Split (1)

train · 500k rows

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
omr-ws-ad12bd2e166419cd3e63d90cdfac24e55080f9981bcdb7b5c2c36ab9f5047c9f	A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the lengths of sides AB and AC.	[ "AB = 13 cm, AC = 15 cm" ]	[ "A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the lengths of sides AB and AC." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 6, "tir": 24 }	30
omr-ws-effebc203b659f6944bdba8b52c1a9b7fa57fec027f131f32a3aaa77240d2943	A triangle $ ABC $ has $ M $ and $ N $ on sides $ BC $ and $ AC $, respectively, such that $ AM $ and $ BN $ intersect at $ P $ and the areas of triangles $ ANP $, $ APB $, and $ PMB $ are $ 5 $, $ 10 $, and $ 8 $ respectively. If $ R $ and $ S $ are the midpoints of $ MC $ and \( NC \...	[ "\\dfrac{9}{2}" ]	[ "A triangle \$ ABC \$ has \$ M \$ and \$ N \$ on sides \$ BC \$ and \$ AC \$, respectively, such that \$ AM \$ and \$ BN \$ intersect at \$ P \$ and the areas of triangles \$ ANP \$, \$ APB \$, and \$ PMB \$ are \$ 5 \$, \$ 10 \$, and \$ 8 \$ respectively. If \$ R \$ and \$ S \$ are ...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-81f3dcf05f459d5c2e9a3e57d0d5601d39fa82305eeab2cd393dd4cfadae18aa	A triangle $ ABC $ has side lengths $ AB = 10 $, $ AC = 12 $, and $ BC = 17 $. There is a point $ D $ such that $ BD = 7 $ and $ DC = 10 $. Find $ AD $.	[ "\$\\frac{818 \\sqrt{17}}{17}\$" ]	[ "A triangle \$ ABC \$ has side lengths \$ AB = 10 \$, \$ AC = 12 \$, and \$ BC = 17 \$. There is a point \$ D \$ such that \$ BD = 7 \$ and \$ DC = 10 \$. Find \$ AD \$." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-30d2ac3998454bf44271c90e01a1f606938fa97bb86d4d69ed50c4df81840971	A triangle $ T $ in the $ uv $ plane has vertices $ (0,0) $, $ (2,0) $, and $ (0,2) $. Find the image $ S $ of $ T $ on the plane $ xy $.	[ "\$ 0 \\leq x \\leq 2 \$ and \$ -x^2 \\leq y \\leq x \$" ]	[ "A triangle \$ T \$ in the \$ uv \$ plane has vertices \$ (0,0) \$, \$ (2,0) \$, and \$ (0,2) \$. Find the image \$ S \$ of \$ T \$ on the plane \$ xy \$." ]	[ "aops_c7_college_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-939e428de98d17ba23e90aee28d0a65358c8e449c16877d38d97297ef3c0446e	A triangle $ \triangle ABC $ is inscribed in a conic $ \omega $. Let the poles of $ BC $ and $ BA $ be $ D $ and $ E $ with respect to the conic $ \omega $. Let a line $ \alpha $ passing through $ D $ intersect $ AC $ at $ F $. If the pole of $ \alpha $ is $ P $, and the coordinates of $ D $...	[ "(7, 8)" ]	[ "A triangle \$ \\triangle ABC \$ is inscribed in a conic \$ \\omega \$. Let the poles of \$ BC \$ and \$ BA \$ be \$ D \$ and \$ E \$ with respect to the conic \$ \\omega \$. Let a line \$ \\alpha \$ passing through \$ D \$ intersect \$ AC \$ at \$ F \$. If the pole of \$ \\alpha \$ is \\( P...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-c79bebf2232e3d07dca439255e8cf8da4effac5a5b575e003a4c08baa919f950	A triangle $ABC$ and a positive number $k$ are given. Find the locus of a point $M$ inside the triangle such that the projections of $M$ on the sides of $\Delta ABC$ form a triangle of area $k$.	[ "\\text{A circle}" ]	[ "A triangle \$ABC\$ and a positive number \$k\$ are given. Find the locus of a point \$M\$ inside the triangle such that the projections of \$M\$ on the sides of \$\\Delta ABC\$ form a triangle of area \$k\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 11, "genselect": 3 }	14
omr-ws-2283f4739fc8a3b8f7108b394b978cdfbf947b3f5a6355b3e4a5b2463efeb9c9	A triangle $ABC$ can be divided into two isosceles triangles by a line through $A$. Given that one of the angles of the triangles is $30^\circ$, find all possible values of the other two angles.	[ "\$60^\\circ, 90^\\circ; 75^\\circ, 90^\\circ; 135^\\circ, 30^\\circ; 67.5^\\circ, 75^\\circ; 45^\\circ, 120^\\circ; 112.5^\\circ, 30^\\circ\$" ]	[ "A triangle \$ABC\$ can be divided into two isosceles triangles by a line through \$A\$. Given that one of the angles of the triangles is \$30^\\circ\$, find all possible values of the other two angles." ]	[ "aops_c6_high_school_olympiads" ]	[ "tir" ]	{ "tir": 2 }	2
omr-ws-9748eb10a1d0fd53a69e8f594f83930a990a7d5a2389944067139f208e5313ca	A triangle $ABC$ has an area of 20. $M$, $N$, and $P$ are the midpoints of sides $AB$, $BC$, and $CA$ respectively. $H_a$, $H_b$, and $H_c$ are the orthocenters of triangles $AMP$, $BMN$, and $CNP$ respectively. What is the area of triangle $H_aH_bH_c$?	[ "5" ]	[ "A triangle \$ABC\$ has an area of 20. \$M\$, \$N\$, and \$P\$ are the midpoints of sides \$AB\$, \$BC\$, and \$CA\$ respectively. \$H_a\$, \$H_b\$, and \$H_c\$ are the orthocenters of triangles \$AMP\$, \$BMN\$, and \$CNP\$ respectively. What is the area of triangle \$H_aH_bH_c\$?" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-2d1c1be58c93b198a31fa306b815bccf92417019d617b2bdf3b248ab435de22d	A triangle $ABC$ has another triangle $DEF$ inscribed in it with $D$ on $BC$, $E$ on $CA$, and $F$ on $AB$. The angles are defined as follows: - $\angle BAC = a$, - $\angle ABC = b$, - $\angle ACB = c$, - $\angle FDE = d$, - $\angle DEF = e$, - $\angle EFD = f$, - $\angle AFE = g$, - \(\an...	[ "30" ]	[ "A triangle \$ABC\$ has another triangle \$DEF\$ inscribed in it with \$D\$ on \$BC\$, \$E\$ on \$CA\$, and \$F\$ on \$AB\$. The angles are defined as follows:\n- \$\\angle BAC = a\$,\n- \$\\angle ABC = b\$,\n- \$\\angle ACB = c\$,\n- \$\\angle FDE = d\$,\n- \$\\angle DEF = e\$,\n- \\(\\an...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 7, "genselect": 1 }	8
omr-ws-4731d53fe29223082f145cc06b1fb7083fa128c96356ce410e00525dceb3a54f	A triangle $ABC$ has integer sides, $\angle A = 2 \angle B$, and $\angle C > 90^\circ$. Find the minimum possible perimeter of this triangle.	[ "39" ]	[ "A triangle \$ABC\$ has integer sides, \$\\angle A = 2 \\angle B\$, and \$\\angle C > 90^\\circ\$. Find the minimum possible perimeter of this triangle." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-2bf3fa09cc02b62e68c92dbb70e3e8955b4ae40e0b5358dd6d4391a7e6edae67	A triangle $ABC$ has its vertices lying on a circle $C$ with radius 1 and $\angle BAC = 60^\circ$. A circle with center $I$ is inscribed in triangle $ABC$. The line $AI$ meets circle $C$ again at $D$. Find the length of the segment $ID$.	[ "1" ]	[ "A triangle \$ABC\$ has its vertices lying on a circle \$C\$ with radius 1 and \$\\angle BAC = 60^\\circ\$. A circle with center \$I\$ is inscribed in triangle \$ABC\$. The line \$AI\$ meets circle \$C\$ again at \$D\$. Find the length of the segment \$ID\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 17 }	17
