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-429e3a9e256a0012a83584f963ce9e78d35cfa369971535ddfd780eb00dafb7c | Triangle $ABC$ lies in the cartesian plane and has an area of $70$. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20),$ respectively, and the coordinates of $A$ are $(p,q).$ The line containing the median to side $BC$ has slope $-5.$ Find the largest possible value of $p+q.$ [asy]defaultpen(fontsize(8)); size(1... | [
"47"
] | [
"Triangle $ABC$ lies in the cartesian plane and has an area of $70$. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20),$ respectively, and the coordinates of $A$ are $(p,q).$ The line containing the median to side $BC$ has slope $-5.$ Find the largest possible value of $p+q.$\n[asy]defaultpen(fontsize(8)); ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b17313efa3098db180bfdcd931957e40a34bbdefd601939107c3f5ad904191e9 | Triangle $ABC$ satisfies $AB = 5$ and $AC = 7$. The incircle of $\triangle ABC$ is tangent to $\overline{AB}$ at $F$. Given that the reflection of $F$ over $\overline{BC}$ lies on the circumcircle of $\triangle ABC$, find the square of the area of $\triangle ABC$. | [
"300"
] | [
"Triangle $ABC$ satisfies $AB = 5$ and $AC = 7$. The incircle of $\\triangle ABC$ is tangent to $\\overline{AB}$ at $F$. Given that the reflection of $F$ over $\\overline{BC}$ lies on the circumcircle of $\\triangle ABC$, find the square of the area of $\\triangle ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 19
} | 40 | 0 | 0 |
omr-ws-958ee8966318b2e632316af0d2dab07a0377cb30fb97c5d6ed3c9f70ae1bb8d1 | Triangle $ABC$ satisfies $AB = 7$, $BC = 12$, and $CA = 13$. If the area of $ABC$ can be expressed in the form $m\sqrt{n}$, where $n$ is not divisible by the square of a prime, then determine $m + n$. | [
"27"
] | [
"Triangle $ABC$ satisfies $AB = 7$, $BC = 12$, and $CA = 13$. If the area of $ABC$ can be expressed in the form $m\\sqrt{n}$, where $n$ is not divisible by the square of a prime, then determine $m + n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-43003d98154eedadf4a56cedbe52bbac3f5c4982078d9441bc33577bb0e8bbbd | Triangle $ABC$ with $AB < BC$ is inscribed in a circle $\omega$ and circumscribed about a circle $\gamma$ with center $I$. A line $\ell$ parallel to $AC$ touches the circle $\gamma$ and intersects the arcs $BAC$ and $BCA$ at points $P$ and $Q$, respectively. Given that $PQ = 2BI$, calculate the value of $\frac{AP + 2PB... | [
"2"
] | [
"Triangle $ABC$ with $AB < BC$ is inscribed in a circle $\\omega$ and circumscribed about a circle $\\gamma$ with center $I$. A line $\\ell$ parallel to $AC$ touches the circle $\\gamma$ and intersects the arcs $BAC$ and $BCA$ at points $P$ and $Q$, respectively. Given that $PQ = 2BI$, calculate the value of $\\fra... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-020c537dcc63ee87dce8d57c27e6dfebc2d0c5d84c067b1fd78617138b10585e | Triangle $ABC$ with $AB = AC$ is inscribed in circle $o$. Circles $o_1$ and $o_2$ are internally tangent to circle $o$ at points $P$ and $Q$, respectively, and they are tangent to segments $AB$ and $AC$, respectively, and are disjoint from the interior of triangle $ABC$. Let $m$ be a line tangent to circles $o_1$ and $... | [
"\\dfrac{1}{3}"
] | [
"Triangle $ABC$ with $AB = AC$ is inscribed in circle $o$. Circles $o_1$ and $o_2$ are internally tangent to circle $o$ at points $P$ and $Q$, respectively, and they are tangent to segments $AB$ and $AC$, respectively, and are disjoint from the interior of triangle $ABC$. Let $m$ be a line tangent to circles $o_1$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-66d7f59f13d0139a0cb8bb4538c6fe78a67b09f1221267cf7c91fd7938e22a5e | Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? | [
"75"
] | [
"Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\\overline{AB}$, and let $E$ be the midpoint of $\\overline{AC}$. The angle bisector of $\\angle BAC$ intersects $\\overline{DE}$ and $\\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-980c23272ea97772735ba6a8c545b2d1e3f7934b18ba30d4691d6d47eecff235 | Triangle $ABC$ with $\angle ACB > 90^\circ$ is inscribed in a circle with center $S$. Line $CS$ intersects segment $AB$ at point $D$. Given that $AC + BC = 2CS$, calculate the ratio of the radii of the incircles of triangles $ADC$ and $BDC$. | [
"1"
] | [
"Triangle $ABC$ with $\\angle ACB > 90^\\circ$ is inscribed in a circle with center $S$. Line $CS$ intersects segment $AB$ at point $D$. Given that $AC + BC = 2CS$, calculate the ratio of the radii of the incircles of triangles $ADC$ and $BDC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-70e2eae08c5ed5acaff9406a95e94d8e9d2d6ea64c78799f3f9256c07e19c8e3 | Triangle $ABC$ with $\measuredangle A = 90^\circ$ has incenter $I$. A circle centered at $I$ and passing through $A$ intersects $\overline{BC}$ at points $E$ and $F$ such that $BE < BF$. Given that $\tfrac{BE}{EF} = \tfrac{2}{3}$, find $\tfrac{CF}{FE}$ in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime ... | [
"7"
] | [
"Triangle $ABC$ with $\\measuredangle A = 90^\\circ$ has incenter $I$. A circle centered at $I$ and passing through $A$ intersects $\\overline{BC}$ at points $E$ and $F$ such that $BE < BF$. Given that $\\tfrac{BE}{EF} = \\tfrac{2}{3}$, find $\\tfrac{CF}{FE}$ in the form $\\tfrac{m}{n}$, where $m$ and $n$ are relat... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 1
} | 20 | 0 | 0 |
omr-ws-a98152da218c036bc47c01342e49ccdd9902730c2ebc7ddb184175064f6f4c31 | Triangle $ABC$ with an obtuse angle $C$ is inscribed in a circle with center $O$. It is given that $AC + BC = 2OC$. The segment $OC$ intersects the side $AB$ at point $D$. Let $r_1$ and $r_2$ be the radii of the inscribed circles of triangles $ADC$ and $BDC$, respectively. Calculate the value of $r_1 + r_2$ if $OC = 10... | [
"5(\\sqrt{3} - 1)"
] | [
"Triangle $ABC$ with an obtuse angle $C$ is inscribed in a circle with center $O$. It is given that $AC + BC = 2OC$. The segment $OC$ intersects the side $AB$ at point $D$. Let $r_1$ and $r_2$ be the radii of the inscribed circles of triangles $ADC$ and $BDC$, respectively. Calculate the value of $r_1 + r_2$ if $OC... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-c852bb2df9b4edd2d95c24256c97b0c65c214e4a165e1947cf53c8ea9e143ef9 | Triangle $ABC$ with circumcircle $\Gamma$ has side lengths $AB=8$, $BC=6$, and $AC=4$. Let $D$ be a point on side $\overline{BC}$ such that there exists a circle internally tangent to $\Gamma$ at $A$ and tangent to $\overline{BC}$ at $D$. Let $E$ be a point on minor arc $\widehat{BC}$ of $\Gamma$ such that the length $... | [
"129"
] | [
"Triangle $ABC$ with circumcircle $\\Gamma$ has side lengths $AB=8$, $BC=6$, and $AC=4$. Let $D$ be a point on side $\\overline{BC}$ such that there exists a circle internally tangent to $\\Gamma$ at $A$ and tangent to $\\overline{BC}$ at $D$. Let $E$ be a point on minor arc $\\widehat{BC}$ of $\\Gamma$ such that t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-23ea810acb5fdb0e886032147f7f46c0db11bdc9fa756672b0b076d3a5e0cc9e | Triangle $ABC$ with circumcircle $\omega$ has orthocenter $H$ which lies on $\omega,$ and $H$ is closer to $B$ than to either of $A$ and $C.$ The reflection of $C$ over the midpoint $M$ of $AB$ is $C'.$ If $BC'$ intersects $\omega$ again at $D \neq B$ and quadrilateral $AMDC'$ is cyclic, find $\angle BAC.$ | [
