC = 5$, and $AC = 6$, find the product of the side lengths of pentagon $BAECD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-64d68fafead7fa53bc4ddf095ed2d2107f1bd3014d858d7820e498cb9c32cbc7 | Triangle $ABC$ has fixed vertices $B$ and $C$ with $BC = 2$, and vertex $A$ is variable. Denote by $H$ and $G$ the orthocenter and centroid of triangle $ABC$, respectively. Let $F \in (HG)$ such that $\frac{HF}{FG} = 3$. Find the locus of point $A$ such that $F \in BC$. | [
"\\( x^2 - y^2 = 1 \\)"
] | [
"Triangle $ABC$ has fixed vertices $B$ and $C$ with $BC = 2$, and vertex $A$ is variable. Denote by $H$ and $G$ the orthocenter and centroid of triangle $ABC$, respectively. Let $F \\in (HG)$ such that $\\frac{HF}{FG} = 3$. Find the locus of point $A$ such that $F \\in BC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 39,
"genselect": 4,
"tir": 5
} | 48 | 0 | 0 |
omr-ws-dc8ac9275839750a71de0fc9488fb2de7afbcec12afcccd66f9bdfcbd91699c1 | Triangle $ABC$ has incenter $I$, circumcenter $O$, and inradius $3$. The line from $I$ perpendicular to $BC$ meets the circumcircle at $E$ such that $IE$ does not pass through either $AB$ or $AC$. Given that $IA = 4$, $IB = 5$, and $IC = 6$, compute $\frac{IO}{IE}$. | [
"\\dfrac{1}{2}"
] | [
"Triangle $ABC$ has incenter $I$, circumcenter $O$, and inradius $3$. The line from $I$ perpendicular to $BC$ meets the circumcircle at $E$ such that $IE$ does not pass through either $AB$ or $AC$. Given that $IA = 4$, $IB = 5$, and $IC = 6$, compute $\\frac{IO}{IE}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-a07a3d3257dbd3f3f25e13828062dd6b0fdfd6532673b004ea7f6678de08d9f9 | Triangle $ABC$ has incenter $I$, circumcircle $\omega$, and circumcenter $O$. The circle with diameter $AI$ intersects $\omega$ again at $Q$. Let $N$ be the midpoint of arc $BAC$. Let the line tangent to $\omega$ passing through $A$ intersect $QN$ at another point $M$. If $AI = 12$, $AO = 13$, and $AN = 15$, calculate ... | [
"5"
] | [
"Triangle $ABC$ has incenter $I$, circumcircle $\\omega$, and circumcenter $O$. The circle with diameter $AI$ intersects $\\omega$ again at $Q$. Let $N$ be the midpoint of arc $BAC$. Let the line tangent to $\\omega$ passing through $A$ intersect $QN$ at another point $M$. If $AI = 12$, $AO = 13$, and $AN = 15$, ca... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-62c032c1425997870920a7c502c076d70920300525a1f1893f66ccfd3b5092c0 | Triangle $ABC$ has incenter $I$. Point $A$ is reflected over $I$ to get $A^{\prime}$. Similarly, $B^{\prime}$ and $C^{\prime}$ are constructed. The lines $AB^{\prime}, BC^{\prime},$ and $CA^{\prime}$ are drawn. The circumcircle $\omega_2$ of $\triangle A^{\prime}B^{\prime}C^{\prime}$ intersects the lines at $D, E,$ and... | [
"\\(\\frac{1365}{16}\\)"
] | [
"Triangle $ABC$ has incenter $I$. Point $A$ is reflected over $I$ to get $A^{\\prime}$. Similarly, $B^{\\prime}$ and $C^{\\prime}$ are constructed. The lines $AB^{\\prime}, BC^{\\prime},$ and $CA^{\\prime}$ are drawn. The circumcircle $\\omega_2$ of $\\triangle A^{\\prime}B^{\\prime}C^{\\prime}$ intersects the line... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c6408b96fb5b5c44fe661bae07379a482b76e61124d0c6affba9652d73527111 | Triangle $ABC$ has incentre $I$. Let $D$ be the foot of the perpendicular from $A$ to side $BC$. Let $X$ be a point such that segment $AX$ is a diameter of the circumcircle of triangle $ABC$. Given that $ID = 2$, $IA = 3$, and $IX = 4$, find the inradius of triangle $ABC$. | [
"\\(\\frac{11}{12}\\)"
] | [
"Triangle $ABC$ has incentre $I$. Let $D$ be the foot of the perpendicular from $A$ to side $BC$. Let $X$ be a point such that segment $AX$ is a diameter of the circumcircle of triangle $ABC$. Given that $ID = 2$, $IA = 3$, and $IX = 4$, find the inradius of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-5cfcaaeb3babb911c5bad317dcf6da472b19977fe77d492459fa8eada2d81dbf | Triangle $ABC$ has incentre $I$. The line $AI$ meets the circumcircle of $ABC$ again at $D$. The feet of the perpendiculars from $I$ to $BD$ and $CD$ are denoted $E$ and $F$ respectively. Given that $IE + IF = \frac{1}{2} AD$, determine the possible values of $\angle BAC$. | [
"\\(30^\\circ\\) or \\(150^\\circ\\)"
] | [
"Triangle $ABC$ has incentre $I$. The line $AI$ meets the circumcircle of $ABC$ again at $D$. The feet of the perpendiculars from $I$ to $BD$ and $CD$ are denoted $E$ and $F$ respectively. Given that $IE + IF = \\frac{1}{2} AD$, determine the possible values of $\\angle BAC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-0ac0a8e4793e57cfc0b287b0db766cb0b2f3c83cedc0a50cdf32156abf8f221f | Triangle $ABC$ has incircle $(I)$ and points $P, Q$ in the plane. Let $QA, QB, QC$ meet $BA, CA, AB$ at $D, E, F$ respectively. The tangent at $D$, other than $BC$, of the circle $(I)$ meets $PA$ at $X$. The points $Y$ and $Z$ are defined similarly. The tangent at $X$, other than $XD$, of the circle $(I)$ meets $BC, (I... | [
"2"
] | [
"Triangle $ABC$ has incircle $(I)$ and points $P, Q$ in the plane. Let $QA, QB, QC$ meet $BA, CA, AB$ at $D, E, F$ respectively. The tangent at $D$, other than $BC$, of the circle $(I)$ meets $PA$ at $X$. The points $Y$ and $Z$ are defined similarly. The tangent at $X$, other than $XD$, of the circle $(I)$ meets $B... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-2ec9f68f1d24fcbd202e5f9ac9d0139f41ed58d650f007d190db39eea5cdba10 | Triangle $ABC$ has incircle $O$ that is tangent to $AC$ at $D$. Let $M$ be the midpoint of $AC$. Point $E$ lies on $BC$ such that line $AE$ is perpendicular to the extension of $BO$. Given $AC = 2013$, $AB = 2014$, and $DM = 249$, find the length of $CE$. | [
"498"
] | [
"Triangle $ABC$ has incircle $O$ that is tangent to $AC$ at $D$. Let $M$ be the midpoint of $AC$. Point $E$ lies on $BC$ such that line $AE$ is perpendicular to the extension of $BO$. Given $AC = 2013$, $AB = 2014$, and $DM = 249$, find the length of $CE$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-805ed97344104ff49ada1b69da555ebdbc2d1e1f654f05c716229a9ce44bd163 | Triangle $ABC$ has incircle $\omega$ and incenter $I$. Points $P$ and $Q$ are chosen on sides $AB$ and $BC$ respectively such that $PI = QI$ and $PB > QB$. Line segment $QI$ intersects $\omega$ at $T$. A tangent line to $\omega$ at $T$ intersects sides $AB$ and $BC$ at $U$ and $V$ respectively. If $PU = 10$ and $VQ = 6... | [