omr-ws-f720767400ca7e78af18134a3f49d7e244caf7fd590750d11d8f416bcb87f01c	A triangle $ABC$ has its vertices lying on a circle $\Gamma$ of radius $1$, with $\angle BAC = 60^\circ$. A circle with center $I$ is inscribed in $\triangle ABC$. The line $AI$ meets circle $\Gamma$ again at $D$. Find the length of the segment $ID$.	[ "1" ]	[ "A triangle \$ABC\$ has its vertices lying on a circle \$\\Gamma\$ of radius \$1\$, with \$\\angle BAC = 60^\\circ\$. A circle with center \$I\$ is inscribed in \$\\triangle ABC\$. The line \$AI\$ meets circle \$\\Gamma\$ again at \$D\$. Find the length of the segment \$ID\$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-77fcca223e7a889f8c00fa392a8500f4c6e803a6105fbfbe867f1c2172a37f74	A triangle $ABC$ has positive integer side lengths and $\angle{A} = 2 \angle{B}$ and $\angle{C} > 90^\circ$. Find the minimum length of the perimeter.	[ "77" ]	[ "A triangle \$ABC\$ has positive integer side lengths and \$\\angle{A} = 2 \\angle{B}\$ and \$\\angle{C} > 90^\\circ\$. Find the minimum length of the perimeter." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 12, "genselect": 6, "tir": 11 }	29
omr-ws-4609465dd7555e2725affe34f1cff889e31adff135c814215cfb98e0a31641df	A triangle $ABC$ is drawn to circumscribe a circle with a radius of 4 cm. The segments $BD$ and $DC$, where $D$ is the point of contact of the circle with side $BC$, are 8 cm and 6 cm, respectively. Find the lengths of sides $AB$ and $AC$.	[ "AB = 15 cm, AC = 13 cm" ]	[ "A triangle \$ABC\$ is drawn to circumscribe a circle with a radius of 4 cm. The segments \$BD\$ and \$DC\$, where \$D\$ is the point of contact of the circle with side \$BC\$, are 8 cm and 6 cm, respectively. Find the lengths of sides \$AB\$ and \$AC\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 9, "tir": 15 }	24
omr-ws-2658a2dd0547ed0fbd2445227aa205fc3ff6d4909c9b843d5405bd61022a0df4	A triangle $ABC$ is formed with $A$ being the current location of Darth Vader’s spaceship, $B$ being the location of the Rebel Base, and $C$ being the location of the Death Star. Let $D$, $E$, and $F$ be the locations of the spaceships of Luke Skywalker, Han Solo, and Princess Leia, respectively. It is tr...	[ "2\\sqrt{6}" ]	[ "A triangle \$ABC\$ is formed with \$A\$ being the current location of Darth Vader’s spaceship, \$B\$ being the location of the Rebel Base, and \$C\$ being the location of the Death Star. Let \$D\$, \$E\$, and \$F\$ be the locations of the spaceships of Luke Skywalker, Han Solo, and Princess Leia, res...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-95ec9f8948184c0b607b1c435d38aac8e3a15995fc57d46cfde8b10d99425fb4	A triangle $ABC$ is inscribed in a circle $(O, R)$. A circle $(O', R')$ is internally tangent to $(O)$ at $I$ such that $R < R'$. Let $P$ be a point on circle $(O)$. Rays $PA$, $PB$, and $PC$ meet circle $(O')$ at points $A_1$, $B_1$, and $C_1$ respectively. Let $A_2B_2C_2$ be the triang...	[ "R' - R" ]	[ "A triangle \$ABC\$ is inscribed in a circle \$(O, R)\$. A circle \$(O', R')\$ is internally tangent to \$(O)\$ at \$I\$ such that \$R < R'\$. Let \$P\$ be a point on circle \$(O)\$. Rays \$PA\$, \$PB\$, and \$PC\$ meet circle \$(O')\$ at points \$A_1\$, \$B_1\$, and \$C_1\$ respective...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 1 }	6
omr-ws-fbf4ca732a25262c3b97449bb07f53a1a93d5615c3663304892eb743e8a48514	A triangle $ABC$ is inscribed in a circle $S$. Let $A_0$ and $C_0$ be the midpoints of the arcs $BC$ and $AB$ on $S$, not containing the opposite vertex, respectively. The circle $S_1$ centered at $A_0$ is tangent to $BC$, and the circle $S_2$ centered at $C_0$ is tangent to $AB$. Let $d$ be...	[ "4" ]	[ "A triangle \$ABC\$ is inscribed in a circle \$S\$. Let \$A_0\$ and \$C_0\$ be the midpoints of the arcs \$BC\$ and \$AB\$ on \$S\$, not containing the opposite vertex, respectively. The circle \$S_1\$ centered at \$A_0\$ is tangent to \$BC\$, and the circle \$S_2\$ centered at \$C_0\$ is ta...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-e8e03f28d99cf4723f7be404cd132017fd2c928cdcdce6f8a30a25051247d037	A triangle $ABC$ is inscribed in a circle $\omega$. A variable line $\ell$ parallel to $BC$ intersects segments $AB$ and $AC$ at points $D$ and $E$ respectively, and intersects $\omega$ at points $K$ and $L$ (with $D$ between $K$ and $E$). Circle $\gamma_1$ is tangent to segments $KD$ an...	[ "The internal angle bisector of \$\\angle BAC\$." ]	[ "A triangle \$ABC\$ is inscribed in a circle \$\\omega\$. A variable line \$\\ell\$ parallel to \$BC\$ intersects segments \$AB\$ and \$AC\$ at points \$D\$ and \$E\$ respectively, and intersects \$\\omega\$ at points \$K\$ and \$L\$ (with \$D\$ between \$K\$ and \$E\$). Circle \\(\\gamm...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 2, "genselect": 1 }	3
omr-ws-7c69981e0d0e7e6ce64eda866005c493c556bf43721221686dad124c44a9e061	A triangle $ABC$ is inscribed in a circle $\omega$. A variable line $l$ parallel to $BC$ intersects segments $AB$ and $AC$ at points $D$ and $E$ respectively, and intersects $\omega$ at points $K$ and $L$ (where $D$ lies between $K$ and $E$). Circle $\gamma_1$ is tangent to segments $KD$...	[ "\\text{The A-symmedian of triangle } ABC" ]	[ "A triangle \$ABC\$ is inscribed in a circle \$\\omega\$. A variable line \$l\$ parallel to \$BC\$ intersects segments \$AB\$ and \$AC\$ at points \$D\$ and \$E\$ respectively, and intersects \$\\omega\$ at points \$K\$ and \$L\$ (where \$D\$ lies between \$K\$ and \$E\$). Circle \\(\\ga...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 6, "genselect": 7 }	13