"\\(\\arctan\\left(\\frac{\\sqrt{2}}{2}\\right)\\)"
] | [
"Triangle $ABC$ with circumcircle $\\omega$ has orthocenter $H$ which lies on $\\omega,$ and $H$ is closer to $B$ than to either of $A$ and $C.$ The reflection of $C$ over the midpoint $M$ of $AB$ is $C'.$ If $BC'$ intersects $\\omega$ again at $D \\neq B$ and quadrilateral $AMDC'$ is cyclic, find $\\angle BAC.$"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 7
} | 9 | 0 | 0 |
omr-ws-3f908b86a67563d9e9c11dcc29bace7e63e252ef406c2de4959ca0f4a90973d7 | Triangle $ABC$ with right angle at $C$, $\angle BAC < 45^\circ$ and $AB = 4$. Point $P$ on $\overline{AB}$ is chosen such that $\angle APC = 2\angle ACP$ and $CP = 1$. The ratio $\frac{AP}{BP}$ can be represented in the form $p + q\sqrt{r}$, where $p$, $q$, $r$ are positive integers and $r$ is not divisible by the squa... | [
"007",
"7"
] | [
"Triangle $ABC$ with right angle at $C$, $\\angle BAC < 45^\\circ$ and $AB = 4$. Point $P$ on $\\overline{AB}$ is chosen such that $\\angle APC = 2\\angle ACP$ and $CP = 1$. The ratio $\\frac{AP}{BP}$ can be represented in the form $p + q\\sqrt{r}$, where $p$, $q$, $r$ are positive integers and $r$ is not divisi... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 3,
"tir": 2
} | 30 | 0 | 0 |
omr-ws-70ac81560f5aa0b261f57e8ab9c186627842c9450d1314508b6c92bb8425ccf3 | Triangle $ABC$ with vertices $A(-2, 0)$, $B(1, 4)$ and $C(-3, 2)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $C$ to $C'$? | [
"6"
] | [
"Triangle $ABC$ with vertices $A(-2, 0)$, $B(1, 4)$ and $C(-3, 2)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $C$ to $C'$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-88d4e07d27c2c31b189eebadd62da28c77f7c2203040b547c81a102115b45317 | Triangle $ABC$ with vertices $A(1, -3)$, $B(-2, 0)$ and $C(4, 3)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $A$ to $A'$? | [
"2"
] | [
"Triangle $ABC$ with vertices $A(1, -3)$, $B(-2, 0)$ and $C(4, 3)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $A$ to $A'$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ed994cb57e23e12dfad6f919aac2def0a113936c82e2fae971ceb5d5f56a442 | Triangle $ABC$ with vertices of $A(6,2)$, $B(2,5)$, and $C(2,2)$ is reflected over the x-axis to triangle $A'B'C'$. This triangle is reflected over the y-axis to triangle $A''B''C''$. What are the coordinates of point $C''$? | [
"(-2, -2)"
] | [
"Triangle $ABC$ with vertices of $A(6,2)$, $B(2,5)$, and $C(2,2)$ is reflected over the x-axis to triangle $A'B'C'$. This triangle is reflected over the y-axis to triangle $A''B''C''$. What are the coordinates of point $C''$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ecdc957a07e1de40a8eccbbc549305edfeab7bcc99b67cc6b342f3a182c1f852 | Triangle $ABC$, right-angled at $A$, has circumcircle $\Gamma$. Point $D$ on arc $\widehat{AB}$ and point $E$ on arc $\widehat{AC}$ of $\Gamma$ are such that $AD=AE$. Lines $BD$ and $CE$ meet at $K$. The tangents to $\Gamma$ at $D$ and $E$ meet at $T$. If the circumcircle of $\triangle DKT$ meets $\Gamma$ again at $M$,... | [
"\\dfrac{9}{13}"
] | [
"Triangle $ABC$, right-angled at $A$, has circumcircle $\\Gamma$. Point $D$ on arc $\\widehat{AB}$ and point $E$ on arc $\\widehat{AC}$ of $\\Gamma$ are such that $AD=AE$. Lines $BD$ and $CE$ meet at $K$. The tangents to $\\Gamma$ at $D$ and $E$ meet at $T$. If the circumcircle of $\\triangle DKT$ meets $\\Gamma$ a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-55aa26118ac38c3b519b0d2ac48448c393d89aa65e1646d733b97c79ad4dd6d3 | Triangle $ABC$, with obtuse angle $B$, is inscribed in a circle. Altitude $\overline{CH}$ of the triangle is tangent to the circle at point $C$. If $AB=6$ and $BH=8$, then the area of the circle can be written as $k\pi$. Find the value of $k$. | [
"121"
] | [
"Triangle $ABC$, with obtuse angle $B$, is inscribed in a circle. Altitude $\\overline{CH}$ of the triangle is tangent to the circle at point $C$. If $AB=6$ and $BH=8$, then the area of the circle can be written as $k\\pi$. Find the value of $k$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-50ef61e679d781fc4e535d66ff9b8c538f779bdf64a44ebd52bfc3aab55e9ded | Triangle $ABC^{}_{}$ has $AB=9^{}_{}$ and $BC: AC=40: 41^{}_{}$. What's the largest area that this triangle can have? | [
"820"
] | [
"Triangle $ABC^{}_{}$ has $AB=9^{}_{}$ and $BC: AC=40: 41^{}_{}$. What's the largest area that this triangle can have?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-9e243468e9a091303e8940dac5641cbf17c4fd4f0d1197790fb06aed55a5d709 | Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n \geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\... | [
"182"
] | [
"Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\\overline{AB}$, and for $n \\geq 2$, let $C_n$ be the foot of the altitude to $\\overline{C_{n-2}B}$ in $\\triangle C_{n-2}C_{n-1}B$. T... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-5f15fb45d35c4dd497cfb2e3608441454ba05dc30ecbe90be355f3154ae4101c | Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n \geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\... | [
"182"
] | [
"Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\\overline{AB}$, and for $n \\geq 2$, let $C_n$ be the foot of the altitude to $\\overline{C_{n-2}B}$ in $\\triangle C_{n-2}C_{n-1}B$. T... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50679dd07e9f80f17b9052d3a26b698ac13f7c565c0fa21c97a8ee9f1455ddf6 | Triangle $ABC_1$ has an area of $24$. For all $n > 1$, let point $C_n$ be the centroid of triangle $ABC_{n-1}$. Let $X_n$ be the area of the quadrilateral $C_nAC_{n+1}B$. Compute $\left( \sum^{\infty}_{i=0} X_{2i+1}\right)^2$. | [
"324"
] | [
"Triangle $ABC_1$ has an area of $24$. For all $n > 1$, let point $C_n$ be the centroid of triangle $ABC_{n-1}$. Let $X_n$ be the area of the quadrilateral $C_nAC_{n+1}B$. Compute $\\left( \\sum^{\\infty}_{i=0} X_{2i+1}\\right)^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 1
} | 24 | 0 | 0 |
omr-ws-f7f61dbaed2c24334cbea4a14b859b7caf3d453d897a2d0f7d2f6bc76067b931 | Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1... | [
"961"
] | [
"Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\\overline{AB_{n-1}}$ and $\\overline{AC_{n-1}}$, respectively, creating three similar triangles $\\triangle AB_nC_n \\sim \\triangle B_{n-1}C_nC_{n-1} \\sim \\tria... | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 83,
"genselect": 14,
"tir": 19
} | 116 | 0 | 0 |
omr-ws-2871627be2f67be369ded48d9cadcb4a174d500e02988e440a52338658ee7e23 | Triangle $AOB$ is rotated $90^\circ$ around point $O$ to form triangle $A_1OB_1$ ($A$ maps to $A_1$, $B$ maps to $B_1$). Let $M$ be the midpoint of segment $AB_1$ and let $N$ be the midpoint of segment $A_1B$. If $\overrightarrow{OM} = (x, y)$ and $\overrightarrow{A_1N} = (a, b)$, find the value of $ax + by$. | [
"0"
] | [
"Triangle $AOB$ is rotated $90^\\circ$ around point $O$ to form triangle $A_1OB_1$ ($A$ maps to $A_1$, $B$ maps to $B_1$). Let $M$ be the midpoint of segment $AB_1$ and let $N$ be the midpoint of segment $A_1B$. If $\\overrightarrow{OM} = (x, y)$ and $\\overrightarrow{A_1N} = (a, b)$, find the value of $ax + by$."