"16"
] | [
"Triangle $ABC$ has incircle $\\omega$ and incenter $I$. Points $P$ and $Q$ are chosen on sides $AB$ and $BC$ respectively such that $PI = QI$ and $PB > QB$. Line segment $QI$ intersects $\\omega$ at $T$. A tangent line to $\\omega$ at $T$ intersects sides $AB$ and $BC$ at $U$ and $V$ respectively. If $PU = 10$ and... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9911f2eac2e0351e3ec5fb6123b15d623f5b2694cff447ecd8cd66053b9e50d1 | Triangle $ABC$ has inradius $8$ and circumradius $20$. Let $M$ be the midpoint of side $BC$, and let $N$ be the midpoint of arc $BC$ on the circumcircle not containing $A$. Let $s_A$ denote the length of segment $MN$, and define $s_B$ and $s_C$ similarly with respect to sides $CA$ and $AB$. Evaluate the product $s_A s_... | [
"640"
] | [
"Triangle $ABC$ has inradius $8$ and circumradius $20$. Let $M$ be the midpoint of side $BC$, and let $N$ be the midpoint of arc $BC$ on the circumcircle not containing $A$. Let $s_A$ denote the length of segment $MN$, and define $s_B$ and $s_C$ similarly with respect to sides $CA$ and $AB$. Evaluate the product $s... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-018b94edaa5aaa72cdf70f70a1f27b3d19abc27c937d8d12137a0c4e87f51a31 | Triangle $ABC$ has integer side lengths $a, b, c$ such that $a + b + c = 35.$ $G$ is the centroid and $I$ is the incenter. If $\angle CIG = 90^\circ,$ find $c.$ | [
"11"
] | [
"Triangle $ABC$ has integer side lengths $a, b, c$ such that $a + b + c = 35.$ $G$ is the centroid and $I$ is the incenter. If $\\angle CIG = 90^\\circ,$ find $c.$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-cdda115102bde13a29b91cec4957d308347654884099834c8eca1d708007e075 | Triangle $ABC$ has integer-length sides and $AC = 2007$. The internal bisector of $\angle BAC$ meets $BC$ at $D$. Given that $AB = CD$, determine the lengths of $AB$ and $BC$. | [
"\\( AB = 1338, BC = 2230 \\)"
] | [
"Triangle $ABC$ has integer-length sides and $AC = 2007$. The internal bisector of $\\angle BAC$ meets $BC$ at $D$. Given that $AB = CD$, determine the lengths of $AB$ and $BC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-33c8c62cf3eac7d70271d4a69399365fed728d831161b4d2bb93a27457d834b5 | Triangle $ABC$ has integer-length sides, and $AC=2007$. The internal bisector of $\angle BAC$ meets $BC$ at $D$. Given that $AB=CD$, determine the lengths of $AB$ and $BC$. | [
"\\( AB = 1338 \\), \\( BC = 2230 \\)"
] | [
"Triangle $ABC$ has integer-length sides, and $AC=2007$. The internal bisector of $\\angle BAC$ meets $BC$ at $D$. Given that $AB=CD$, determine the lengths of $AB$ and $BC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 26
} | 28 | 0 | 0 |
omr-ws-68c6e1a5ab547def864cb13e891935403a6aff922210ff7a8a8f28884f65d8e5 | Triangle $ABC$ has integer-length sides, with $AC = 2007$. The internal bisector of $\angle BAC$ meets $BC$ at $D$. Given that $AB = CD$, determine the lengths of $AB$ and $BC$. | [
"\\( AB = 1338 \\), \\( BC = 2230 \\)"
] | [
"Triangle $ABC$ has integer-length sides, with $AC = 2007$. The internal bisector of $\\angle BAC$ meets $BC$ at $D$. Given that $AB = CD$, determine the lengths of $AB$ and $BC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 19
} | 20 | 0 | 0 |
omr-ws-838eb460003cc3182198f0cdaf9df15830e7254fcd7206a0983e1b560e43e1c8 | Triangle $ABC$ has orthocenter $H$, circumcenter $O$, and $M$ as the midpoint of $BC$. Given that $OM = MH = HO = 12$, find $AB + BC + CA$. | [
"\\(\\sqrt{1296 - 144\\sqrt{6}} + 24\\sqrt{2} + \\sqrt{1296 + 144\\sqrt{6}}\\)"
] | [
"Triangle $ABC$ has orthocenter $H$, circumcenter $O$, and $M$ as the midpoint of $BC$. Given that $OM = MH = HO = 12$, find $AB + BC + CA$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 8
} | 11 | 0 | 0 |
omr-ws-aa49d4b78e4986def8459cdaecf80042d32c640a03ed61bddff95304d893f421 | Triangle $ABC$ has orthocenter $H$, circumcenter $O$, and $M$ as the midpoint of $BC$. Given that $OM = MH = HO = 12$, find $AB^2 + BC^2 + CA^2$. | [
"\\( 3744 \\)"
] | [
"Triangle $ABC$ has orthocenter $H$, circumcenter $O$, and $M$ as the midpoint of $BC$. Given that $OM = MH = HO = 12$, find $AB^2 + BC^2 + CA^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-d0d1ca973f1f4f67c980514f9bc55a3b598c5e69463d571641e46a98cafe4365 | Triangle $ABC$ has orthocenter $H$, circumcenter $O$, and the foot of the altitude from $A$ is $T$. Given that $HOT$ is an equilateral triangle with area $1$, the area of $ABC$ can be expressed as $\frac{a \sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer that is not... | [
"40"
] | [
"Triangle $ABC$ has orthocenter $H$, circumcenter $O$, and the foot of the altitude from $A$ is $T$. Given that $HOT$ is an equilateral triangle with area $1$, the area of $ABC$ can be expressed as $\\frac{a \\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer that... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 5
} | 18 | 0 | 0 |
omr-ws-2bcb8d8b7d5cdcb1844012caaa51469c4549c1926e5ef3e47b9af44fc84263aa | Triangle $ABC$ has orthocenter $H$. If $AH=13$, $BH=40$, and $CH=68$, find the circumradius of triangle $ABC$. | [
"\\(\\frac{85}{2}\\)"
] | [
"Triangle $ABC$ has orthocenter $H$. If $AH=13$, $BH=40$, and $CH=68$, find the circumradius of triangle $ABC$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-10c3e02c78d0861ce2785a81599c919c6c97b002163128c7a1ff7c5676ab6b48 | Triangle $ABC$ has points $D$, $E$, and $F$ on segment $BC$ in that order, where $D$ is between $B$ and $E$. Given that $AD$ and $AE$ trisect angle $BAF$, $\angle BAF = 60^{\circ}$, $\frac{EF}{EC} = \frac{2}{3}$, and $\frac{AE}{AC} = 2$, find $\angle BAC$. | [
"80^\\circ"
] | [
"Triangle $ABC$ has points $D$, $E$, and $F$ on segment $BC$ in that order, where $D$ is between $B$ and $E$. Given that $AD$ and $AE$ trisect angle $BAF$, $\\angle BAF = 60^{\\circ}$, $\\frac{EF}{EC} = \\frac{2}{3}$, and $\\frac{AE}{AC} = 2$, find $\\angle BAC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-38d9dd57247042676a6ea953aea6e5bd9ff1bef15f69638ef68a73b1eaadb4c0 | Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the angle bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$. | [
"108"
] | [
"Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the angle bisectors of $\\angle B$ and $\\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\\triangle ABC$."