omr-ws-48db92f1d23790678b2b5237d445bb0ee284f8a6609db69c951a7c9af2d365f4	A triangle $ABC$ is inscribed in a circle with center $U$ and radius $r$. A tangent $c'$ to a larger circle $K(U, 2r)$ is drawn so that $C$ lies between the lines $c = AB$ and $c'$. Lines $a'$ and $b'$ are analogously defined. The triangle formed by $a', b', c'$ is denoted $A'B'C'$. Let \(M, N, ...	[ "\\dfrac{9}{4}" ]	[ "A triangle \$ABC\$ is inscribed in a circle with center \$U\$ and radius \$r\$. A tangent \$c'\$ to a larger circle \$K(U, 2r)\$ is drawn so that \$C\$ lies between the lines \$c = AB\$ and \$c'\$. Lines \$a'\$ and \$b'\$ are analogously defined. The triangle formed by \$a', b', c'\$ is denot...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-685c7c66265504cc39695da021af694b2b6739ee491b9950f91ad24b7190cce9	A triangle $ABC$ is inscribed in a circle with center $U$ and radius $r$. A tangent $c'$ to a larger circle $K(U, 2r)$ is drawn so that $C$ lies between the lines $c = AB$ and $c'$. Lines $a'$ and $b'$ are analogously defined. The triangle formed by $a', b', c'$ is denoted $A'B'C'$. Let $M_1$,...	[ "\\dfrac{3}{4}" ]	[ "A triangle \$ABC\$ is inscribed in a circle with center \$U\$ and radius \$r\$. A tangent \$c'\$ to a larger circle \$K(U, 2r)\$ is drawn so that \$C\$ lies between the lines \$c = AB\$ and \$c'\$. Lines \$a'\$ and \$b'\$ are analogously defined. The triangle formed by \$a', b', c'\$ is denot...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-a38f03ff34e0117235faa082a59a7f00f86c3f3059b211f6f5d10bdb5991a8e7	A triangle $ABC$ is inscribed in a circle with radius $4 \text{ cm}$. Given that $\angle A = 60^\circ$, $AC - AB = 4 \text{ cm}$, and the area of $\triangle ABC$ is $x \text{ cm}^2$, find the value of $\left(\frac{x}{2}\right)^2$.	[ "48" ]	[ "A triangle \$ABC\$ is inscribed in a circle with radius \$4 \\text{ cm}\$. Given that \$\\angle A = 60^\\circ\$, \$AC - AB = 4 \\text{ cm}\$, and the area of \$\\triangle ABC\$ is \$x \\text{ cm}^2\$, find the value of \$\\left(\\frac{x}{2}\\right)^2\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 6 }	22
omr-ws-719cb8559461b6395723e50abd8cbba8546dc8a08bb4fa2ba41a040427e6e72f	A triangle $ABC$ with $AB = BC = CA = 1$ is covered with 6 circles of radius $r$. Calculate the minimum value of $r$ such that the entire area of the triangle is covered by these circles. Express your answer in the form $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are positive integers and $a$ is not divisib...	[ "9" ]	[ "A triangle \$ABC\$ with \$AB = BC = CA = 1\$ is covered with 6 circles of radius \$r\$. Calculate the minimum value of \$r\$ such that the entire area of the triangle is covered by these circles. Express your answer in the form \$\\frac{\\sqrt{a}}{b}\$, where \$a\$ and \$b\$ are positive integers and...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 3, "genselect": 4 }	7
omr-ws-0f2ad6a18cf2e58bd9d2a39f79d70f344884dcd846d9077452562e36b283c919	A triangle $ABC$ with $AC = 20$ is inscribed in a circle $\omega$. A tangent $t$ to $\omega$ is drawn through $B$. The distance from $A$ to $t$ is 25 and the distance from $C$ to $t$ is 16. If $S$ denotes the area of triangle $ABC$, find the largest integer not exceeding $\frac{S}{20}$.	[ "10" ]	[ "A triangle \$ABC\$ with \$AC = 20\$ is inscribed in a circle \$\\omega\$. A tangent \$t\$ to \$\\omega\$ is drawn through \$B\$. The distance from \$A\$ to \$t\$ is 25 and the distance from \$C\$ to \$t\$ is 16. If \$S\$ denotes the area of triangle \$ABC\$, find the largest integer not exc...	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 14, "tir": 21 }	35
omr-ws-4f517e056b686a55860b1f13967759958adb6eb9a747f0e63f1b16b2665f13f0	A triangle $ABC$ with an obtuse angle at vertex $C$ is inscribed in a circle with center $O$. The circumcircle of triangle $AOB$ centered at point $P$ intersects line $AC$ at points $A$ and $A_1$, line $BC$ at points $B$ and $B_1$, and the perpendicular bisector of segment $PC$ at points $D$ a...	[ "2" ]	[ "A triangle \$ABC\$ with an obtuse angle at vertex \$C\$ is inscribed in a circle with center \$O\$. The circumcircle of triangle \$AOB\$ centered at point \$P\$ intersects line \$AC\$ at points \$A\$ and \$A_1\$, line \$BC\$ at points \$B\$ and \$B_1\$, and the perpendicular bisector of segme...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-435947484d0e248e45ac1a68db95e2512348865de5e5ea7c5237a0a51c75deaa	A triangle and a quadrilateral are situated in the plane such that they have a finite number of intersection points $ I $. Find the sum of all possible values of $ I $.	[ "21" ]	[ "A triangle and a quadrilateral are situated in the plane such that they have a finite number of intersection points \$ I \$. Find the sum of all possible values of \$ I \$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 8, "genselect": 5 }	13
omr-ws-e7af6802e01391a45c7effb89c12fbf2db72e7a633f56fe2128be5e83a9c4422	A triangle and a square are circumscribed about a circle of diameter 1. Let $L$ be the length of the part of the boundary of the square which lies outside the triangle. Find the maximum possible value of $10L$.	[ "20" ]	[ "A triangle and a square are circumscribed about a circle of diameter 1. Let $L$ be the length of the part of the boundary of the square which lies outside the triangle. Find the maximum possible value of $10L$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-ce7d5547fd7106653ef5394c0ed8cb4fa8a79cff3a139dd2038bb20e671ce4d5	A triangle can be cut into two isosceles triangles. One of the angles of the original triangle is 36 degrees. Determine all possible values of the largest angle of the original triangle.	[ "\$72^\\circ, 90^\\circ, 108^\\circ, 126^\\circ, 132^\\circ\$" ]	[ "A triangle can be cut into two isosceles triangles. One of the angles of the original triangle is 36 degrees. Determine all possible values of the largest angle of the original triangle." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-2a4438de8213b1031366a350f8dcd97deeea53413df92353a93376cd0719c4c4	A triangle can be cut into two isosceles triangles. One of the angles of the original triangle is 36˚. Determine all possible values of the largest angle of the original triangle.	[ "132, 90, 108, 72" ]	[ "A triangle can be cut into two isosceles triangles. One of the angles of the original triangle is 36˚. Determine all possible values of the largest angle of the original triangle." ]	[ "aops_c6_high_school_olympiads" ]	[ "tir" ]	{ "tir": 1 }	1