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4d311cf9519c69041360f4ecf8b55badcbb2f9979bb56efe792fbb34c859a1b1 | Triangle $A_1A_2A_3$ is right-angled at vertex $A_3$. For all $n \ge 3$, define $A_{n+1}$ as the foot of the altitude from $A_n$ to $A_{n-1}A_{n-2}$. If $A_1$ and $A_3$ are kept fixed, find the locus of point $P$ as $A_2$ varies. | [
"\\( A_3 \\)"
] | [
"Triangle $A_1A_2A_3$ is right-angled at vertex $A_3$. For all $n \\ge 3$, define $A_{n+1}$ as the foot of the altitude from $A_n$ to $A_{n-1}A_{n-2}$. If $A_1$ and $A_3$ are kept fixed, find the locus of point $P$ as $A_2$ varies."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-86480663866228dd793bad633efa3b81a5351a95974b19b7887dc1fc5d53aa5a | Triangle $A_1A_2A_3$ is right-angled at vertex $A_3$. For all $n \ge 3$, define $A_{n+1}$ as the foot of the altitude from $A_n$ to $A_{n-1}A_{n-2}$. Let $P$ be the unique point that lies in all of the triangles $A_{i-1}A_iA_{i+1}$. If the coordinates of $A_1$, $A_2$, and $A_3$ are $(0,0)$, $(1,0)$, and $(0,1)$ respect... | [
"\\left(\\dfrac{1}{3}, \\dfrac{1}{3}\\right)"
] | [
"Triangle $A_1A_2A_3$ is right-angled at vertex $A_3$. For all $n \\ge 3$, define $A_{n+1}$ as the foot of the altitude from $A_n$ to $A_{n-1}A_{n-2}$. Let $P$ be the unique point that lies in all of the triangles $A_{i-1}A_iA_{i+1}$. If the coordinates of $A_1$, $A_2$, and $A_3$ are $(0,0)$, $(1,0)$, and $(0,1)$ r... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a836f6864939114275b259b096bb1a842501ce2b2496968a8d084241c357b5d7 | Triangle $A_1B_1C_1$ is the orthogonal projection of a regular triangle $ABC$ with side length $1$. Given that $A_1B_1 < \frac{\sqrt{3}}{2}$ and $A_1C_1 < \frac{1}{2}$, calculate the minimum possible value of the cosine of the angle between the planes $ABC$ and $A_1B_1C_1$. | [
"\\sqrt{\\dfrac{\\sqrt{3} - 1}{3}}"
] | [
"Triangle $A_1B_1C_1$ is the orthogonal projection of a regular triangle $ABC$ with side length $1$. Given that $A_1B_1 < \\frac{\\sqrt{3}}{2}$ and $A_1C_1 < \\frac{1}{2}$, calculate the minimum possible value of the cosine of the angle between the planes $ABC$ and $A_1B_1C_1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc280ca7c0d700e7c0e1f44c0ed42e1385b48b31bfc98eec7a48a835c7840ddf | Triangle $BAD$ is right-angled at $B$. On $AD$ there is a point $C$ for which $AC=CD$ and $AB=BC$. What is the magnitude of angle $DAB$ in degrees? | [
"60"
] | [
"Triangle $BAD$ is right-angled at $B$. On $AD$ there is a point $C$ for which $AC=CD$ and $AB=BC$. What is the magnitude of angle $DAB$ in degrees?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f4f2ea69d2b085a21d95e2d79d82c85018420378dfbc97ce26f6781217c26dc4 | Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA=DC$ and $AC$ is the bisector of $\angle{DAB}$. Point $E$ is chosen so that $EA=ED$ and $AD$ is the bisector of $\angle{EAC}$. Let $M$ be the midpoint of $CF$. Le... | [
"\\left( \\dfrac{2 - \\sqrt{2}}{2}, \\dfrac{\\sqrt{2}}{2} \\right)"
] | [
"Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA=DC$ and $AC$ is the bisector of $\\angle{DAB}$. Point $E$ is chosen so that $EA=ED$ and $AD$ is the bisector of $\\angle{EAC}$. Let $M$ be the midpoint of $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-11d34f9bce301c4090c67c680d6d4edbf4f31976572047f679789de23c33f77c | Triangle $CAP$ is an isosceles triangle with base $15$ and $\cos(\angle CAP) = \frac{-1}{8}$. Let $M$ and $N$ be the feet of the medians from $P$ and $A$, respectively. Let $T$ be a point on the circumcircle of triangle $CAP$ such that $TN$ is tangent to the circumcircle, and $Q$ be the foot of the perpendicular from $... | [
"\\dfrac{15}{4}"
] | [
"Triangle $CAP$ is an isosceles triangle with base $15$ and $\\cos(\\angle CAP) = \\frac{-1}{8}$. Let $M$ and $N$ be the feet of the medians from $P$ and $A$, respectively. Let $T$ be a point on the circumcircle of triangle $CAP$ such that $TN$ is tangent to the circumcircle, and $Q$ be the foot of the perpendicula... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-c301dde4e6e963cd0af128063708661bd46bb087aad6d0d5e6c920177ef2308c | Triangle $DEF$ has $m \angle D = 2 \times m \angle E$. Find the least possible perimeter of $\triangle DEF$ if the side lengths are integers. | [
"66"
] | [
"Triangle $DEF$ has $m \\angle D = 2 \\times m \\angle E$. Find the least possible perimeter of $\\triangle DEF$ if the side lengths are integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-db363d3118bbdfeea2cedf5e13eb385055fccc32fe57b7fe9d20dfc2ba22ce6b | Triangle $DEF$ is the circumcevian triangle of a point $P$ with respect to triangle $ABC$. If $K_3K_6$ cuts $BC$ at $X_1$, $K_1K_4$ cuts $CA$ at $Y_1$, and $K_2K_5$ cuts $AB$ at $Z_1$, then let $\ell_1$ be the line passing through $X_1, Y_1, Z_1$. If $P$ is the orthocenter of triangle $ABC$, calculate the ratio $\frac{... | [
"\\dfrac{1}{4}"
] | [
"Triangle $DEF$ is the circumcevian triangle of a point $P$ with respect to triangle $ABC$. If $K_3K_6$ cuts $BC$ at $X_1$, $K_1K_4$ cuts $CA$ at $Y_1$, and $K_2K_5$ cuts $AB$ at $Z_1$, then let $\\ell_1$ be the line passing through $X_1, Y_1, Z_1$. If $P$ is the orthocenter of triangle $ABC$, calculate the ratio $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-65d57260a46357bf1df41f61913d56f769f4382019b710d567df1e18eab122a2 | Triangle $DEF$ is the circumcevian triangle of a point $P$ with respect to triangle $ABC$. If $K_3K_6$ cuts $EF$ at $X_2$, $K_1K_4$ cuts $FD$ at $Y_2$, and $K_2K_5$ cuts $DE$ at $Z_2$, calculate the ratio $\frac{X_2Y_2}{Y_2Z_2}$ given that $X_2, Y_2, Z_2$ are collinear on line $\ell_2$. | [
"1"
] | [
"Triangle $DEF$ is the circumcevian triangle of a point $P$ with respect to triangle $ABC$. If $K_3K_6$ cuts $EF$ at $X_2$, $K_1K_4$ cuts $FD$ at $Y_2$, and $K_2K_5$ cuts $DE$ at $Z_2$, calculate the ratio $\\frac{X_2Y_2}{Y_2Z_2}$ given that $X_2, Y_2, Z_2$ are collinear on line $\\ell_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c06135b83aa20328664e208dc2e79e74aad1776ee983bff233d4fe20b5c58809 | Triangle $HJK$ is isosceles with $HJ = HK$ and $JK = 4\sqrt{5}$. The coordinates of $H$ are $(-4, 1)$, $J$ are $(j, 15)$ where $j < 0$, and $K$ are $(6, k)$. $M$ is the midpoint of $JK$. The gradient of $HM$ is $2$. Find the values of $j$ and $k$. | [
"\\( j = -2 \\), \\( k = 11 \\)"
] | [
"Triangle $HJK$ is isosceles with $HJ = HK$ and $JK = 4\\sqrt{5}$. The coordinates of $H$ are $(-4, 1)$, $J$ are $(j, 15)$ where $j < 0$, and $K$ are $(6, k)$. $M$ is the midpoint of $JK$. The gradient of $HM$ is $2$. Find the values of $j$ and $k$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-f8371c726860de308fcee7db0c2be4f6df8b69df9c51b1e8e1296981267f44c1 | Triangle $KMN$ is cut from triangle $ABC$ so that segment $KM$ lies on one side $AB$, and lines $KN$ and $MN$ are parallel to sides $BC$ and $AC$, respectively. The perimeter of the resulting hexagon is $20\%$ larger than the perimeter of triangle $ABC$. If the same triangle $KMN$ is cut out with side $KN$ attached to ... | [
"81"
] | [
"Triangle $KMN$ is cut from triangle $ABC$ so that segment $KM$ lies on one side $AB$, and lines $KN$ and $MN$ are parallel to sides $BC$ and $AC$, respectively. The perimeter of the resulting hexagon is $20\\%$ larger than the perimeter of triangle $ABC$. If the same triangle $KMN$ is cut out with side $KN$ attach... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-046e8f9b81700073f68f4f21afa169aca0a406e7e55c473e220364232b95841e | Triangle $MNO$ has side lengths $MN = 11$, $NO = 21$, and $MO = 23$. The points $E$ and $F$ trisect $MO$ such that $ME < MF$. Segments $NE$ and $NF$ are extended to points $E'$ and $F'$ such that $ME' \parallel NF$ and $OF' \parallel NE$. The area of pentagon $MNOF'E'$ is $p\sqrt{q}$, where $p$ and $q$ are relatively p... | [
"217"
] | [