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 29,
"tir": 14
} | 43 | 0 | 0 |
omr-ws-891c061ad4b6e731d82f37e4d6d20bd89d06801a8e1291cb6bb4851594f41d36 | Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$. | [
"108"
] | [
"Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\\angle B$ and $\\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\\triangle ABC$.",
"Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of th... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 10
} | 41 | 0 | 0 |
omr-ws-a3fec2842b626b39f6ad483d89d98a4bea652bfd865b8f6db9e42739fdae40ad | Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\angle APB = \angle BPC = \angle CPA$. Find $PC$. | [
"33"
] | [
"Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\\angle APB = \\angle BPC = \\angle CPA$. Find $PC$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-04b0d5d9f920c233ba105cd55bf8916b4dc435ebdbd2abdcad34c02dda2c30b2 | Triangle $ABC$ has side lengths $1, 1, \sqrt{2}$. Compute the distance from the incentre to the circumcentre. | [
"\\(1 - \\frac{\\sqrt{2}}{2}\\)"
] | [
"Triangle $ABC$ has side lengths $1, 1, \\sqrt{2}$. Compute the distance from the incentre to the circumcentre."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-45a47e92b99f00050acf26479c5732cee774891f4e393d58e74e653ab1253c68 | Triangle $ABC$ has side lengths $7$, $8$, and $9$. A line containing the center of the inscribed circle and parallel to the shortest side of the triangle intersects the other two sides at points $D$ and $E$. What is the length of the line segment $DE$? | [
"\\(\\frac{119}{24}\\)"
] | [
"Triangle $ABC$ has side lengths $7$, $8$, and $9$. A line containing the center of the inscribed circle and parallel to the shortest side of the triangle intersects the other two sides at points $D$ and $E$. What is the length of the line segment $DE$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-c28354cde0757089a187125281f9d06a1be7e0a6af0fdead54620b88dc31d92b | Triangle $ABC$ has side lengths $7$, $8$, and $9$. A line containing the center of the inscribed circle and parallel to the shortest side of the triangle intersects the other two sides at points $D$ and $E$. What is the length of the segment $DE$? | [
"\\(\\frac{49}{24}\\)"
] | [
"Triangle $ABC$ has side lengths $7$, $8$, and $9$. A line containing the center of the inscribed circle and parallel to the shortest side of the triangle intersects the other two sides at points $D$ and $E$. What is the length of the segment $DE$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-50f6e5dae1e65cdb7ef76de9f44d3c025724c6367df2411d00a22b7d109b0926 | Triangle $ABC$ has side lengths $AB = 10$, $AC = 14$, and $BC = 16$. The angle bisectors of the triangle intersect at point $I$. If the area of triangle $AIC$ can be expressed as $m\sqrt{n}$ in simplest form, what is the value of $m + n$? | [
"17"
] | [
"Triangle $ABC$ has side lengths $AB = 10$, $AC = 14$, and $BC = 16$. The angle bisectors of the triangle intersect at point $I$. If the area of triangle $AIC$ can be expressed as $m\\sqrt{n}$ in simplest form, what is the value of $m + n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c8e4d68ef9445a488eb64f96cfb320194d064ca68c196584cbfda4bde4fd56a4 | Triangle $ABC$ has side lengths $AB = 11$, $BC = 24$, and $CA = 20$. The bisector of $\angle BAC$ intersects $\overline{BC}$ at point $D$ and intersects the circumcircle of $\triangle ABC$ at point $E \ne A$. The circumcircle of $\triangle BED$ intersects the line $AB$ at points $B$ and $F \ne B$. What is the length of... | [
"30"
] | [
"Triangle $ABC$ has side lengths $AB = 11$, $BC = 24$, and $CA = 20$. The bisector of $\\angle BAC$ intersects $\\overline{BC}$ at point $D$ and intersects the circumcircle of $\\triangle ABC$ at point $E \\ne A$. The circumcircle of $\\triangle BED$ intersects the line $AB$ at points $B$ and $F \\ne B$. What is th... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 6
} | 25 | 0 | 0 |
omr-ws-09096318187cb7a4dc4a91c3613508377c673380f0b38c73a5c67f7784e76a7a | Triangle $ABC$ has side lengths $AB = 12$, $AC = 5$, and $BC = 13$. $AE$ is the altitude from $A$ to $BC$, and $D$ is a point on $BC$ such that $AD = AC$. If the length of $ED$ can be expressed in the form $m/n$ with $m$ and $n$ relatively prime, find $m + n$. | [
"38"
] | [
"Triangle $ABC$ has side lengths $AB = 12$, $AC = 5$, and $BC = 13$. $AE$ is the altitude from $A$ to $BC$, and $D$ is a point on $BC$ such that $AD = AC$. If the length of $ED$ can be expressed in the form $m/n$ with $m$ and $n$ relatively prime, find $m + n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-766a495c9b4c209f15e64f7e53ba94b8aa1d23197ed3c9fb253d42724785ce64 | Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ = \omega$, the area of $PQRS$ can be expressed as the quadratic polynomial $$Area(PQRS) = \... | [
"161"
] | [
"Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\\overline{AB}$, vertex $Q$ on $\\overline{AC}$, and vertices $R$ and $S$ on $\\overline{BC}$. In terms of the side length $PQ = \\omega$, the area of $PQRS$ can be expressed as the quadratic polynomial $$Area(... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 21
} | 44 | 0 | 0 |
omr-ws-c813b96585b49141aa80988dfd78fc6b51d8d70a853ea7942be2f7d3934ba185 | Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ = \omega$, the area of $PQRS$ can be expressed as the quadratic polynomial\[Area(PQRS) = \a... | [
"161"
] | [
"Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\\overline{AB}$, vertex $Q$ on $\\overline{AC}$, and vertices $R$ and $S$ on $\\overline{BC}$. In terms of the side length $PQ = \\omega$, the area of $PQRS$ can be expressed as the quadratic polynomial\\[Area(... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ca33003a789991b11be11584e649ff5de7663231e02a3ba1fa8851cc449bf625 | Triangle $ABC$ has side lengths $AB = 14$, $BC = 8$, and $AC = 9$. $AC$ is extended to point $D$ and $AB$ is extended to point $E$ such that $BD$ is an angle bisector of $\angle CBE$. Find the length of $AD$. | [
"21"
] | [
"Triangle $ABC$ has side lengths $AB = 14$, $BC = 8$, and $AC = 9$. $AC$ is extended to point $D$ and $AB$ is extended to point $E$ such that $BD$ is an angle bisector of $\\angle CBE$. Find the length of $AD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 3
} | 25 | 0 | 0 |
omr-ws-8050057e8086ca502c0538edabcf744b7d16890d5ebdd4434fd899ff769d2a83 | Triangle $ABC$ has side lengths $AB = 2$, $AC = 6.5$, and $BC = 7.5$. Points $M$ and $N$ lie on $AB$ and $AC$ respectively so that $MC$ and $NB$ intersect at point $O$. If triangles $MBO$ and $NCO$ both have area $1$, the product of all possible areas of triangle $AMN$ can be written as $\frac{p}{q}$ for relatively pri... | [