omr-ws-23216c1872d21b21ea0965245da0eced49e26e6314af4f2c2858a062cda69a3b	A triangle formed by three lattice points on a Cartesian plane is called "fit" if the coordinates of the centroid are lattice points. Determine the smallest positive integer $ k $ such that for any $ k $ distinct lattice points chosen on a Cartesian plane, one can always choose three of them such that they form a "...	[ "9" ]	[ "A triangle formed by three lattice points on a Cartesian plane is called \"fit\" if the coordinates of the centroid are lattice points. Determine the smallest positive integer \$ k \$ such that for any \$ k \$ distinct lattice points chosen on a Cartesian plane, one can always choose three of them such that th...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 11, "genselect": 2 }	13
omr-ws-77634f063482937c1b17beb05d32e1b9c11e9658e80a6e6e5c445c67d99f7063	A triangle garden O has sides of lengths 30 ft, 28 ft, and 28 ft. A curve A is to go around the garden such that each point on A is exactly 8 ft away from the nearest point on O. What is the approximate length of curve A in feet?	[ "\$ 86 + 16\\pi \$" ]	[ "A triangle garden O has sides of lengths 30 ft, 28 ft, and 28 ft. A curve A is to go around the garden such that each point on A is exactly 8 ft away from the nearest point on O. What is the approximate length of curve A in feet?" ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-c1a04f280583497cdb861e2c4832506f7d1d2d963f35362de6de2ef1a74a8f4a	A triangle has a 60° angle, and the two adjacent sides are 12 and $12\sqrt{3}$. Find the radius of a circle with the same vertex as the center, if the arc inside the triangle divides it into two regions of equal area.	[ "\\dfrac{18}{\\sqrt{\\pi}}" ]	[ "A triangle has a 60° angle, and the two adjacent sides are 12 and \$12\\sqrt{3}\$. Find the radius of a circle with the same vertex as the center, if the arc inside the triangle divides it into two regions of equal area." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-3faa90ea7987d7e5cc815f7271a407ac8692ece7bdb97d560ec65a9e55030e54	A triangle has a base of 6 and altitudes of lengths 2 and $2\sqrt{3}$ drawn to the remaining sides. Find the lengths of the sides of the triangle.	[ "\$ 6, 3\\sqrt{2}, \\sqrt{6} \$" ]	[ "A triangle has a base of 6 and altitudes of lengths 2 and \$2\\sqrt{3}\$ drawn to the remaining sides. Find the lengths of the sides of the triangle." ]	[ "aops_c4_high_school_math" ]	[ "tir" ]	{ "tir": 9 }	9
omr-ws-79db7f1527d3dbc0d44ce16c58e68df8613782f2eb2468071c5f26d3f57851c8	A triangle has a base of length $8$ and an area of $12$. What is the radius of the largest circle that can be inscribed in this triangle?	[ "\$\\frac{4}{3}\$" ]	[ "A triangle has a base of length $8$ and an area of $12$. What is the radius of the largest circle that can be inscribed in this triangle?" ]	[ "aops_c6_high_school_olympiads", "c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 32 }	48
omr-ws-4c549cb68c631269752b005555849e4a1bb1ef7cda712ca0419b4a37c1a9f4c5	A triangle has a base of length 8 and an area of 12. What is the radius of the largest circle that can be inscribed in this triangle?	[ "\$\\frac{4}{3}\$" ]	[ "A triangle has a base of length 8 and an area of 12. What is the radius of the largest circle that can be inscribed in this triangle?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 28 }	44
omr-ws-d4f33233371a7fa49482857ea7b35499ed1038742fe993d8bb9df2a43e3a8e2b	A triangle has a fixed base AB that is 2 inches long. The median from A to side BC is 1½ inches long and can have any position emanating from A. What is the locus of the vertex C of the triangle?	[ "a circle with radius 3 in. and center 4 in. from B along BA" ]	[ "A triangle has a fixed base AB that is 2 inches long. The median from A to side BC is 1½ inches long and can have any position emanating from A. What is the locus of the vertex C of the triangle?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 17 }	33
omr-ws-ced42fa3af2d683d1031034cbfe0d25a9451ed1e92b89f1cfc9ff1b95340a1c9	A triangle has a fixed base $AB$ that is 2 inches long. The median from $A$ to side $BC$ is $1 \dfrac{1}{2}$ inches long and can have any position emanating from $A$. Determine the locus of the vertex $C$ of the triangle.	[ "A circle with radius 3 inches and center 4 inches from \$B\$ along \$BA\$." ]	[ "A triangle has a fixed base \$AB\$ that is 2 inches long. The median from \$A\$ to side \$BC\$ is \$1 \\dfrac{1}{2}\$ inches long and can have any position emanating from \$A\$. Determine the locus of the vertex \$C\$ of the triangle." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 13 }	28
omr-ws-2867d96181e4447633dfc4508bb60924eb83fc8c1f3518e637c6b118a47fa0a1	A triangle has a side of length 6 cm, a side of length 8 cm and a right angle. What is the shortest possible length of the remaining side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth.	[ "5.29" ]	[ "A triangle has a side of length 6 cm, a side of length 8 cm and a right angle. What is the shortest possible length of the remaining side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth." ]	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-960a50dfd267aa68284612cb50c70506e6244b9a42d656d8f1ee48edfed4f5e2	A triangle has altitudes of length 5 and 7. What is the maximum length of the third altitude?	[ "17.5" ]	[ "A triangle has altitudes of length 5 and 7. What is the maximum length of the third altitude?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 11 }	27
omr-ws-9705746b12ba83b7461deb8f87dc88a573f765f056667ef6e58fd32dfddc3ce9	A triangle has altitudes of lengths 15, 12, and 20. Find the perimeter of the triangle.	[ "60" ]	[ "A triangle has altitudes of lengths 15, 12, and 20. Find the perimeter of the triangle." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 7 }	23
omr-ws-8b3d7312d1b2afd3b4f9de1bff9fd87f543f211d28631632b3c295b90c3fdbbc	A triangle has altitudes with measures 8, 8, and $4\sqrt{5}$. What is its area?	[ "40" ]	[ "A triangle has altitudes with measures 8, 8, and \$4\\sqrt{5}\$. What is its area?" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-2a3257d68cd9e072988aaa5d69ef55156f0d4de105c33976a0aed0aa53de4f3f	A triangle has an area of $30$, one side of length $10$, and the median to that side of length $9$. Let $\theta$ be the acute angle formed by that side and the median. What is $\sin{\theta}$?	[ "\$\\frac{2}{3}\$" ]	[ "A triangle has an area of $30$, one side of length $10$, and the median to that side of length $9$. Let $\\theta$ be the acute angle formed by that side and the median. What is $\\sin{\\theta}$?" ]	[ "aops_c5_contests_amp_programs" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 30 }	46