"Triangle $MNO$ has side lengths $MN = 11$, $NO = 21$, and $MO = 23$. The points $E$ and $F$ trisect $MO$ such that $ME < MF$. Segments $NE$ and $NF$ are extended to points $E'$ and $F'$ such that $ME' \\parallel NF$ and $OF' \\parallel NE$. The area of pentagon $MNOF'E'$ is $p\\sqrt{q}$, where $p$ and $q$ are rela... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 4
} | 27 | 0 | 0 |
omr-ws-dce091f02a7998ede726fb411dde1bff4d0d719131ab29dcfb70ac1225ed4d0f | Triangle $MOH$ is equilateral with side length $6$. Let triangle $ABC$ be such that $O$ and $H$ are its circumcenter and orthocenter, and $M$ is the midpoint of side $\overline{BC}$. If the area of $\triangle{ABC}$ can be expressed as $a\sqrt{b}$ for positive integers $a$ and $b$ with $b$ squarefree, then find $a+b$. | [
"092"
] | [
"Triangle $MOH$ is equilateral with side length $6$. Let triangle $ABC$ be such that $O$ and $H$ are its circumcenter and orthocenter, and $M$ is the midpoint of side $\\overline{BC}$. If the area of $\\triangle{ABC}$ can be expressed as $a\\sqrt{b}$ for positive integers $a$ and $b$ with $b$ squarefree, then find ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-d3231bb13e35ef42845c9339260fc3af5885cb1e9d3011758be22c4127004cb8 | Triangle $O_1MO_2$ is an isosceles right triangle with a right angle at $M$ and $O_1M = 2$. The midpoints of $O_1M$, $MO_2$, and $O_2O_1$, together with point $M$, form a square $MATH$. The midpoints of $MA$, $AT$, $TH$, and $HM$ are $P$, $Q$, $R$, and $S$, respectively. The midpoints of $MP$, $PQ$, $QR$, $RS$, and $SM... | [
"\\(\\frac{13}{32}\\)"
] | [
"Triangle $O_1MO_2$ is an isosceles right triangle with a right angle at $M$ and $O_1M = 2$. The midpoints of $O_1M$, $MO_2$, and $O_2O_1$, together with point $M$, form a square $MATH$. The midpoints of $MA$, $AT$, $TH$, and $HM$ are $P$, $Q$, $R$, and $S$, respectively. The midpoints of $MP$, $PQ$, $QR$, $RS$, an... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 24
} | 42 | 0 | 0 |
omr-ws-94278075f55a115613e713006dd08f3b165752c31775bcfd511453969cbe38fa | Triangle $PQR$ has sides $PQ$, $PR$, and $QR$ with lengths $7$, $8$, and $9$, respectively. The interior angle bisector of angle $R$ intersects the incircle of $\triangle PQR$ at points $S$ and $G$. Find the square of the distance $SG$. | [
"20"
] | [
"Triangle $PQR$ has sides $PQ$, $PR$, and $QR$ with lengths $7$, $8$, and $9$, respectively. The interior angle bisector of angle $R$ intersects the incircle of $\\triangle PQR$ at points $S$ and $G$. Find the square of the distance $SG$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-e8926fd6da1945378e0d0e14cfa30509a1f0a9f29f9b2a1ee6149ebfc503a645 | Triangle $QRS$ has the following properties: - $Q$ is a right angle. - $9\sin S = 8 \cot R$. - The length of one of the legs is 13. Let the two possible areas of the triangle be $a$ and $b$. The expression $\left|\frac{1}{a} - \frac{1}{b}\right|$ can be written as $\frac{t \sqrt{u}}{v}$, where $t$ and $v$ are relativel... | [
"48"
] | [
"Triangle $QRS$ has the following properties:\n- $Q$ is a right angle.\n- $9\\sin S = 8 \\cot R$.\n- The length of one of the legs is 13.\n\nLet the two possible areas of the triangle be $a$ and $b$. The expression $\\left|\\frac{1}{a} - \\frac{1}{b}\\right|$ can be written as $\\frac{t \\sqrt{u}}{v}$, where $t$ an... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5deaafe9d0167dc158ce4c4735316e1ae3ca3f4241e97e5c6160b5826ebc9793 | Triangle $RHS$ has $RH = HS = 25$ and $RS = 14$. Points $A$ and $B$ are placed on $RH$ and $HS$ respectively, and lines are drawn from $A$ and $B$ perpendicular to $RS$. These lines intersect $RS$ at $P$ and $Q$ respectively. If $PQ = 3$ and $AP - BQ = 5$, then $QS$ can be written in the form $i/j$ where $i$ and $j$ ar... | [
"277"
] | [
"Triangle $RHS$ has $RH = HS = 25$ and $RS = 14$. Points $A$ and $B$ are placed on $RH$ and $HS$ respectively, and lines are drawn from $A$ and $B$ perpendicular to $RS$. These lines intersect $RS$ at $P$ and $Q$ respectively. If $PQ = 3$ and $AP - BQ = 5$, then $QS$ can be written in the form $i/j$ where $i$ and $... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-8a0b8b4ae437cc25f20c88a185a3a053d5bf62b7002da7c2739b122a2d8f76a4 | Triangle $T$ has side lengths $1$, $2$, and $\sqrt{7}$. It turns out that one can arrange three copies of triangle $T$ to form two equilateral triangles, one inside the other. Compute the ratio of the area of the outer equilateral triangle to the area of the inner equilateral triangle. | [
"7"
] | [
"Triangle $T$ has side lengths $1$, $2$, and $\\sqrt{7}$. It turns out that one can arrange three copies of triangle $T$ to form two equilateral triangles, one inside the other. Compute the ratio of the area of the outer equilateral triangle to the area of the inner equilateral triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-16804a15cd01728c9296058f27d58173e87080ffaf6c86383621abdd9943e6ff | Triangle $TWO$ has centroid $H$ and midpoints $U, N, D$ of $\overline{TW}, \overline{WO}, \overline{TO}$ respectively. If $TW = 25, WO = 17, TO = 26$, find the area of $\triangle HUT$. | [
"51"
] | [
"Triangle $TWO$ has centroid $H$ and midpoints $U, N, D$ of $\\overline{TW}, \\overline{WO}, \\overline{TO}$ respectively. If $TW = 25, WO = 17, TO = 26$, find the area of $\\triangle HUT$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-33d7b96fc76fdc73b1407c35599394db0055732a6a84f72294ab6875acd1513b | Triangle $XYZ$ has $XY = 5$, $YZ = 6$, and $XZ = 7$. The altitude drawn from $Y$ to $XZ$ has length $\frac{\sqrt{p}}{q}$ where $p$ and $q$ are positive integers. What is $p + q$? | [
"871"
] | [
"Triangle $XYZ$ has $XY = 5$, $YZ = 6$, and $XZ = 7$. The altitude drawn from $Y$ to $XZ$ has length $\\frac{\\sqrt{p}}{q}$ where $p$ and $q$ are positive integers. What is $p + q$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-be9bdce326441ba6a68cee55c500ad7da962578f7c229017a36307b17a44e8c9 | Triangle $XYZ$ has all angles less than $90^{\circ}$ with orthocenter $H$. A point $P$ is randomly chosen on side $YZ$ and points $C$ and $D$ are then chosen on $XY$ and $XZ$ such that $PY=PC$ and $PD=PZ$. If $\angle CHD = 180^{\circ} - \angle X$, find the value of $\cos(\angle X)$ given that $\angle Y = 60^{\circ}$ an... | [
"\\dfrac{\\sqrt{2}}{2}"
] | [
"Triangle $XYZ$ has all angles less than $90^{\\circ}$ with orthocenter $H$. A point $P$ is randomly chosen on side $YZ$ and points $C$ and $D$ are then chosen on $XY$ and $XZ$ such that $PY=PC$ and $PD=PZ$. If $\\angle CHD = 180^{\\circ} - \\angle X$, find the value of $\\cos(\\angle X)$ given that $\\angle Y = 60... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f2261da66232efe1e5a42ecb7ecb5030b50f483d0da601d1366cfbbf81185cfa | Triangle $XYZ$ has sides $XY=3$, $YZ=4$, and $ZX=5$. Let $M$ be the midpoint of $ZX$ and let $\ell$ be the line passing through points $M$ and $Y$. Triangle $XYZ$ is then reflected about $\ell$ to form triangle $X'YZ'$. Compute the area of the union of triangles $XYZ$ and $X'YZ'$. | [
"\\dfrac{106}{13}"
] | [
"Triangle $XYZ$ has sides $XY=3$, $YZ=4$, and $ZX=5$. Let $M$ be the midpoint of $ZX$ and let $\\ell$ be the line passing through points $M$ and $Y$. Triangle $XYZ$ is then reflected about $\\ell$ to form triangle $X'YZ'$. Compute the area of the union of triangles $XYZ$ and $X'YZ'$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-2251208baff1bd10a06685024be550de4b2b3cfb175f7c4bcbf4418f5753c16b | Triangle $XYZ$ is inside square $KLMN$ with the area of the square being $1$. The vertices of the triangle divide three sides of the square in the following ratios: $KX : XL = 3 : 2$ $KY : YN = 4 : 1$ $NZ : ZM = 2 : 3$ What is the area of triangle $XYZ$? | [
"\\(\\frac{11}{50}\\)"
] | [