"2"
] | [
"Triangle $ABC$ has side lengths $AB = 2$, $AC = 6.5$, and $BC = 7.5$. Points $M$ and $N$ lie on $AB$ and $AC$ respectively so that $MC$ and $NB$ intersect at point $O$. If triangles $MBO$ and $NCO$ both have area $1$, the product of all possible areas of triangle $AMN$ can be written as $\\frac{p}{q}$ for relative... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-5d0c273215e23472bf79b28e23a67c6b2ac45eff131cdf63feba7c3be1a3aa7b | Triangle $ABC$ has side lengths $AB = 231$, $BC = 160$, and $AC = 281$. Point $D$ is constructed on the opposite side of line $AC$ as point $B$ such that $AD = 178$ and $CD = 153$. Compute the distance from $B$ to the midpoint of segment $AD$. | [
"208"
] | [
"Triangle $ABC$ has side lengths $AB = 231$, $BC = 160$, and $AC = 281$. Point $D$ is constructed on the opposite side of line $AC$ as point $B$ such that $AD = 178$ and $CD = 153$. Compute the distance from $B$ to the midpoint of segment $AD$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94ece5bd7471b700b0ef3745662afc8f31a973a8c86e48911137578439b0dfb4 | Triangle $ABC$ has side lengths $AB = 2\sqrt{5}$, $BC = 1$, and $CA = 5$. Point $D$ is on side $AC$ such that $CD = 1$, and point $F$ is such that $BF = 2$ and $CF = 3$. Let $E$ be the intersection of lines $AB$ and $DF$. Find the area of quadrilateral $CDEB$. | [
"$\\frac{22}{35}$"
] | [
"Triangle $ABC$ has side lengths $AB = 2\\sqrt{5}$, $BC = 1$, and $CA = 5$. Point $D$ is on side $AC$ such that $CD = 1$, and point $F$ is such that $BF = 2$ and $CF = 3$. Let $E$ be the intersection of lines $AB$ and $DF$. Find the area of quadrilateral $CDEB$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-88c89a8fdad873e245995a75bce92dc282f67766beaeed9655ac654a1e287cdd | Triangle $ABC$ has side lengths $AB = 3$, $BC = 4$, and $CA = 5$. Draw line $\ell_A$ such that $\ell_A$ is parallel to $BC$ and splits the triangle into two polygons of equal area. Define lines $\ell_B$ and $\ell_C$ analogously. The intersection points of $\ell_A$, $\ell_B$, and $\ell_C$ form a triangle. Determine its ... | [
"\\(\\frac{3}{2}\\Bigl(3\\sqrt{2} - 4\\Bigr)^{2}\\)"
] | [
"Triangle $ABC$ has side lengths $AB = 3$, $BC = 4$, and $CA = 5$. Draw line $\\ell_A$ such that $\\ell_A$ is parallel to $BC$ and splits the triangle into two polygons of equal area. Define lines $\\ell_B$ and $\\ell_C$ analogously. The intersection points of $\\ell_A$, $\\ell_B$, and $\\ell_C$ form a triangle. De... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 26
} | 45 | 0 | 0 |
omr-ws-bd10077547a53e66ac2aa42ac133fe96334e7cea889e345cba6229528867503a | Triangle $ABC$ has side lengths $AB = 5$, $AC = 10$, and $BC = 9$. The median of $\triangle ABC$ from $A$ intersects the circumcircle of the triangle again at point $D$. What is $BD + CD$? | [
"\\(\\frac{135}{13}\\)"
] | [
"Triangle $ABC$ has side lengths $AB = 5$, $AC = 10$, and $BC = 9$. The median of $\\triangle ABC$ from $A$ intersects the circumcircle of the triangle again at point $D$. What is $BD + CD$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-cb104e65fb285fe5617c99703611ddd32f87222b4a8eb1ab6adeced826e7f846 | Triangle $ABC$ has side lengths $AB = 5$, $BC = 12$, and $CA = 13$. Circle $\Gamma$ is centered at point $X$ exterior to triangle $ABC$ and passes through points $A$ and $C$. Let $D$ be the second intersection of $\Gamma$ with segment $\overline{BC}$. If $\angle BDA = \angle CAB$, the radius of $\Gamma$ can be expresse... | [
"193"
] | [
"Triangle $ABC$ has side lengths $AB = 5$, $BC = 12$, and $CA = 13$. Circle $\\Gamma$ is centered at point $X$ exterior to triangle $ABC$ and passes through points $A$ and $C$. Let $D$ be the second intersection of $\\Gamma$ with segment $\\overline{BC}$. If $\\angle BDA = \\angle CAB$, the radius of $\\Gamma$ can ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-0b402ac38837872f363c06645cdb46fa43debfc80777104eb1b178a5214e862b | Triangle $ABC$ has side lengths $AB = 5$, $BC = 12$, and $CA = 13$. Triangle $DEF$ is equilateral such that $A$ lies on side $\overline{EF}$, $B$ lies on side $\overline{FD}$, and $C$ lies on side $\overline{DE}$. The maximum area of $\triangle DEF$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ for positive integers $a$,... | [
"319"
] | [
"Triangle $ABC$ has side lengths $AB = 5$, $BC = 12$, and $CA = 13$. Triangle $DEF$ is equilateral such that $A$ lies on side $\\overline{EF}$, $B$ lies on side $\\overline{FD}$, and $C$ lies on side $\\overline{DE}$. The maximum area of $\\triangle DEF$ can be expressed as $\\frac{a+b\\sqrt{c}}{d}$ for positive in... | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-8baafd17b4f4df01295f968a8ed1efcb98e595d5630a4d22dad6eb4c4fa03913 | Triangle $ABC$ has side lengths $AB = 65$, $BC = 33$, and $AC = 56$. Find the radius of the circle tangent to sides $AC$ and $BC$ and to the circumcircle of triangle $ABC$. | [
"24"
] | [
"Triangle $ABC$ has side lengths $AB = 65$, $BC = 33$, and $AC = 56$. Find the radius of the circle tangent to sides $AC$ and $BC$ and to the circumcircle of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1e0732d69aa565b42ffd29cc5d519556c1e73e4091f062fbd25a0091f560e8d | Triangle $ABC$ has side lengths $AB = 69$, $BC = 96$, and $CA = 42$. Let $I$ be the incenter of $\triangle ABC$, let $D$ be the foot of the altitude from $I$ to $\overline{BC}$, and let $X$ be the midpoint of $\overline{IA}$. Let $P$ be a point such that $\overline{PD} \perp \overline{XD}$ and $PD = XI$. Finally, let $... | [
"173"
] | [
"Triangle $ABC$ has side lengths $AB = 69$, $BC = 96$, and $CA = 42$. Let $I$ be the incenter of $\\triangle ABC$, let $D$ be the foot of the altitude from $I$ to $\\overline{BC}$, and let $X$ be the midpoint of $\\overline{IA}$. Let $P$ be a point such that $\\overline{PD} \\perp \\overline{XD}$ and $PD = XI$. Fin... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-26b300c2ed68d288e67b30584ba4975172beed6e4cf157495137c57a1d8e2b96 | Triangle $ABC$ has side lengths $AB = 9$ and $BC = 16$. Given that $\angle ABC$ is a right angle, determine the length of $AC$. | [
"\\(\\sqrt{337}\\)"
] | [
"Triangle $ABC$ has side lengths $AB = 9$ and $BC = 16$. Given that $\\angle ABC$ is a right angle, determine the length of $AC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-f90282e06f7d2d7e209eccffba709c3364ab370ed28fa45d16f2ce35ab072840 | Triangle $ABC$ has side lengths $AB = 9$, $BC =$ $5\sqrt{3}$, and $AC = 12$. Points $A = P_{0}, P_{1}, P_{2},..., P_{2450} = B$ are on segment $\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2,..., 2449$, and points $A = Q_{0}, Q_{1}, Q_{2},..., Q_{2450} = C$ are on segment $\overline{AC}$ with... | [
"020"
] | [