omr-ws-b9cdb613180c17d4ab3d59242242d8af299bebe380d9a14e050c8790fcf52f5e	A triangle has an area of 12 and two medians of lengths 5 and 6. Find the sum of all possible lengths of the third median to the nearest integer.	[ "14" ]	[ "A triangle has an area of 12 and two medians of lengths 5 and 6. Find the sum of all possible lengths of the third median to the nearest integer." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 2 }	17
omr-ws-361d20ef2b841157c076b546c5a33f567e298348fd3458f504b85006c2be493b	A triangle has an area of 2, a perimeter of 8, and an angle of 30 degrees. What is the length of the side opposite the 30-degree angle?	[ "\$ 3 - \\frac{\\sqrt{3}}{2} \$" ]	[ "A triangle has an area of 2, a perimeter of 8, and an angle of 30 degrees. What is the length of the side opposite the 30-degree angle?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 26 }	42
omr-ws-f913ed12bf794eff5ae2785f62218b0e1d7e0d0aa1290cd49bc23353a72dfdef	A triangle has an area of $16 \text{ in.}^2$, and two of its sides have lengths $5 \text{ in.}$ and $7 \text{ in.}$. Find the angle included by these two sides.	[ "\$ \\theta \\approx 66^\\circ \$ or \$ \\theta \\approx 114^\\circ \$" ]	[ "A triangle has an area of \$16 \\text{ in.}^2\$, and two of its sides have lengths \$5 \\text{ in.}\$ and \$7 \\text{ in.}\$. Find the angle included by these two sides." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 2, "genselect": 7, "tir": 16 }	25
omr-ws-aa2dcf200b883369543dc14e9068576ad84cee9df9958b68b1556f095b6f864d	A triangle has an interior angle of 75° and another angle defined by the equation \[3 \sec x + m(\cos x - \sin x) - 3(\sin x + \cos x) = 0.\] Find the value of $ m $.	[ "\$ 3 \\sqrt{3} \$" ]	[ "A triangle has an interior angle of 75° and another angle defined by the equation \n\n\\[3 \\sec x + m(\\cos x - \\sin x) - 3(\\sin x + \\cos x) = 0.\\]\n\nFind the value of \$ m \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-1b1c65086b580aa9d385c30de47cd47a3c12f0c31b3d420e6209c5fcc98a2db4	A triangle has angles $180 - 3x$, $2x$, and $x$, with opposite sides $n+1$, $n+2$, and $n$ respectively. Find the perimeter of the triangle.	[ "15" ]	[ "A triangle has angles $180 - 3x$, $2x$, and $x$, with opposite sides $n+1$, $n+2$, and $n$ respectively. Find the perimeter of the triangle." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 12 }	28
omr-ws-29d76d2dcfe9ef31c807fe785daa5c1eb8ce31e0e48ca7d96d7791da02b3180e	A triangle has coordinates (0,0), (a,0), (0,b) where $ a \geq b $ and $ a $ and $ b $ are integers. The triangle contains 74 lattice points excluding those on its edges. What is the sum of the areas of all such possible triangles?	[ "4311.5" ]	[ "A triangle has coordinates (0,0), (a,0), (0,b) where \$ a \\geq b \$ and \$ a \$ and \$ b \$ are integers. The triangle contains 74 lattice points excluding those on its edges. What is the sum of the areas of all such possible triangles?" ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-f85f281f5074960eaee7a33868c74b8210ac662fbc9c02e7adb25939a06d4364	A triangle has endpoints at $A(5, 5)$, $B(14,-7)$, and $C(21, 17)$. If it gets rotated $60^o$ clockwise about the origin, reflected across the line $y = 2x$, dilated by a factor of $\frac15$ with respect to the origin, and then rotated $30^o$ counter-clockwise about $(1, 2)$, what is the area of the resultant triangle?	[ "6" ]	[ "A triangle has endpoints at $A(5, 5)$, $B(14,-7)$, and $C(21, 17)$. If it gets rotated $60^o$ clockwise about the origin, reflected across the line $y = 2x$, dilated by a factor of $\\frac15$ with respect to the origin, and then rotated $30^o$ counter-clockwise about $(1, 2)$, what is the area of the resultant tri...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-e99bcdaa33b17ea9752805333424aa29168d6d931f56944c319c9497faf87f26	A triangle has integer side lengths and an area of 84. How many different perimeters are possible?	[ "4" ]	[ "A triangle has integer side lengths and an area of 84. How many different perimeters are possible?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 12, "genselect": 8 }	20
omr-ws-ad3d9c0978bdcad3cac96ce4b3e2c2b21d10f14e18f9b8a289d5832a7820fb3e	A triangle has integer sides, a perimeter of 9, and one side of length 3. Determine the possible lengths of the other two sides.	[ "(2, 4) or (3, 3)" ]	[ "A triangle has integer sides, a perimeter of 9, and one side of length 3. Determine the possible lengths of the other two sides." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 7, "tir": 32 }	39
omr-ws-52565c0e4bb1d6d44f016a1493669719e5bccf69c7a3d6f9e9ed4723838e0577	A triangle has integral sides with a perimeter of 2006. One of the sides is sixteen times another side. How many such triangles exist?	[ "3" ]	[ "A triangle has integral sides with a perimeter of 2006. One of the sides is sixteen times another side. How many such triangles exist?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 30, "tir": 8 }	38
omr-ws-11ae39837b1140f145a41c8767cd9b7d7285ebb39f6e34e1c74e5eb0c2314f47	A triangle has integral sides. Its perimeter is 2006, and one of the sides is sixteen times another side. How many such triangles exist?	[ "3" ]	[ "A triangle has integral sides. Its perimeter is 2006, and one of the sides is sixteen times another side. How many such triangles exist?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 3 }	19
omr-ws-20f8653cc79d1163b401cf5cf41a2f93792ce8bb5f3380f9b947581ded426365	A triangle has its orthocenter at the origin and the slopes of its sides are -1, -2, and -3. Find the locus of its circumcenter.	[ "\$ y = \\frac{7x}{12} \$" ]	[ "A triangle has its orthocenter at the origin and the slopes of its sides are -1, -2, and -3. Find the locus of its circumcenter." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 15 }	31
omr-ws-44e7220a145eae4c3002b3a9c9f3e62ae85c7b3665b14483916ce3aeac22672a	A triangle has one constant angle and a constant inradius. Explicitly define all triangles for which the minimum perimeter is achieved.	[ "\\text{The triangle is isosceles with the two equal sides forming the constant angle}" ]	[ "A triangle has one constant angle and a constant inradius. Explicitly define all triangles for which the minimum perimeter is achieved." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 12 }	12