"Triangle $XYZ$ is inside square $KLMN$ with the area of the square being $1$. The vertices of the triangle divide three sides of the square in the following ratios:\n$KX : XL = 3 : 2$\n$KY : YN = 4 : 1$\n$NZ : ZM = 2 : 3$\nWhat is the area of triangle $XYZ$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-e7751810b4b8877b02556424ea204918721a50f8fb8076d3494d701e93ea73ae | Triangle $XYZ$ with $\angle XYZ > 90^\circ$ is inscribed in a unit circle centered at $O$. Let $Z_1$ be the reflection of $Z$ over $XY$. Let $Z_1Y$ intersect $OX$ at $R$ and $XZ_1$ intersect $OY$ at $V$. Define point $S$ on $XZ$ such that $XR = RS \times ZZ_1$. If $OZ_1 \perp XZ$ and $\frac{OS}{SR} = \frac{3}{4}$, the ... | [
"40"
] | [
"Triangle $XYZ$ with $\\angle XYZ > 90^\\circ$ is inscribed in a unit circle centered at $O$. Let $Z_1$ be the reflection of $Z$ over $XY$. Let $Z_1Y$ intersect $OX$ at $R$ and $XZ_1$ intersect $OY$ at $V$. Define point $S$ on $XZ$ such that $XR = RS \\times ZZ_1$. If $OZ_1 \\perp XZ$ and $\\frac{OS}{SR} = \\frac{3... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c3747671178178d8ca7f50efc39cca0f03f1335d7012077d276e65891e73b2c0 | Triangle $\Delta ABC$ has $AB=3$, $BC=4$, and a right angle at $B$. Equilateral triangle $ACD$ is placed such that $\Delta ABC \bigcap \Delta ACD=\emptyset$. Given that $BD^2=a-b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime, find $a+b+c$. | [
"40"
] | [
"Triangle $\\Delta ABC$ has $AB=3$, $BC=4$, and a right angle at $B$. Equilateral triangle $ACD$ is placed such that $\\Delta ABC \\bigcap \\Delta ACD=\\emptyset$. Given that $BD^2=a-b\\sqrt{c}$ where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime, find $a+b+c$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-e3069bf1a38d3083835b50d18897f4cfc3845918dc745da628c5dccafdab21b1 | Triangle $\Delta ABC$ is inscribed in circle $\Omega$. The tangency point of $\Omega$ and the $A$-mixtilinear circle of $\Delta ABC$ is $T$. Points $E$, $F$ are chosen on $AC$, $AB$ respectively so that $EF \parallel BC$ and $(TEF)$ is tangent to $\Omega$. Let $\omega$ denote the $A$-excircle of $\Delta AEF$, which is ... | [
"\\dfrac{1}{2}"
] | [
"Triangle $\\Delta ABC$ is inscribed in circle $\\Omega$. The tangency point of $\\Omega$ and the $A$-mixtilinear circle of $\\Delta ABC$ is $T$. Points $E$, $F$ are chosen on $AC$, $AB$ respectively so that $EF \\parallel BC$ and $(TEF)$ is tangent to $\\Omega$. Let $\\omega$ denote the $A$-excircle of $\\Delta AE... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-2699c346ac5f7e7aca15af2d14e2c0443f0069bd26e65263bef0feef3e39feaf | Triangle $\Delta GRB$ is dissected into 25 small triangles. The vertices are painted in three colors such that vertex $G$ is green, vertex $R$ is red, and vertex $B$ is blue. Each vertex on side $GR$ is either green or red, each vertex on side $RB$ is either red or blue, and each vertex on side $GB$ is either green or ... | [
"1"
] | [
"Triangle $\\Delta GRB$ is dissected into 25 small triangles. The vertices are painted in three colors such that vertex $G$ is green, vertex $R$ is red, and vertex $B$ is blue. Each vertex on side $GR$ is either green or red, each vertex on side $RB$ is either red or blue, and each vertex on side $GB$ is either gre... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-8b8b4f20b718fcb45c873faa442ac77022ae268bb62e4638dccf1100ddcba5fe | Triangle $\bigtriangleup ABC$ has side lengths $AB = 13$, $BC = 14$, and $AC = 15$. Let $P$ be a point on line segment $BC$, and $H_1$ and $H_2$ be the orthocenters of triangles $\bigtriangleup ABP$ and $\bigtriangleup ACP$ respectively. There are exactly two possible points $P$ such that $H_1H_2 = 1$. The distance bet... | [
"19"
] | [
"Triangle $\\bigtriangleup ABC$ has side lengths $AB = 13$, $BC = 14$, and $AC = 15$. Let $P$ be a point on line segment $BC$, and $H_1$ and $H_2$ be the orthocenters of triangles $\\bigtriangleup ABP$ and $\\bigtriangleup ACP$ respectively. There are exactly two possible points $P$ such that $H_1H_2 = 1$. The dist... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-764d5637498bf7ae42dccc51992829ebe68c2173efe87805dc0d71ba3d3752d0 | Triangle $\overline{ABC}$ has two sides of length $p$ inches that meet at a $36^{\circ}$ angle. The length of $\left|\overline{BC}\right|$ is 1 inch. Find the length of the angle bisector of $\angle C$. | [
"1"
] | [
"Triangle $\\overline{ABC}$ has two sides of length $p$ inches that meet at a $36^{\\circ}$ angle. The length of $\\left|\\overline{BC}\\right|$ is 1 inch. Find the length of the angle bisector of $\\angle C$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-8a9f0117556b1c1f5ed6b12b10b4606d9fc1fd82f9c49e1c239201c1720eeb00 | Triangle $\triangle ABC$ has $AB = \sqrt{10}$, $BC = \sqrt{17}$, and $CA = \sqrt{41}$. The area of $\triangle ABC$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. | [
"13"
] | [
"Triangle $\\triangle ABC$ has $AB = \\sqrt{10}$, $BC = \\sqrt{17}$, and $CA = \\sqrt{41}$. The area of $\\triangle ABC$ can be expressed in the form $\\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-f66cf0810e922b35c6747abeffc85dc25becb67efa017eba6ef0183980742713 | Triangle $\triangle ABC$ has $AB=11,$ $BC=13,$ and $AC=20.$ A circle is drawn with diameter $AC.$ Line $AB$ intersects the circle at $D \neq A$ and line $BC$ intersects the circle at $E \neq B.$ What is the length of $DE$? | [
"\\(\\frac{100}{13}\\)"
] | [
"Triangle $\\triangle ABC$ has $AB=11,$ $BC=13,$ and $AC=20.$ A circle is drawn with diameter $AC.$ Line $AB$ intersects the circle at $D \\neq A$ and line $BC$ intersects the circle at $E \\neq B.$ What is the length of $DE$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-e0d20492127b6feaf62c402b37975f8b1265003dbcc317ad818e9be7eef01e21 | Triangle $\triangle ABC$ has $AB=3$, $AC=4$, and $BC=5$. Pairwise externally tangent circles $\Gamma_A$, $\Gamma_B$, and $\Gamma_C$ are constructed such that each $\Gamma_P$ is internally tangent to the circumcircle of $\triangle ABC$ at point $P$. If the minimum of the sum of the radii of $\Gamma_A$, $\Gamma_B$, and $... | [
"20"
] | [
"Triangle $\\triangle ABC$ has $AB=3$, $AC=4$, and $BC=5$. Pairwise externally tangent circles $\\Gamma_A$, $\\Gamma_B$, and $\\Gamma_C$ are constructed such that each $\\Gamma_P$ is internally tangent to the circumcircle of $\\triangle ABC$ at point $P$. If the minimum of the sum of the radii of $\\Gamma_A$, $\\Ga... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6fb49c9da1e55a80d542b54781ac8fbe1cbb9ef07295a9185368d453805746c9 | Triangle $\triangle ABC$ has $BC=6$, $AB = 9$, and $AC=10$. Let $H$ be its orthocenter. Suppose the circle with diameter $AH$ intersects $AB$ and $AC$ for the second time at $X$ and $Y$ respectively. Then $XY$ can be written in the form $\tfrac mn$ where $m$ and $n$ are relatively prime positive integers. What is $m-n$... | [
"23"
] | [
"Triangle $\\triangle ABC$ has $BC=6$, $AB = 9$, and $AC=10$. Let $H$ be its orthocenter. Suppose the circle with diameter $AH$ intersects $AB$ and $AC$ for the second time at $X$ and $Y$ respectively. Then $XY$ can be written in the form $\\tfrac mn$ where $m$ and $n$ are relatively prime positive integers. What i... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-fd3fb23478f42a4cd8f35130ab0b873390cdd11f246c931cd560d8efc85b7821 | Triangle $\triangle ABC$ has $\angle A = 45^\circ$, $\angle B = 60^\circ$, $\angle C = 75^\circ$, with $BC = 1$. $\overline{BC}$ is extended to a point $D$ such that $\overline{AC}$ is the angle bisector of $\angle BAD$. The value of $\frac{[ABC]}{[ABD]}$ can be written in the form $\frac{\sqrt{a} - b}{c}$, where $a$, ... | [
"312"
] | [
"Triangle $\\triangle ABC$ has $\\angle A = 45^\\circ$, $\\angle B = 60^\\circ$, $\\angle C = 75^\\circ$, with $BC = 1$. $\\overline{BC}$ is extended to a point $D$ such that $\\overline{AC}$ is the angle bisector of $\\angle BAD$. The value of $\\frac{[ABC]}{[ABD]}$ can be written in the form $\\frac{\\sqrt{a} - b... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 10
} | 32 | 0 | 0 |