"Triangle $ABC$ has side lengths $AB = 9$, $BC =$ $5\\sqrt{3}$, and $AC = 12$. Points $A = P_{0}, P_{1}, P_{2},..., P_{2450} = B$ are on segment $\\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2,..., 2449$, and points $A = Q_{0}, Q_{1}, Q_{2},..., Q_{2450} = C$ are on segment $\\overline{... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 4
} | 27 | 0 | 0 |
omr-ws-ab4d2be40c26deab4aedced3633285c4d98f95bd94b219c218720777ab5110ab | Triangle $ABC$ has side lengths $AB=1$, $BC=\sqrt{3}$, and $AC=2$. Points $D$, $E$, and $F$ are chosen on $AB$, $BC$, and $AC$ respectively, such that $\angle EDF = \angle DFA = 90^{\circ}$. Given that the maximum possible value of $[DEF]^2$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\gcd (a, ... | [
"67"
] | [
"Triangle $ABC$ has side lengths $AB=1$, $BC=\\sqrt{3}$, and $AC=2$. Points $D$, $E$, and $F$ are chosen on $AB$, $BC$, and $AC$ respectively, such that $\\angle EDF = \\angle DFA = 90^{\\circ}$. Given that the maximum possible value of $[DEF]^2$ can be expressed as $\\frac{a}{b}$ for positive integers $a, b$ with ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-59ea43362057334a283290981f8d57bf9b95b46c7b64138e53dc749aeb429a83 | Triangle $ABC$ has side lengths $AB=12$, $BC=25$, and $CA=17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ=w$, the area of $PQRS$ can be expressed as the quadratic polynomial \[\text{Area}(PQRS)=\alpha w-... | [
"161"
] | [
"Triangle $ABC$ has side lengths $AB=12$, $BC=25$, and $CA=17$. Rectangle $PQRS$ has vertex $P$ on $\\overline{AB}$, vertex $Q$ on $\\overline{AC}$, and vertices $R$ and $S$ on $\\overline{BC}$. In terms of the side length $PQ=w$, the area of $PQRS$ can be expressed as the quadratic polynomial\n\\[\\text{Area}(PQRS... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-16b1ed057c734caa406cb1372d200670c58ab483b68c6150598b463ab422c8dc | Triangle $ABC$ has side lengths $AB=120$, $BC=220$, and $AC=180$. Lines $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ are drawn parallel to $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$, respectively, such that the intersection of $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ with the interior of $\triangle ABC$ are segments... | [
"715"
] | [
"Triangle $ABC$ has side lengths $AB=120$, $BC=220$, and $AC=180$. Lines $\\ell_{A}$, $\\ell_{B}$, and $\\ell_{C}$ are drawn parallel to $\\overline{BC}$, $\\overline{AC}$, and $\\overline{AB}$, respectively, such that the intersection of $\\ell_{A}$, $\\ell_{B}$, and $\\ell_{C}$ with the interior of $\\triangle AB... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-3932806ad024c7c99757388dd558ea46883e1168d25f042f6c522aa28c3d31c1 | Triangle $ABC$ has side lengths $AB=120,BC=220$, and $AC=180$. Lines $\ell_A,\ell_B$, and $\ell_C$ are drawn parallel to $\overline{BC},\overline{AC}$, and $\overline{AB}$, respectively, such that the intersections of $\ell_A,\ell_B$, and $\ell_C$ with the interior of $\triangle ABC$ are segments of lengths $55,45$, an... | [
"715"
] | [
"Triangle $ABC$ has side lengths $AB=120,BC=220$, and $AC=180$. Lines $\\ell_A,\\ell_B$, and $\\ell_C$ are drawn parallel to $\\overline{BC},\\overline{AC}$, and $\\overline{AB}$, respectively, such that the intersections of $\\ell_A,\\ell_B$, and $\\ell_C$ with the interior of $\\triangle ABC$ are segments of leng... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-f420b242039dc4347e61bf7d4b76e1c9aab616c7613df5a8119fbd88f960e742 | Triangle $ABC$ has side lengths $AB=13$, $AC=14$, and $BC=15$. Let $A'$, $B'$, and $C'$ be the excenters of triangle $ABC$. Find the area of triangle $A'B'C'$. | [
"341.25"
] | [
"Triangle $ABC$ has side lengths $AB=13$, $AC=14$, and $BC=15$. Let $A'$, $B'$, and $C'$ be the excenters of triangle $ABC$. Find the area of triangle $A'B'C'$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-729b5edac0301b9126203a72428887dd520ba618eb556c451dbeb7b7dd608891 | Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Points $A'$ and $A''$ are on segment $BC$ such that $BA'=A'A''=A''C$ with $A'$ closer to $B$ than $A''$. Points $B'$ and $B''$ are on segment $CA$ such that $CB'=B'B''=B''A$ with $B'$ closer to $C$ than $B''$. Points $C'$ and $C''$ are on segment $AB$ such ... | [
"59"
] | [
"Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Points $A'$ and $A''$ are on segment $BC$ such that $BA'=A'A''=A''C$ with $A'$ closer to $B$ than $A''$. Points $B'$ and $B''$ are on segment $CA$ such that $CB'=B'B''=B''A$ with $B'$ closer to $C$ than $B''$. Points $C'$ and $C''$ are on segment $AB$ ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e136e764982b6725fdb6a8364d3beff23d90f1e531bc51c20f320b817c585630 | Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. The angle bisectors of $\angle BAC$, $\angle ABC$, and $\angle ACB$ intersect the circumcircle of $\triangle ABC$ again at points $D$, $E$, and $F$, respectively. Compute the area of hexagon $AFBDCE$. | [
"\\(\\frac{1365}{8}\\)"
] | [
"Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. The angle bisectors of $\\angle BAC$, $\\angle ABC$, and $\\angle ACB$ intersect the circumcircle of $\\triangle ABC$ again at points $D$, $E$, and $F$, respectively. Compute the area of hexagon $AFBDCE$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-4f8edff4f2a1613f9ad816be8c9f83ffe5c5a89db1864b2bed96a4ab40974293 | Triangle $ABC$ has side lengths $AB=14$, $BC=15$, and $CA=13$. A circle with diameter $AB$ intersects $BC$ at $D$ and $CA$ at $E$. Find the circumradius of triangle $CDE$. | [
"\\(\\frac{33}{8}\\)"
] | [
"Triangle $ABC$ has side lengths $AB=14$, $BC=15$, and $CA=13$. A circle with diameter $AB$ intersects $BC$ at $D$ and $CA$ at $E$. Find the circumradius of triangle $CDE$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-244e0a26490196d9468df33132b136ff67e8af1183d19e1b1445b1f126813e9b | Triangle $ABC$ has side lengths $AB=15$, $BC=13$, and $AC=14$. A semicircle is inscribed in $\triangle ABC$ with its base on $\overline{AC}$ and center $D$. Another semicircle is tangent to $\overline{AB}$, $\overline{BC}$, and the first semicircle with its base parallel to $\overline{AC}$. This sequence is continued i... | [
"36"
] | [
"Triangle $ABC$ has side lengths $AB=15$, $BC=13$, and $AC=14$. A semicircle is inscribed in $\\triangle ABC$ with its base on $\\overline{AC}$ and center $D$. Another semicircle is tangent to $\\overline{AB}$, $\\overline{BC}$, and the first semicircle with its base parallel to $\\overline{AC}$. This sequence is c... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31ee73ae273ccf7d7184b8420c084ff17fd0ef2824db5d033fdb85ef15e05e06 | Triangle $ABC$ has side lengths $AB=2\sqrt5$, $BC=1$, and $CA=5$. Point $D$ is on side $AC$ such that $CD=1$, and point $F$ is such that $BF=2$ and $CF=3$. Let $E$ be the intersection of $AB$ and $DF$. Compute the area of quadrilateral $CDEB$. | [