omr-ws-522c11449f9edac1378346678285001aad5a8d94f5b29d07e508826d64442637	A triangle has side lengths $4$, $5$, and $a$. The length of the median to the side of length $a$ is a quarter of the perimeter of the triangle. Compute $a$.	[ "\\dfrac{2\\sqrt{329} - 9}{5}" ]	[ "A triangle has side lengths $4$, $5$, and $a$. The length of the median to the side of length $a$ is a quarter of the perimeter of the triangle. Compute $a$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-51e3149f1225704dbd14c443ab8f52c6c584b2202fc6d32d4cc18b77340264f3	A triangle has side lengths $7, 8, 9$. The angle bisectors of two of the three angles intersect at points $m$ and $n$. Compute all possible lengths of the distance from $m$ to $n$.	[ "3\\sqrt{14}" ]	[ "A triangle has side lengths $7, 8, 9$. The angle bisectors of two of the three angles intersect at points $m$ and $n$. Compute all possible lengths of the distance from $m$ to $n$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-68767e3485c8ca62fbd3b3b40f766f72103702bca399da14d6a43b8d45febeea	A triangle has side lengths 5, $\frac{5\sqrt{2}}{2}$, and $\frac{5(\sqrt{3}+1)}{2}$. The angles are denoted $\theta_1$, $\theta_2$, and $\theta_3$. Given $f(n) = \frac{n}{\theta_n}$, find the maximum value of $f(1) + f(2) + f(3)$.	[ "\\dfrac{194}{7\\pi}" ]	[ "A triangle has side lengths 5, \$\\frac{5\\sqrt{2}}{2}\$, and \$\\frac{5(\\sqrt{3}+1)}{2}\$. The angles are denoted \$\\theta_1\$, \$\\theta_2\$, and \$\\theta_3\$. Given \$f(n) = \\frac{n}{\\theta_n}\$, find the maximum value of \$f(1) + f(2) + f(3)\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 12, "genselect": 8 }	20
omr-ws-bbda9f23711ad89e5a7fa7099766d7b44fc645bf07c938f7a49ed3c8132b53b5	A triangle has side lengths 7, 8, and 9. There are exactly two lines that simultaneously bisect the perimeter and area of the triangle. Let $\theta$ be the acute angle between these two lines. Find $\tan \theta.$ [asy] unitsize(0.5 cm); pair A, B, C, P, Q, R, S, X; B = (0,0); C = (8,0); A = intersectionpoint(arc(B,7,0,...	[ "3 \\sqrt{5} + 2 \\sqrt{10}" ]	[ "A triangle has side lengths 7, 8, and 9. There are exactly two lines that simultaneously bisect the perimeter and area of the triangle. Let $\\theta$ be the acute angle between these two lines. Find $\\tan \\theta.$\n\n[asy]\nunitsize(0.5 cm);\n\npair A, B, C, P, Q, R, S, X;\n\nB = (0,0);\nC = (8,0);\nA = inter...	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-ea502c956da4c4989ef8bb23f840af3b085526dc21f6432409edadb62150c08b	A triangle has side lengths 888, 925, and $ x > 0 $. Find the value of $ x $ that minimizes the area of the circle circumscribed about the triangle.	[ "259" ]	[ "A triangle has side lengths 888, 925, and \$ x > 0 \$. Find the value of \$ x \$ that minimizes the area of the circle circumscribed about the triangle." ]	[ "aops_c5_contests_amp_programs" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 20 }	36
omr-ws-1e937e9358d37bf647bf4a3b2ea5795ebed80e280591cecd53a0d60fb979039c	A triangle has side lengths $a$, $b$, and $c$, all of which are integers. If the perimeter of the triangle is 2018, find the greatest possible value of $a$.	[ "1008" ]	[ "A triangle has side lengths \$a\$, \$b\$, and \$c\$, all of which are integers. If the perimeter of the triangle is 2018, find the greatest possible value of \$a\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-2afc3a3290c9a6b3ea5f3ad2cf0eacdc23581953846d86fcfa71c2c412687d69	A triangle has side lengths of $x, 75, 100$ where $x < 75$ and altitudes of lengths $y, 28, 60$ where $y < 28$. What is the value of $x + y$?	[ "56" ]	[ "A triangle has side lengths of $x, 75, 100$ where $x < 75$ and altitudes of lengths $y, 28, 60$ where $y < 28$. What is the value of $x + y$?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 22 }	38
omr-ws-af24d2d2f58da5c3eab28f7e5034367a8046ab1c7402583235c0df8396339a2a	A triangle has side lengths of 8, 15 and 17 units. What is the area of the triangle, in square units?	[ "60" ]	[ "A triangle has side lengths of 8, 15 and 17 units. What is the area of the triangle, in square units?" ]	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-b8fbaaa65dff04f3b10869ec90c4419ef661e28ade771c9a9a1efbf12d603eeb	A triangle has side lengths that form both an arithmetic and geometric sequence. If the triangle's inradius is 2, find the area of the circumcircle.	[ "\$ 16\\pi \$" ]	[ "A triangle has side lengths that form both an arithmetic and geometric sequence. If the triangle's inradius is 2, find the area of the circumcircle." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 28 }	44
omr-ws-a7c94018f686782e9bdff0b312a8480fb10384b609951f67ad32ddff6b87c491	A triangle has sides 10, 17, and 21. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the other two vertices of the square touch the two shorter sides of the triangle. What is the length of the side of the square? Give an exact answer.	[ "\$\\frac{168}{29}\$" ]	[ "A triangle has sides 10, 17, and 21. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the other two vertices of the square touch the two shorter sides of the triangle. What is the length of the side of the square? Give an exact answer." ]	[ "aops_c5_contests_amp_programs", "c5_contests_amp_programs" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 26 }	42
omr-ws-85f523b0ab1dbcc3787b74177715133f26b45e9c9af04ba8599ed1d2bc2fa919	A triangle has sides 13, 14, 15. It is rotated through $180^{\circ}$ about its centroid to form an overlapping triangle. Find the area of the union of the two triangles.	[ "112" ]	[ "A triangle has sides 13, 14, 15. It is rotated through $180^{\\circ}$ about its centroid to form an overlapping triangle. Find the area of the union of the two triangles." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 20 }	36
omr-ws-e68b0e72c1c16ec2212e66015cb4253239b2a18193ae336a75ac7a8bcc519b96	A triangle has sides 13, 14, and 15. It is rotated through 180 degrees about its centroid to form an overlapping triangle. Find the area of the union of the two triangles.	[ "112" ]	[ "A triangle has sides 13, 14, and 15. It is rotated through 180 degrees about its centroid to form an overlapping triangle. Find the area of the union of the two triangles." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 13, "genselect": 4, "tir": 16 }	33