omr-ws-493f2fcb639c294275fc590ccb578266d04aec4c905b337dee36db5bb9fe54eb | Triangle $\triangle ABC$ has a perimeter of 4. Points $X$ and $Y$ lie on rays $AB$ and $AC$, respectively, such that $AX = AY = 1$. Segments $BC$ and $XY$ intersect at point $M$. If $P_1$ and $P_2$ are the perimeters of $\triangle ABM$ and $\triangle ACM$ respectively, find the value of $P_1 + P_2$. | [
"6"
] | [
"Triangle $\\triangle ABC$ has a perimeter of 4. Points $X$ and $Y$ lie on rays $AB$ and $AC$, respectively, such that $AX = AY = 1$. Segments $BC$ and $XY$ intersect at point $M$. If $P_1$ and $P_2$ are the perimeters of $\\triangle ABM$ and $\\triangle ACM$ respectively, find the value of $P_1 + P_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-a4c8d4ba0537982dee1b71cf92f97fd17a750eb7d559b332ef58ac64a2f1604b | Triangle $\triangle ABC$ has a right angle at $A$, and $\angle B = 30^\circ$. Cevians $AN$ and $CM$ are such that $BN = 3$, $CN = 7$, and $\angle MAN = \angle MCN$. The quantity $AN \cdot CM$ can be expressed as $a\sqrt{b}$ where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Com... | [
"29"
] | [
"Triangle $\\triangle ABC$ has a right angle at $A$, and $\\angle B = 30^\\circ$. Cevians $AN$ and $CM$ are such that $BN = 3$, $CN = 7$, and $\\angle MAN = \\angle MCN$. The quantity $AN \\cdot CM$ can be expressed as $a\\sqrt{b}$ where $a$ and $b$ are positive integers and $b$ is not divisible by the square of an... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 4
} | 51 | 0 | 0 |
omr-ws-99a5a438a7d4084727e40a9dde07afcaf8ea32e7da0c2e1d80eaaf5dc71238f0 | Triangle $\triangle ABC$ has a right angle at $C$, $\angle A = 60^\circ$, and $AC=10$. Find the radius of the incircle of $\triangle ABC$. | [
"5(\\sqrt{3}-1)"
] | [
"Triangle $\\triangle ABC$ has a right angle at $C$, $\\angle A = 60^\\circ$, and $AC=10$. Find the radius of the incircle of $\\triangle ABC$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cde9bfeb3eef072868e79ad241ce34bc1adfb824ac61b346bff74a6333038acf | Triangle $\triangle ABC$ has an area of $10$. Points $D$, $E$, and $F$ are on sides $AB$, $BC$, and $CA$ respectively, with $AD = 2$ and $DB = 3$. If triangle $\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then what is that area? | [
"6"
] | [
"Triangle $\\triangle ABC$ has an area of $10$. Points $D$, $E$, and $F$ are on sides $AB$, $BC$, and $CA$ respectively, with $AD = 2$ and $DB = 3$. If triangle $\\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then what is that area?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 38,
"genselect": 5
} | 43 | 0 | 0 |
omr-ws-f420303227b8cc82a5f2cafd03d33044d7bcedc02b13551c42061aab2208c6a7 | Triangle $\triangle ABC$ has area $10$. Points $D$, $E$, and $F$, all distinct from $A$, $B$, and $C$, are on sides $AB$, $BC$, and $CA$ respectively, and $AD = 2$, $DB = 3$. If triangle $\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then that area is? | [
"6"
] | [
"Triangle $\\triangle ABC$ has area $10$. Points $D$, $E$, and $F$, all distinct from $A$, $B$, and $C$, are on sides $AB$, $BC$, and $CA$ respectively, and $AD = 2$, $DB = 3$. If triangle $\\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then that area is?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-68f5b4b28750524d59cb14e028fec0e335d8cfdfe931eb6306fc2fbeafe44842 | Triangle $\triangle ABC$ has dimensions $BC = a$, $CA = b$, $AB = c$. If $a, b, c$ forms a geometric progression, and $\sin(\angle B - \angle A)$, $\sin(\angle A)$, $\sin(\angle C)$ forms an arithmetic progression, determine $\cos(\angle B)$. | [
"\\(\\frac{\\sqrt{5} - 1}{2}\\)"
] | [
"Triangle $\\triangle ABC$ has dimensions $BC = a$, $CA = b$, $AB = c$. If $a, b, c$ forms a geometric progression, and $\\sin(\\angle B - \\angle A)$, $\\sin(\\angle A)$, $\\sin(\\angle C)$ forms an arithmetic progression, determine $\\cos(\\angle B)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-fab9fd13ead41013049b686621b00fbd1c6a4ba6cf8b5eb2dd3ce726ff4bd650 | Triangle $\triangle ABC$ has medians $\overline{AM}$ and $\overline{BN}$. Point $P$ lies on $AC$ such that triangle $\triangle AMP$ is right with $m\angle APM = \tfrac{\pi}{2}$. A square $PQRS$ is inscribed in triangle $\triangle AMP$ with $Q$ on $\overline{AC}$, $R$ on $\overline{AM}$, and $S$ on $\overline{MP}$. All ... | [
"no_valid_answer_found"
] | [
"Triangle $\\triangle ABC$ has medians $\\overline{AM}$ and $\\overline{BN}$. Point $P$ lies on $AC$ such that triangle $\\triangle AMP$ is right with $m\\angle APM = \\tfrac{\\pi}{2}$. A square $PQRS$ is inscribed in triangle $\\triangle AMP$ with $Q$ on $\\overline{AC}$, $R$ on $\\overline{AM}$, and $S$ on $\\ove... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d2b77f413bcc1bf0ddfaa632944f1e41e4f90bf30ba4aa975815846262e24a3c | Triangle $\triangle ABC$ has side lengths $AB = 13$, $BC = 14$, and $CA = 15$. The incircle of $\triangle ABC$ has center $I$, and line $AI$ is extended to intersect the circumcircle $\omega$ at point $P$. What is the area of quadrilateral $ABPC$? | [
"\\( 108 \\frac{1}{3} \\)"
] | [
"Triangle $\\triangle ABC$ has side lengths $AB = 13$, $BC = 14$, and $CA = 15$. The incircle of $\\triangle ABC$ has center $I$, and line $AI$ is extended to intersect the circumcircle $\\omega$ at point $P$. What is the area of quadrilateral $ABPC$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72ac4cd8b3a66fc27f33bfff455e477b0d473d334244b988043dad3275debe6e | Triangle $\triangle ABC$ has sides $\overline{AB}$, $\overline{AC}$, and $\overline{BC}$ of length $8$, $15$, and $17$, respectively. A point $P$ is randomly chosen inside of $\triangle ABC$. What is the probability that $\overline{BP}$ is longer than $\overline{CP}$? | [
"\\dfrac{289}{900}"
] | [
"Triangle $\\triangle ABC$ has sides $\\overline{AB}$, $\\overline{AC}$, and $\\overline{BC}$ of length $8$, $15$, and $17$, respectively. A point $P$ is randomly chosen inside of $\\triangle ABC$. What is the probability that $\\overline{BP}$ is longer than $\\overline{CP}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7dbfdeb760c5781bc9e90924607db9c758f127f3ec99cb05a562cb9840234b43 | Triangle $\triangle ABC$ is given. Let $\Gamma$ be its circumscribed circle, and $\gamma$ be its Euler circle. The Euler line of triangle $\triangle ABC$ intersects $\gamma$ at points $X$ and $Y$. Tangents to $\gamma$ at $X$ and $Y$ intersect $\Gamma$ at pairs of points $M, N$ and $R, S$. Let $P$ be the intersection po... | [
"5"
] | [
"Triangle $\\triangle ABC$ is given. Let $\\Gamma$ be its circumscribed circle, and $\\gamma$ be its Euler circle. The Euler line of triangle $\\triangle ABC$ intersects $\\gamma$ at points $X$ and $Y$. Tangents to $\\gamma$ at $X$ and $Y$ intersect $\\Gamma$ at pairs of points $M, N$ and $R, S$. Let $P$ be the int... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9d4fd8620ea1e53f1f770a05bac1f75593d3286b629b67b38474668eaaf52fca | Triangle $\triangle ABC$ is inscribed in a circle $\omega$ of radius $12$ so that $\angle B = 68^\circ$ and $\angle C = 64^\circ$. The perpendicular from $A$ to $BC$ intersects $\omega$ at $D$, and the angle bisector of $\angle B$ intersects $\omega$ at $E$. What is the value of $DE^2$? | [
"432"
] | [
"Triangle $\\triangle ABC$ is inscribed in a circle $\\omega$ of radius $12$ so that $\\angle B = 68^\\circ$ and $\\angle C = 64^\\circ$. The perpendicular from $A$ to $BC$ intersects $\\omega$ at $D$, and the angle bisector of $\\angle B$ intersects $\\omega$ at $E$. What is the value of $DE^2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-eb1da568133049c66ded8c80a217c3a2c0e076cea539c447074cee7042240465 | Triangle $\triangle ABC$ is inscribed in circle $\Omega$. Let $I$ denote its incenter and $I_A$ its $A$-excenter. Let $N$ denote the midpoint of arc $BAC$. Line $NI_A$ meets $\Omega$ a second time at $T$. The perpendicular to $AI$ at $I$ meets sides $AC$ and $AB$ at $E$ and $F$ respectively. The circumcircle of $\trian... | [