"$\\frac{22}{35}$"
] | [
"Triangle $ABC$ has side lengths $AB=2\\sqrt5$, $BC=1$, and $CA=5$. Point $D$ is on side $AC$ such that $CD=1$, and point $F$ is such that $BF=2$ and $CF=3$. Let $E$ be the intersection of $AB$ and $DF$. Compute the area of quadrilateral $CDEB$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-bf26eb9ae5f082f130795a1a01666c4c8e7db1a522c7e4af57647855dd99049d | Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \neq C$ is a point of intersection of the circumcircles of $\triangle ACD$ and $\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expressed as $\tfrac{a+b\sqrt{c}}{d}$, where $a$, $b$... | [
"032",
"32"
] | [
"Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \\neq C$ is a point of intersection of the circumcircles of $\\triangle ACD$ and $\\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expressed as $\\tfrac{a+b\\sqrt{c}}{d}$, wher... | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"additional_problems",
"cot"
] | {
"additional_problems": 1,
"cot": 1
} | 2 | 0 | 0 |
omr-ws-ca9da0b853820c0547397823ac27acc9ff09f972b833912414146588cd01dff4 | Triangle $ABC$ has side lengths $AB=5$, $BC=6$, and $AC=7$. Two bugs start simultaneously from $A$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $D$. What is $BD$? | [
"4"
] | [
"Triangle $ABC$ has side lengths $AB=5$, $BC=6$, and $AC=7$. Two bugs start simultaneously from $A$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $D$. What is $BD$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-28e2ec9bf4b24977e8ac1c60248fabf2c5c2658dade15dd51ffa2db60389d404 | Triangle $ABC$ has side lengths $AB=5$, $BC=6$, and $CA=7$. The point $P$ lies inside the triangle, and let $D$, $E$, and $F$ be the feet of the perpendiculars from $P$ to $BC$, $CA$, and $AB$, respectively. Given that \[PA^2 + PB^2 + PC^2 + BD \cdot DC + CE \cdot EA + AF \cdot FB = \frac{3457}{55},\] find the product ... | [
"337"
] | [
"Triangle $ABC$ has side lengths $AB=5$, $BC=6$, and $CA=7$. The point $P$ lies inside the triangle, and let $D$, $E$, and $F$ be the feet of the perpendiculars from $P$ to $BC$, $CA$, and $AB$, respectively. Given that\n\n\\[PA^2 + PB^2 + PC^2 + BD \\cdot DC + CE \\cdot EA + AF \\cdot FB = \\frac{3457}{55},\\]\n\n... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-9aee7b934d9f46651c8857e9d5b6b8bb54dc33699b77f3547b8f62087b782ca2 | Triangle $ABC$ has side lengths $AB=5$, $BC=7$, and $AC=8$. Points $M$ and $N$ are chosen on $AB$ and $AC$, respectively. Given that the probability that triangle $AMN$ is an acute triangle can be expressed as $\frac{m}{n}$, what is $m+n$? | [
"231"
] | [
"Triangle $ABC$ has side lengths $AB=5$, $BC=7$, and $AC=8$. Points $M$ and $N$ are chosen on $AB$ and $AC$, respectively. Given that the probability that triangle $AMN$ is an acute triangle can be expressed as $\\frac{m}{n}$, what is $m+n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-173ab5f21f78437cdcfcf1676d5f1f68998b33197bff8e361f5073d1f80f5e95 | Triangle $ABC$ has side lengths $AB=5$, $BC=9$, and $AC=6$. Define the incircle of $ABC$ to be $C_1$. Then define $C_i$ for $i \geq 1$ to be externally tangent to $C_{i-1}$ and tangent to $AB$ and $BC$. Compute the sum of the areas of all circles $C_n$. | [
"\\(\\frac{8\\pi}{3}\\)"
] | [
"Triangle $ABC$ has side lengths $AB=5$, $BC=9$, and $AC=6$. Define the incircle of $ABC$ to be $C_1$. Then define $C_i$ for $i \\geq 1$ to be externally tangent to $C_{i-1}$ and tangent to $AB$ and $BC$. Compute the sum of the areas of all circles $C_n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-9bbf33e17d9087adeeca94b7d205ce2ff8cae56354c29b37fce5bd43de32123a | Triangle $ABC$ has side lengths $AB=65$, $BC=33$, and $AC=56$. Find the radius of the circle tangent to sides $AC$ and $BC$ and to the circumcircle of triangle $ABC$. | [
"24"
] | [
"Triangle $ABC$ has side lengths $AB=65$, $BC=33$, and $AC=56$. Find the radius of the circle tangent to sides $AC$ and $BC$ and to the circumcircle of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-bd39ec0188490ce54ebee8c3b4f7e791208d35ab592e0d89815cf9e350de3a0c | Triangle $ABC$ has side lengths $AB=7$, $BC=8$, and $CA=9$. Let $E$ be the foot of the perpendicular from $B$ to $AC$ and $F$ be the foot of the perpendicular from $C$ to $AB$. Denote $M$ as the midpoint of $BC$. The circumcircles of $\triangle BMF$ and $\triangle CME$ intersect at another point $G$. Compute the length... | [
"\\(\\frac{36}{7}\\)"
] | [
"Triangle $ABC$ has side lengths $AB=7$, $BC=8$, and $CA=9$. Let $E$ be the foot of the perpendicular from $B$ to $AC$ and $F$ be the foot of the perpendicular from $C$ to $AB$. Denote $M$ as the midpoint of $BC$. The circumcircles of $\\triangle BMF$ and $\\triangle CME$ intersect at another point $G$. Compute the... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 10
} | 27 | 0 | 0 |
omr-ws-12871092f8ea325557beb21aa15520ec3939eca61f00d62282a5dd922fd7b6c8 | Triangle $ABC$ has side lengths $AB=7, BC=8, CA=9.$ Define $M, N, P$ to be the midpoints of sides $BC, CA, AB,$ respectively. The circumcircles of $\triangle APN$ and $\triangle ABM$ intersect at another point $K.$ Find the length of $NK.$ | [
"\\(\\frac{17}{14}\\)"
] | [
"Triangle $ABC$ has side lengths $AB=7, BC=8, CA=9.$ Define $M, N, P$ to be the midpoints of sides $BC, CA, AB,$ respectively. The circumcircles of $\\triangle APN$ and $\\triangle ABM$ intersect at another point $K.$ Find the length of $NK.$"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-21cc760fc73076caad68d8391b9f084d1401d2177c1901621cc6ad9891b9bfa5 | Triangle $ABC$ has side lengths $AB=7, BC=8,$ and $CA=9.$ Circle $\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\omega_1$ and $\omega_2$ not equal to $A.$ Then $AK=\tfrac mn,$ where $m$ an... | [
"011",
"11"
] | [
"Triangle $ABC$ has side lengths $AB=7, BC=8,$ and $CA=9.$ Circle $\\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\\omega_1$ and $\\omega_2$ not equal to $A.$ Then $AK=\\tfrac mn,$ wh... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 62,
"genselect": 8,
"tir": 9
} | 79 | 0 | 0 |
omr-ws-fc38d80d824f31758defbf5d72b055e45eb456d8ea897e73c3a324f97fe6ecbf | Triangle $ABC$ has side lengths $AB=7, BC=8,$ and $CA=9.$ Circle $\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\omega_1$ and $\omega_2$ not equal to $A.$ Then $AK=\tfrac{m}{n},$ where $m$... | [
"011"
] | [
"Triangle $ABC$ has side lengths $AB=7, BC=8,$ and $CA=9.$ Circle $\\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\\omega_1$ and $\\omega_2$ not equal to $A.$ Then $AK=\\tfrac{m}{n},$... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 7,