omr-ws-1a6eff486ec05a1f5cce9ee132e557b48d1cec6936f05c190b37fe5989d83f78	A triangle has sides 20, 21, and 22. The line through the incenter parallel to the shortest side meets the other two sides at $X$ and $Y$. Find $XY$.	[ "\\dfrac{860}{63}" ]	[ "A triangle has sides 20, 21, and 22. The line through the incenter parallel to the shortest side meets the other two sides at \$X\$ and \$Y\$. Find \$XY\$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-bc2d62e3b9fc0ff35d0b55dd1523950fb403b16885d844c2bf9d40f309f60d28	A triangle has sides 36, 40, and 52. If the sides are increased by $ x $, $ 2x $, and $ 3x $, where $ x $ is a positive integer, then the newly formed triangle is a right triangle. Find the sum of all distinct possible perimeters of the newly formed triangle.	[ "448" ]	[ "A triangle has sides 36, 40, and 52. If the sides are increased by \$ x \$, \$ 2x \$, and \$ 3x \$, where \$ x \$ is a positive integer, then the newly formed triangle is a right triangle. Find the sum of all distinct possible perimeters of the newly formed triangle." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 15, "genselect": 8 }	23
omr-ws-1387ae863d26866a3961b32ab124a07d4cd02a30a7213181943e6a4e22f80cf2	A triangle has sides $ b = 4 $ and $ c = 7 $. Given that its area is $ 6\sqrt{5} $, find all possible values for the length of $ a $.	[ "\$ a = 7 \$ or \$ a = 9 \$" ]	[ "A triangle has sides \$ b = 4 \$ and \$ c = 7 \$. Given that its area is \$ 6\\sqrt{5} \$, find all possible values for the length of \$ a \$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 1, "tir": 27 }	28
omr-ws-fd1fe812c70744e66332abf22e637ab74946ab319a44c6c16bb13a71fd0b7622	A triangle has sides measuring 13, 14, and 15. A line perpendicular to the side of length 14 divides the interior of the triangle into two regions of equal area. Find the length of the segment of the perpendicular that lies within the triangle.	[ "\$4\\sqrt{7}\$" ]	[ "A triangle has sides measuring 13, 14, and 15. A line perpendicular to the side of length 14 divides the interior of the triangle into two regions of equal area. Find the length of the segment of the perpendicular that lies within the triangle." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 26 }	41
omr-ws-7023376c590d9425170d51c5c07d089de2d28826652891ca02962a39f59cbe35	A triangle has sides of length $48$, $55$, and $73$. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ and $d$ are relatively prime positive integers such ...	[ "200689" ]	[ "A triangle has sides of length $48$, $55$, and $73$. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ and $d$ are relatively prime positive integers ...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 7, "genselect": 6, "tir": 22 }	35
omr-ws-2f47c17c71e3b2d69c9e915ebf5d40a4a620c6d24c22570d7c9215b4754bf0ee	A triangle has sides of length $48$, $55$, and $73$. Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the length of the shortest altitude of the triangle. Find the value of $a+b$.	[ "2713" ]	[ "A triangle has sides of length $48$, $55$, and $73$. Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the length of the shortest altitude of the triangle. Find the value of $a+b$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 18 }	33
omr-ws-920485bc7b8b5fb842583603a6fa25f27bd9489e8ec04bdcb28f5b569cbf9743	A triangle has sides of length $\sqrt{2}$, $3 + \sqrt{3}$, and $2\sqrt{2} + \sqrt{6}$. Compute the area of the smallest regular polygon that has three vertices coinciding with the vertices of the given triangle.	[ "12 + 6\\sqrt{3}" ]	[ "A triangle has sides of length $\\sqrt{2}$, $3 + \\sqrt{3}$, and $2\\sqrt{2} + \\sqrt{6}$. Compute the area of the smallest regular polygon that has three vertices coinciding with the vertices of the given triangle." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-3b3f7ed15a1b57b9a91b8e97e612ffc3e876aaefbc983fc3b18790322c44ac28	A triangle has sides of length 15, 20, and 25 units. It is inscribed in a circle. What is the circle's radius?	[ "\\dfrac{25}{2}" ]	[ "A triangle has sides of length 15, 20, and 25 units. It is inscribed in a circle. What is the circle's radius?" ]	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-5c83666b91bb177854f81d45fb6267763bbcd1e984a6b3a1361e875f787b6896	A triangle has sides of length 5 and 6 units. The length of the third side is $x$ units, where $x$ is an integer. What is the largest possible perimeter of the triangle?	[ "21\\text{ units}" ]	[ "A triangle has sides of length 5 and 6 units. The length of the third side is $x$ units, where $x$ is an integer. What is the largest possible perimeter of the triangle?" ]	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-d082f2e504890df0def7c62ed4510e73df8a35ac3de3b2dac5d23925043a09ec	A triangle has sides of length 5, 6, and 7. Find the greatest integer less than the area of its circumcircle.	[ "40" ]	[ "A triangle has sides of length 5, 6, and 7. Find the greatest integer less than the area of its circumcircle." ]	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-6e388a9b983d61f634233d83bb85d9c90326b00790bdaa67a7704a9729f28d60	A triangle has sides of lengths $6, 7,$ and $8.$ What is the probability that a random point selected inside the triangle is closest to the side of length $8?$ Express your answer as a common fraction in reduced form.	[ "$\\frac{8}{21}$" ]	[ "A triangle has sides of lengths $6, 7,$ and $8.$ What is the probability that a random point selected inside the triangle is closest to the side of length $8?$ Express your answer as a common fraction in reduced form." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 27 }	43
omr-ws-73f5c559bb632263d2208f25f159268117c1a68aacfa269bcaa436e76b6dfbba	A triangle has sides of lengths $7$, $9$, and $x$, and an area of $6\sqrt{5}$. What is the value of $x$?	[ "4 and 2\\sqrt{61}" ]	[ "A triangle has sides of lengths $7$, $9$, and $x$, and an area of $6\\sqrt{5}$. What is the value of $x$?" ]	[ "aops_c4_high_school_math" ]	[ "tir" ]	{ "tir": 19 }	19