"1"
] | [
"Triangle $\\triangle ABC$ is inscribed in circle $\\Omega$. Let $I$ denote its incenter and $I_A$ its $A$-excenter. Let $N$ denote the midpoint of arc $BAC$. Line $NI_A$ meets $\\Omega$ a second time at $T$. The perpendicular to $AI$ at $I$ meets sides $AC$ and $AB$ at $E$ and $F$ respectively. The circumcircle of... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fa174f630b99ab16892d7fe8c3b6a4d8b2baec7c30d67d27805d01dce84e8686 | Triangle $\triangle A_0B_0C_0$ has $\angle A_0 = 40^\circ$, $\angle B_0 = 60^\circ$, and $\angle C_0 = 80^\circ$. It is rotated about vertex $A_0$ clockwise by some angle $\alpha$ to get $\triangle A_1B_1C_1$. Then, $\triangle A_1B_1C_1$ is rotated clockwise about $B_1$ by some angle $\beta$ to get $\triangle A_2B_2C_2... | [
"80"
] | [
"Triangle $\\triangle A_0B_0C_0$ has $\\angle A_0 = 40^\\circ$, $\\angle B_0 = 60^\\circ$, and $\\angle C_0 = 80^\\circ$. It is rotated about vertex $A_0$ clockwise by some angle $\\alpha$ to get $\\triangle A_1B_1C_1$. Then, $\\triangle A_1B_1C_1$ is rotated clockwise about $B_1$ by some angle $\\beta$ to get $\\t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-36b3e932d1dbfc6b3a93827ca30cd68b03d276da71d4adb03b7e2d393e354209 | Triangle $\triangle MTE$ has $MT = 13$, $TE = 14$, $ME = 15$. $\gamma$ is the orthocenter of $\triangle MTE$, and $\omega$ is the circumcenter of $\triangle \gamma TE$. Triangles $\triangle T\gamma\omega$ and $\triangle E\gamma\omega$ have centroids at $C_1$ and $C_2$, respectively. The distance between $C_1$ and $C_2$... | [
"17"
] | [
"Triangle $\\triangle MTE$ has $MT = 13$, $TE = 14$, $ME = 15$. $\\gamma$ is the orthocenter of $\\triangle MTE$, and $\\omega$ is the circumcenter of $\\triangle \\gamma TE$. Triangles $\\triangle T\\gamma\\omega$ and $\\triangle E\\gamma\\omega$ have centroids at $C_1$ and $C_2$, respectively. The distance betwee... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-00648e2a9d1661485da5323e47458c33992560bbb1d6aa6059f9c3b8ca12f251 | Triangle $\vartriangle ABC$ has $AB = 7$, $BC = 8$, and $CA = 9$. The angle bisector of $\angle B$ cuts the incircle of $\vartriangle ABC$ with center $O$ at $D$. The perpendicular to $\overline{OA}$ passing through the midpoint of $\overline{AB}$ hits $BC$ at $E$. What is the area of $\vartriangle ODE$? | [
"\\(\\frac{25\\sqrt{14}}{14}\\)"
] | [
"Triangle $\\vartriangle ABC$ has $AB = 7$, $BC = 8$, and $CA = 9$. The angle bisector of $\\angle B$ cuts the incircle of $\\vartriangle ABC$ with center $O$ at $D$. The perpendicular to $\\overline{OA}$ passing through the midpoint of $\\overline{AB}$ hits $BC$ at $E$. What is the area of $\\vartriangle ODE$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-1eeb578b7c6bf35fdb799882c3ee99da1d43a90614c7db2d2948feb913da4279 | Triangle $\vartriangle ABC$ has side lengths $AB = 51$, $BC = 52$, and $CA = 53$. For all points $P$ in its interior, denote $d(P)$ to be the sum of the distances from $P$ to the sides of $\vartriangle ABC$. Consider the region in which $d(P) \ge \frac{772}{17}$. The area of this region can be represented as $\frac{m}{... | [
"853"
] | [
"Triangle $\\vartriangle ABC$ has side lengths $AB = 51$, $BC = 52$, and $CA = 53$. For all points $P$ in its interior, denote $d(P)$ to be the sum of the distances from $P$ to the sides of $\\vartriangle ABC$. Consider the region in which $d(P) \\ge \\frac{772}{17}$. The area of this region can be represented as $... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-750c83b2321ff1a0f3ba3f662c2d8cfc66e02f3e2ce6d127eb8974fd86444009 | Triangle 1 has sides of lengths $2$, $3$, and $\frac{9}{2}$. Triangle 2 has the same angles as Triangle 1 and has two sides of lengths $2$ and $3$ (not necessarily in the same order as the sides of Triangle 1). Determine the two possible lengths of the remaining side of Triangle 2. | [
"\\(\\frac{9}{2}\\) and \\(\\frac{4}{3}\\)"
] | [
"Triangle 1 has sides of lengths $2$, $3$, and $\\frac{9}{2}$. Triangle 2 has the same angles as Triangle 1 and has two sides of lengths $2$ and $3$ (not necessarily in the same order as the sides of Triangle 1). Determine the two possible lengths of the remaining side of Triangle 2."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 12
} | 13 | 0 | 0 |
omr-ws-a5fdb1c93c8b927c2e7888bad91cd0409d7abfd032225ac208956324aadbf97a | Triangle ABC and ABD are isosceles with AB = AC = BD. AC intersects BD at E, and AC is perpendicular to BD. Find the sum of angles C and D. | [
"\\(135^\\circ\\)"
] | [
"Triangle ABC and ABD are isosceles with AB = AC = BD. AC intersects BD at E, and AC is perpendicular to BD. Find the sum of angles C and D."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 15
} | 29 | 0 | 0 |
omr-ws-b897879d369eedd05f6cc38296676fd0dd28f18a6e34c2581e29413c0b9d01a9 | Triangle ABC covers 90% of the area of a circle, with the diameter of the circle serving as the altitude of the triangle. The covered region has an area of 87 square units. If the base of Triangle ABC is 13 units, what is the area of the triangle to the nearest whole number? | [
"72"
] | [
"Triangle ABC covers 90% of the area of a circle, with the diameter of the circle serving as the altitude of the triangle. The covered region has an area of 87 square units. If the base of Triangle ABC is 13 units, what is the area of the triangle to the nearest whole number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-40410d9e7e10375d24f0fdef3708621746a8cb8f8ec56cf45d43acc9f6bf65c6 | Triangle ABC has $AC=3$, $BC=4$, $AB=5$. Point D is on AB, and CD bisects the right angle. The inscribed circles of triangles ADC and BCD have radii $r_a$ and $r_b$ respectively. What is $r_a/r_b$? | [
"\\dfrac{3(10 - \\sqrt{2})}{28}"
] | [
"Triangle ABC has $AC=3$, $BC=4$, $AB=5$. Point D is on AB, and CD bisects the right angle. The inscribed circles of triangles ADC and BCD have radii $r_a$ and $r_b$ respectively. What is $r_a/r_b$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-bbddcfb2e349b9ad829c676e8ea335a322c593320f92e1ab6e626b82d7513fcd | Triangle ABC has AB = 13, BC = 14, and AC = 15. The internal angle bisector of A intersects BC at D. A perpendicular from C intersects the extension of AD at E. Find CE. | [
"\\(\\frac{40}{3}\\)"
] | [
"Triangle ABC has AB = 13, BC = 14, and AC = 15. The internal angle bisector of A intersects BC at D. A perpendicular from C intersects the extension of AD at E. Find CE."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04978dd2c083154e7a8acef9c18c04e58ef13f272dfe439576d9a231ba64122f | Triangle ABC has AB = 5 units, BC = 12 units, and a right angle at B. A circle with its center on side AB is tangent to AC and BC. What is its radius? | [
"\\(\\frac{12}{5}\\)"
] | [
"Triangle ABC has AB = 5 units, BC = 12 units, and a right angle at B. A circle with its center on side AB is tangent to AC and BC. What is its radius?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-af9e42db924910fac9549bdeea45e972cc6151430baebd5598ce448f52e2a67b | Triangle ABC has AC = AB and angle ACB = 96 degrees. Point D is within triangle ABC such that angle DAB = 30 degrees. Determine the measure of angle ACD in radians, expressed in terms of pi in simplest terms. | [
"\\dfrac{7\\pi}{30}"
] | [
"Triangle ABC has AC = AB and angle ACB = 96 degrees. Point D is within triangle ABC such that angle DAB = 30 degrees. Determine the measure of angle ACD in radians, expressed in terms of pi in simplest terms."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-c339dd0e84bf464a938da814710f265ffd06a5e902bb53561aaeac8814f1aeea | Triangle ABC has AC = BC and ∠ACB = 106 degrees. M is a point inside the triangle such that ∠MAC = 7 degrees and ∠MCA = 23 degrees. Find ∠CMB. | [
"83"
] | [
"Triangle ABC has AC = BC and ∠ACB = 106 degrees. M is a point inside the triangle such that ∠MAC = 7 degrees and ∠MCA = 23 degrees. Find ∠CMB."