"tir": 4
} | 58 | 0 | 0 |
omr-ws-2355a164d8d00b3a41d811b1cbfc0a134c410102d2077921c23cc252e38d7945 | Triangle $ABC$ has side lengths $BC = 5$, $AC = 12$, and $AB = 13$. Point $D$ is on $AB$ and point $E$ is on $AC$. If $DE$ divides triangle $ABC$ into two equal parts, then find the minimum length of $DE$. | [
"2\\sqrt{3}"
] | [
"Triangle $ABC$ has side lengths $BC = 5$, $AC = 12$, and $AB = 13$. Point $D$ is on $AB$ and point $E$ is on $AC$. If $DE$ divides triangle $ABC$ into two equal parts, then find the minimum length of $DE$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1fffc11e5abddf46de75fadb731b60898b2c11d8b7a327ca8d9c6008dbefa66 | Triangle $ABC$ has side lengths $a=BC$, $b=AC$, and $c=AB$. A circle is constructed outside of $\triangle ABC$ so that it is tangent to $\overline{BC}$, $\overrightarrow{AC}$, and $\overrightarrow{AB}$. Let the circle be tangent to $\overrightarrow{AC}$ at $D$. Express $AD$ in terms of $a$, $b$, and $c$. | [
"\\(\\frac{a + b + c}{2}\\)"
] | [
"Triangle $ABC$ has side lengths $a=BC$, $b=AC$, and $c=AB$. A circle is constructed outside of $\\triangle ABC$ so that it is tangent to $\\overline{BC}$, $\\overrightarrow{AC}$, and $\\overrightarrow{AB}$. Let the circle be tangent to $\\overrightarrow{AC}$ at $D$. Express $AD$ in terms of $a$, $b$, and $c$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 12
} | 26 | 0 | 0 |
omr-ws-db54a4ca3297e3fc5ebbbd6652bf83c6e6be5dfb277b7eddc58438507664d793 | Triangle $ABC$ has side-lengths $AB = 12, BC = 24,$ and $AC = 18.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N.$ What is the perimeter of $\triangle AMN?$ | [
"30"
] | [
"Triangle $ABC$ has side-lengths $AB = 12, BC = 24,$ and $AC = 18.$ The line through the incenter of $\\triangle ABC$ parallel to $\\overline{BC}$ intersects $\\overline{AB}$ at $M$ and $\\overline{AC}$ at $N.$ What is the perimeter of $\\triangle AMN?$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-4483c1a826450209a56e63b25376281e8aa841b9f6b7005696cf8d73e78d3f71 | Triangle $ABC$ has sides $AB = 13$, $BC = 14$, $CA = 15$. Construct squares $ADEB$, $BFGC$, $CHIA$ outside of triangle $ABC$. Determine the area of hexagon $DEFGHI$. | [
"926"
] | [
"Triangle $ABC$ has sides $AB = 13$, $BC = 14$, $CA = 15$. Construct squares $ADEB$, $BFGC$, $CHIA$ outside of triangle $ABC$. Determine the area of hexagon $DEFGHI$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 9
} | 13 | 0 | 0 |
omr-ws-7b155848121a7cf42ebd47e96fefb3824b3471e3272f674c0a711bc2aacc305f | Triangle $ABC$ has sides $AB = 5$, $AC = 4$, and $BC = 3$. Let $O$ be any arbitrary point inside $ABC$, and $D \in BC$, $E \in AC$, $F \in AB$, such that $OD \perp BC$, $OE \perp AC$, $OF \perp AB$. Find the minimum value of $OD^2 + OE^2 + OF^2$. | [
"\\dfrac{72}{25}"
] | [
"Triangle $ABC$ has sides $AB = 5$, $AC = 4$, and $BC = 3$. Let $O$ be any arbitrary point inside $ABC$, and $D \\in BC$, $E \\in AC$, $F \\in AB$, such that $OD \\perp BC$, $OE \\perp AC$, $OF \\perp AB$. Find the minimum value of $OD^2 + OE^2 + OF^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f142eb869de317bd9302f19218d8b86ad035f33bfc7a0b00b10da00512e891bb | Triangle $ABC$ has sides $AB= 7$, $AC= 8$, $BC= 5$. Points $D$ and $E$ are erected outside the triangle such that $\angle DAC = \angle ECB = 20^\circ$ and $\angle ADC = \angle CEB = 90^\circ$. Given that the circle with diameter $DE$ intersects line $AB$ at two distinct points $X_1$ and $X_2$, find $\min(CX_1, CX_2)$. | [
"\\(\\frac{20\\sqrt{3}}{7}\\)"
] | [
"Triangle $ABC$ has sides $AB= 7$, $AC= 8$, $BC= 5$. Points $D$ and $E$ are erected outside the triangle such that $\\angle DAC = \\angle ECB = 20^\\circ$ and $\\angle ADC = \\angle CEB = 90^\\circ$. Given that the circle with diameter $DE$ intersects line $AB$ at two distinct points $X_1$ and $X_2$, find $\\min(CX... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 7
} | 25 | 0 | 0 |
omr-ws-667efbcc6ae10375738cb88da9cf324f3f26f2f5fff5cd10364153bfe761dda8 | Triangle $ABC$ has sides $AB=13$, $BC=14$, and $AC=15$. Circle $w$ is drawn with $AB$ as its diameter. Circle $w$ intersects $BC$ and $AC$ at points $M$ and $N$ respectively. A line $L$ perpendicular to $AC$ and passing through $B$ is drawn. Extend $MN$ to meet $L$ at point $D$. Find the area of triangle $BDM$. | [
"\\(\\frac{84}{5}\\)"
] | [
"Triangle $ABC$ has sides $AB=13$, $BC=14$, and $AC=15$. Circle $w$ is drawn with $AB$ as its diameter. Circle $w$ intersects $BC$ and $AC$ at points $M$ and $N$ respectively. A line $L$ perpendicular to $AC$ and passing through $B$ is drawn. Extend $MN$ to meet $L$ at point $D$. Find the area of triangle $BDM$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 8
} | 21 | 0 | 0 |
omr-ws-d0c5c2dec081bc383f06b8de2bfb4ea5a8b45ac0560ed2b26a64bd3a818ece7e | Triangle $ABC$ has sides $AB=13$, $BC=14$, and $AC=15$. Points $M$ and $N$ are on $BC$ such that both triangles $ABM$ and $ACN$ each have an incircle with radius $2$. The line joining the two incenters intersects $AM$ and $AN$ at $E$ and $F$ respectively. The area of triangle $AEF$ can be expressed as $\frac{m}{n}$. Fi... | [
"187"
] | [
"Triangle $ABC$ has sides $AB=13$, $BC=14$, and $AC=15$. Points $M$ and $N$ are on $BC$ such that both triangles $ABM$ and $ACN$ each have an incircle with radius $2$. The line joining the two incenters intersects $AM$ and $AN$ at $E$ and $F$ respectively. The area of triangle $AEF$ can be expressed as $\\frac{m}{n... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 8,
"tir": 3
} | 57 | 0 | 0 |
omr-ws-201a5484d1f226744cc45c6ca98c365588e5af6db86c69bfeebed153f16d5c3d | Triangle $ABC$ has sides $AB=13$, $BC=14$, and $AC=15$. The angle bisector of $\angle A$ meets the circumcircle of triangle $ABC$ at $D$. Let $AD$ intersect $BC$ at $E$. A point $F$ is on the right side of $AD$ and lies on $BC$ such that $\angle ADF = \frac{1}{2} \angle A$. What is the area of $\triangle EDF$? | [
"\\(\\frac{13}{3}\\)"
] | [
"Triangle $ABC$ has sides $AB=13$, $BC=14$, and $AC=15$. The angle bisector of $\\angle A$ meets the circumcircle of triangle $ABC$ at $D$. Let $AD$ intersect $BC$ at $E$. A point $F$ is on the right side of $AD$ and lies on $BC$ such that $\\angle ADF = \\frac{1}{2} \\angle A$. What is the area of $\\triangle EDF$... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-f6480fe09ab37f0c01ee2eec33252e11d5e0f63b149f53f1c9b122822bfd8cbb | Triangle $ABC$ has sides $AB=5$, $BC=7$, and $AC=8$. A point $X$ is chosen in the interior of $ABC$ such that the triangle formed by the circumcenters of triangles $AXB$, $BXC$, and $AXC$ is equilateral. Find the area of this equilateral triangle. | [
"7\\sqrt{3}"
] | [
"Triangle $ABC$ has sides $AB=5$, $BC=7$, and $AC=8$. A point $X$ is chosen in the interior of $ABC$ such that the triangle formed by the circumcenters of triangles $AXB$, $BXC$, and $AXC$ is equilateral. Find the area of this equilateral triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-9021e7ea1d78da9e7937337650dfc74a9423b4ee916c533cbbee6bcede2ec8d2 | Triangle $ABC$ has sides $AB=9$, $BC = 5\sqrt{3}$, and $AC=12$. Points $A=P_0, P_1, P_2, \dots, P_{2450} = B$ are on segment $\overline{AB}$ with $P_k$ between $P_{k-1}$ and $P_{k+1}$ for $k=1,2,\dots,2449$, and points $A=Q_0, Q_1, Q_2, \dots, Q_{2450} = C$ for $k=1,2,\dots,2449$. Each segment $\overline{P_kQ_k}$ for $... | [
"20"
] | [
"Triangle $ABC$ has sides $AB=9$, $BC = 5\\sqrt{3}$, and $AC=12$. Points $A=P_0, P_1, P_2, \\dots, P_{2450} = B$ are on segment $\\overline{AB}$ with $P_k$ between $P_{k-1}$ and $P_{k+1}$ for $k=1,2,\\dots,2449$, and points $A=Q_0, Q_1, Q_2, \\dots, Q_{2450} = C$ for $k=1,2,\\dots,2449$. Each segment $\\overline{P_... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 8,
"tir": 3
} | 58 | 0 | 0 |
omr-ws-5f733ecebf80bb79757b2578ee47845c7c0596e0885f66780dfe7628646dc32e | Triangle $ABC$ has sides $BC=25$, $AC=20$, and $AB=15$. Point $P$ lies on $AC$ such that $PA=5$. If points $Q$ and $R$ lie on $AB$ and $BC$ respectively, what is the minimum perimeter of $\triangle PQR$? | [
"\\(8\\sqrt{10}\\)"
] | [
"Triangle $ABC$ has sides $BC=25$, $AC=20$, and $AB=15$. Point $P$ lies on $AC$ such that $PA=5$. If points $Q$ and $R$ lie on $AB$ and $BC$ respectively, what is the minimum perimeter of $\\triangle PQR$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-1c2b5a8ae7633d8d1628079d437dd94cac67c1969cec332eece9d2b1f04cd76d | Triangle $ABC$ has sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ of length 43, 13, and 48, respectively. Let $\omega$ be the circle circumscribed around $\triangle ABC$ and let $D$ be the intersection of $\omega$ and the perpendicular bisector of $\overline{AC}$ that is not on the same side of $\overline{... | [
"12"
] | [
"Triangle $ABC$ has sides $\\overline{AB}$, $\\overline{BC}$, and $\\overline{CA}$ of length 43, 13, and 48, respectively. Let $\\omega$ be the circle circumscribed around $\\triangle ABC$ and let $D$ be the intersection of $\\omega$ and the perpendicular bisector of $\\overline{AC}$ that is not on the same side of... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-421fd049ef97bbbae5774393d9468e5b8aefee44d1cf280a31291bf2074863b5 | Triangle $ABC$ has sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ of lengths $43$, $13$, and $48$, respectively. Let $\omega$ be the circumcircle of $\triangle ABC$ and let $D$ be the intersection of $\omega$ and the perpendicular bisector of $\overline{AC}$ that is not on the same side of $\overline{AC}$ ... | [
"12"
] | [
"Triangle $ABC$ has sides $\\overline{AB}$, $\\overline{BC}$, and $\\overline{CA}$ of lengths $43$, $13$, and $48$, respectively. Let $\\omega$ be the circumcircle of $\\triangle ABC$ and let $D$ be the intersection of $\\omega$ and the perpendicular bisector of $\\overline{AC}$ that is not on the same side of $\\o... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ce1e442460afb6ec4290d8b264600783001a59d4f96ef0b341ca7420168575a | Triangle $ABC$ has sides of $6$ units, $8$ units, and $10$ units. The width of a rectangle, whose area is equal to the area of the triangle, is $4$ units. What is the perimeter of this rectangle, in units? | [
"20"
] | [
"Triangle $ABC$ has sides of $6$ units, $8$ units, and $10$ units. The width of a rectangle, whose area is equal to the area of the triangle, is $4$ units. What is the perimeter of this rectangle, in units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dacd4c4a4e041920fccf4d2997216b565dfc551d854fe984455143cbf06c5af6 | Triangle $ABC$ has sides of length $13, 14, 15$. Find the area of the locus of all points $O$ for which the circle of radius 1 centered at $O$ intersects triangle $ABC$ exactly 3 times. | [
"42"
] | [
"Triangle $ABC$ has sides of length $13, 14, 15$. Find the area of the locus of all points $O$ for which the circle of radius 1 centered at $O$ intersects triangle $ABC$ exactly 3 times."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-64ff2d7e02c50f980fd9d0e40b03acf2ed1ba299038582e123077984770e9e1b | Triangle $ABC$ has sides of length $3$, $2$, and $\sqrt{19}$. Find its area (in simplest form, try not to use Heron's formula). | [
"\\(\\frac{3\\sqrt{3}}{2}\\)"
] | [
"Triangle $ABC$ has sides of length $3$, $2$, and $\\sqrt{19}$. Find its area (in simplest form, try not to use Heron's formula)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-65d8a7cb48446575d64c49a593185facfb98950ff674a796aed2e79e801da719 | Triangle $ABC$ has sides of length 13, 14, and 15. Set $S$ is the set of all points $O$ for which the circle of radius $1$ centered at $O$ intersects $ABC$ exactly 3 times. The points in $S$ enclose $n$ distinct finite regions, the sum of whose areas can be expressed in the form $\dfrac{p-q\pi}{r}$, where $p$, $q$, and... | [
"32"
] | [
"Triangle $ABC$ has sides of length 13, 14, and 15. Set $S$ is the set of all points $O$ for which the circle of radius $1$ centered at $O$ intersects $ABC$ exactly 3 times. The points in $S$ enclose $n$ distinct finite regions, the sum of whose areas can be expressed in the form $\\dfrac{p-q\\pi}{r}$, where $p$, $... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9fd8cf7bc60c2f2297d9be6894f43267bc511b5b62d1a5221681fc2969d967cb | Triangle $ABC$ has sides of length 5, 12 and 13 units, and triangle $DEF$ has sides of length 8, 15 and 17 units. What is the ratio of the area of triangle $ABC$ to the area of triangle $DEF$? Express your answer as a common fraction. | [
"\\dfrac{1}{2}"
] | [
"Triangle $ABC$ has sides of length 5, 12 and 13 units, and triangle $DEF$ has sides of length 8, 15 and 17 units. What is the ratio of the area of triangle $ABC$ to the area of triangle $DEF$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb4344f07a4cafe5326733678e949464d78869a419a4532441298e920e0e6e0a | Triangle $ABC$ has sides of lengths $3$, $4$, and $5$. Place a point $D$ randomly and uniformly inside the triangle. What is the expected value of the sum of the perpendicular distances from point $D$ to the triangle’s three sides? | [
"\\(\\frac{47}{15}\\)"
] | [
"Triangle $ABC$ has sides of lengths $3$, $4$, and $5$. Place a point $D$ randomly and uniformly inside the triangle. What is the expected value of the sum of the perpendicular distances from point $D$ to the triangle’s three sides?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
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