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 14
} | 46 | 0 | 0 |
omr-ws-385da7b0843de71d0e699e5e308cbe7845497daa2755827ba57a4d1b14772761 | Triangle ABC has \( AB = 5 \) and \( AC = 6 \) and is inscribed in a circle of radius \( 5 \). What is the length of \( BC \)? | [
"\\( 3\\sqrt{3} + 4 \\) or \\( 3\\sqrt{3} - 4 \\)"
] | [
"Triangle ABC has \\( AB = 5 \\) and \\( AC = 6 \\) and is inscribed in a circle of radius \\( 5 \\). What is the length of \\( BC \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 8
} | 11 | 0 | 0 |
omr-ws-4b02bfa389abbc8114b979f1d31c9537d1f29e188029ba29bb23bd4159144a6b | Triangle ABC has a perimeter of 10, and the lengths of its sides are all integers. If \( a \) is the length of side BC, what is the difference between the largest and smallest possible values of \( a \)? | [
"2"
] | [
"Triangle ABC has a perimeter of 10, and the lengths of its sides are all integers. If \\( a \\) is the length of side BC, what is the difference between the largest and smallest possible values of \\( a \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3433e22d9bf65223bdbddc92eca7e2c8f6a5277126e8e47e974fb1ec976a6dcd | Triangle ABC has a perimeter of 152, and angle BAC is a right angle. A circle with radius 19 and center O on AB is drawn so that it is tangent to AC and BC. Given that OP = m/n, where m and n are relatively prime positive integers, find m + n. | [
"98"
] | [
"Triangle ABC has a perimeter of 152, and angle BAC is a right angle. A circle with radius 19 and center O on AB is drawn so that it is tangent to AC and BC. Given that OP = m/n, where m and n are relatively prime positive integers, find m + n."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c831c1bfccf08fc738ce68dd41ef6fe1173ea94fa8d8cf1c1b66fee8ab918ab3 | Triangle ABC has a right angle at B, with AB = 3 and BC = 4. If D and E are points on AC and BC, respectively, such that CD = DE = 5/3, find the perimeter of quadrilateral ABED. | [
"10.4"
] | [
"Triangle ABC has a right angle at B, with AB = 3 and BC = 4. If D and E are points on AC and BC, respectively, such that CD = DE = 5/3, find the perimeter of quadrilateral ABED."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af72f6cb36ae40684bbf6ea28ad2b16eb88c9ed349faadf156dfbd18ed4b0dd9 | Triangle ABC has a right angle at C. Point P lies on side AB, and segment CP intersects the incircle of triangle ABC at points X and Y, with X closer to C than Y. If CX = XY = YP = 1, compute the area of triangle ABC. | [
"\\( 18 + \\frac{48\\sqrt{7}}{7} \\)"
] | [
"Triangle ABC has a right angle at C. Point P lies on side AB, and segment CP intersects the incircle of triangle ABC at points X and Y, with X closer to C than Y. If CX = XY = YP = 1, compute the area of triangle ABC."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-a4bf9d4319b0a7b75b5feb97c2469a05db2e0fbf933df5472a49ea1b3c71189b | Triangle ABC has an area of 48. Let P be the midpoint of median AM, and let N be the midpoint of side AB. If G is the intersection of MN and BP, find the area of triangle MGP. | [
"4"
] | [
"Triangle ABC has an area of 48. Let P be the midpoint of median AM, and let N be the midpoint of side AB. If G is the intersection of MN and BP, find the area of triangle MGP."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-02f7d069a049dfe17fab5e8d94661c9469ff182a89923f17b3ffd460cc8d75b2 | Triangle ABC has an area of 80. Points K, M, O on side AB and points L, N, P on side AC are such that AK = KM = MO = OB and AL = LN = NP = PC. Find the area of quadrilateral MNPO. | [
"25"
] | [
"Triangle ABC has an area of 80. Points K, M, O on side AB and points L, N, P on side AC are such that AK = KM = MO = OB and AL = LN = NP = PC. Find the area of quadrilateral MNPO."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c61904e010675ea113df3f0b1a4ceb75683d37ac1dc406e30f6154ef17c9574f | Triangle ABC has an area of 84 with sides AB = 13 and AC = 15. Point D is chosen such that BD = DC = 7. E is the midpoint of AC, and F is the midpoint of AB. Let G be the centroid of triangle DEF, and let H be the intersection of FD and BE. Find the area of triangle FGH. | [
"\\(\\frac{7}{2}\\)"
] | [
"Triangle ABC has an area of 84 with sides AB = 13 and AC = 15. Point D is chosen such that BD = DC = 7. E is the midpoint of AC, and F is the midpoint of AB. Let G be the centroid of triangle DEF, and let H be the intersection of FD and BE. Find the area of triangle FGH."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-77d75f7970cf46cc22c706ddcd889defcb27abe4a0f9d547ba8e905a5ca0ca59 | Triangle ABC has an in-radius of 5 and a circum-radius 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... | [
"187"
] | [
"Triangle ABC has an in-radius of 5 and a circum-radius 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 divisi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-539d854d16096a7c067c62c9177d67a1dafff837568a00c194ba5f4c6eea79ea | Triangle ABC has angle BAC = 30 degrees and AB = AC = 2. A semicircle with diameter BC is drawn. Find the probability that a randomly chosen point inside the triangle is in the semicircle. | [
"\\( 2 \\left( \\frac{2 - \\sqrt{3}}{4} \\right) + \\frac{4\\pi}{3} (2 - \\sqrt{3}) \\)"
] | [
"Triangle ABC has angle BAC = 30 degrees and AB = AC = 2. A semicircle with diameter BC is drawn. Find the probability that a randomly chosen point inside the triangle is in the semicircle."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-691e38b720940c169c7873dc9757d8dc5469c15f4e4e5e6d17da32087d3ebaf4 | Triangle ABC has base BC equal to \(a\) and height from A to BC equal to \(h\), with \(h < \frac{\sqrt{3}}{2}a\). Point P is inside the triangle such that \(\angle PAB = \angle PBA = \angle PCB = \alpha\). Calculate the value of \(\alpha\) in degrees for a triangle where \(a = 10\) and \(h = 4\). | [
"11.25"
] | [
"Triangle ABC has base BC equal to \\(a\\) and height from A to BC equal to \\(h\\), with \\(h < \\frac{\\sqrt{3}}{2}a\\). Point P is inside the triangle such that \\(\\angle PAB = \\angle PBA = \\angle PCB = \\alpha\\). Calculate the value of \\(\\alpha\\) in degrees for a triangle where \\(a = 10\\) and \\(h = 4\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40399b0de190be1a0b95e446a1435dcf498f123fb7dba0abb17ca99ef1b1aba3 | Triangle ABC has base BC on line L and point A on line K. Point D is on line K, 2006 units from point A. The height of triangle ABC is equal to the square of the base, and the base is the seventh triangular number. What is the area of triangle DBC? | [
"10976"
] | [
"Triangle ABC has base BC on line L and point A on line K. Point D is on line K, 2006 units from point A. The height of triangle ABC is equal to the square of the base, and the base is the seventh triangular number. What is the area of triangle DBC?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 8
} | 25 | 0 | 0 |
omr-ws-4e032876c40490be13d38c073fee44dbd9d4f4cc85e5ebb93badf342f1e13c2f | Triangle ABC has circumcenter O. Point K is obtained by extending OA in the direction of A by a distance equal to the inradius r of triangle ABC. The circumcircle of triangle KBC has circumradius R. Express R in terms of the sides a, b, and c of triangle ABC. | [
"\\frac{a}{2} \\csc A"
] | [
"Triangle ABC has circumcenter O. Point K is obtained by extending OA in the direction of A by a distance equal to the inradius r of triangle ABC. The circumcircle of triangle KBC has circumradius R. Express R in terms of the sides a, b, and c of triangle ABC."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6ea4dfb603a3155578ad85cdcdca31424113062bee8487cf1a80b39138365fa0 | Triangle ABC has coordinates A(1,2), B(0,5), and C(5,4). Part A: State the coordinates of triangle A'B'C', the image of triangle ABC after the translation which maps (x,y) to (x-6, y+3). Part B: State the coordinates of triangle A''B''C'', the image of triangle A'B'C' after a reflection in the x-axis. Part C: State the... | [
"A'(-5,5), B'(-6,8), C'(-1,7); A''(-5,-5), B''(-6,-8), C''(-1,-7); A'''(5,5), B'''(6,8), C'''(1,7)",
"\\[\nA'(-5, 5), B'(-6, 8), C'(-1, 7)\n\\]\n\\[\nA''(-5, -5), B''(-6, -8), C''(-1, -7)\n\\]\n\\[\nA'''(5, 5), B'''(6, 8), C'''(1, 7)\n\\]"
] | [
"Triangle ABC has coordinates A(1,2), B(0,5), and C(5,4). \n\n Part A: State the coordinates of triangle A'B'C', the image of triangle ABC after the translation which maps (x,y) to (x-6, y+3).\n\n Part B: State the coordinates of triangle A''B''C'', the image of triangle A'B'C' after a reflection in the x-axis.\n... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 30
} | 62 | 0 | 0 |
omr-ws-6f527204733673120e4f0b31e77fce29be5b848ef06151aff60699ea7f0af5d5 | 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. | [
"2"
] | [
"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_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-c125adee1d50466146c9ec678ffbd8f8670315dafe70c92199724cc23aa0d1bc | Triangle ABC has integer side lengths and a perimeter of 7. Determine all possible lengths of side AB. | [
"1, 2, 3"
] | [
"Triangle ABC has integer side lengths and a perimeter of 7. Determine all possible lengths of side AB."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 13
} | 17 | 0 | 0 |
omr-ws-5ee5897238a91f8f362580bc65358a13e6e48ae71b3e1ac4d27cdd071c6db049 | 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. | [
"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"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-6814104458c2bff8957a364da21299d6b91b17093539bbb78358f2105cfc5176 | Triangle ABC has its vertices on a circle of radius \( r \). If the lengths of two of the medians of triangle ABC are equal to \( r \), determine the side lengths of triangle ABC. | [
"\\(x\\), \\(x\\), \\(x\\sqrt{\\frac{7}{2}}\\)"
] | [
"Triangle ABC has its vertices on a circle of radius \\( r \\). If the lengths of two of the medians of triangle ABC are equal to \\( r \\), determine the side lengths of triangle ABC."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 13
} | 30 | 0 | 0 |
omr-ws-da996497207c3f4953b6ffd8de0946a7b0b779205946438d7694598db31e30da | Triangle ABC has positive integer side lengths with AB = AC. Let I be the intersection of the angle bisectors of ∠B and ∠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 ∠B and ∠C. Suppose BI = 8. Find the smallest possible perimeter of triangle ABC."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
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