omr-ws-eae8fd8e011e8247a26b1134f89268f45ea9c293162103036fbd788b0da1cc91	A triangle has sides of lengths 10, 17, and 21. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the other two vertices of the square touch the two shorter sides of the triangle. What is the length of the side of the square?	[ "\$\\frac{168}{29}\$" ]	[ "A triangle has sides of lengths 10, 17, and 21. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the other two vertices of the square touch the two shorter sides of the triangle. What is the length of the side of the square?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 28 }	44
omr-ws-6eb762ec7356e9b5f93efb862dc8a093eb06039617d943b5e5443d250515f409	A triangle has sides of lengths 18', 24', and 26'. What is the length of the inradius to the nearest tenth of a foot?	[ "6.1 ft." ]	[ "A triangle has sides of lengths 18', 24', and 26'. What is the length of the inradius to the nearest tenth of a foot?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 31 }	47
omr-ws-29238bb4edf5cdfd8762b701b42d7d6cdae1bb872f77e896cbea1b156de4163d	A triangle has sides of lengths 20, 24, and 26. A circle is circumscribed about this triangle. Rounded to the nearest integer, find the area of the largest triangle that can be inscribed in this circle.	[ "260" ]	[ "A triangle has sides of lengths 20, 24, and 26. A circle is circumscribed about this triangle. Rounded to the nearest integer, find the area of the largest triangle that can be inscribed in this circle." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-cbfe49e5e747799689e87e540fdb9e7878df1ea106c8646b4e977afdc4355202	A triangle has sides of lengths 3k, 4k, and 5k, where k is a positive integer such that 0 < k < 100. For how many distinct values of k is the distance from the centroid to the vertex of the largest angle an even integer?	[ "16" ]	[ "A triangle has sides of lengths 3k, 4k, and 5k, where k is a positive integer such that 0 < k < 100. For how many distinct values of k is the distance from the centroid to the vertex of the largest angle an even integer?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 1 }	17
omr-ws-456c5407c180cc49e858171a04482e131543eaa4c862062255e683b335c0f84e	A triangle has sides of lengths 4 and 7, and two of its angles are in the ratio 3:1. What is the length of the third side of the triangle?	[ "\$\\frac{9}{4}\$" ]	[ "A triangle has sides of lengths 4 and 7, and two of its angles are in the ratio 3:1. What is the length of the third side of the triangle?" ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-bfc0f0d449834338d532d1480e81d518f85e22a8b8c8a4e3acb101c74b863871	A triangle has sides of lengths $a$, $b$, and $c$ with $a \geq b \geq c$ and an area of 1. Determine the minimum possible value for $b$.	[ "\$\\sqrt{2}\$" ]	[ "A triangle has sides of lengths \$a\$, \$b\$, and \$c\$ with \$a \\geq b \\geq c\$ and an area of 1. Determine the minimum possible value for \$b\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 8 }	23
omr-ws-44c8cbc9a0f3850370946b988079c25d995149038bf3cddf3d1dbb15d3ac93fd	A triangle has sides that form an arithmetic progression, and the angle opposite the longest side is three times the angle opposite the shortest side. The ratio of the longest side to the shortest side can be expressed as $\frac{a+\sqrt{b}}{c}$, where $a, b, c$ are positive integers, $b$ is not divisible by the square ...	[ "50" ]	[ "A triangle has sides that form an arithmetic progression, and the angle opposite the longest side is three times the angle opposite the shortest side. The ratio of the longest side to the shortest side can be expressed as $\\frac{a+\\sqrt{b}}{c}$, where $a, b, c$ are positive integers, $b$ is not divisible by the ...	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 11 }	27
omr-ws-a05cc02d5a7b1e61bab5a61a6de458d5593d4fa6d925003157bd3541f65b9503	A triangle has the property that all angles are integers (in degrees) and one angle is an integral multiple of another. An exterior angle of the triangle is 90 degrees. Describe all such non-similar triangles.	[ "(90^\\circ, 1^\\circ, 89^\\circ)" ]	[ "A triangle has the property that all angles are integers (in degrees) and one angle is an integral multiple of another. An exterior angle of the triangle is 90 degrees. Describe all such non-similar triangles." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 5 }	5
omr-ws-e9a2137a083649ef070ab2c8a932909f4bafc65050c9c09a6ab34783107fc534	A triangle has the property that its sides form an arithmetic progression, and that the angle opposite the longest side is three times the angle opposite the shortest side. The ratio of the longest side to the shortest side can be expressed as $\frac{a+\sqrt{b}}{c}$, where $a, b, c$ are positive integers, $b$ is not di...	[ "50" ]	[ "A triangle has the property that its sides form an arithmetic progression, and that the angle opposite the longest side is three times the angle opposite the shortest side. The ratio of the longest side to the shortest side can be expressed as $\\frac{a+\\sqrt{b}}{c}$, where $a, b, c$ are positive integers, $b$ is...	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-e30ca1754a32ce0767586e5940413e32ed71dda7ef0dbf9f7fb61aab5617c9e0	A triangle has three different integer side lengths and a perimeter of 20 units. What is the maximum length of any one side?	[ "9" ]	[ "A triangle has three different integer side lengths and a perimeter of 20 units. What is the maximum length of any one side?" ]	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-911703e2be369f9141a84cbb466e4508928aaa0aa0ae8cedea19d189c5393377	A triangle has three medians that are in geometric progression. The smallest angle in the triangle is $ \frac{\pi}{4} $. Find the angles of the triangle constructed using these three medians as its sides.	[ "\$ \\frac{\\pi}{4}, \\frac{\\pi}{3}, \\frac{5\\pi}{12} \$", "\\frac{5\\pi}{12}" ]	[ "A triangle has three medians that are in geometric progression. The smallest angle in the triangle is \$ \\frac{\\pi}{4} \$. Find the angles of the triangle constructed using these three medians as its sides." ]	[ "aops_c4_high_school_math", "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 3, "tir": 1 }	4
omr-ws-c7aeb1911aa82709a1f1f8f1e67d052a874353f02be4834b2d609eba32d89e73	A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order.	[ "2, 3, \\text{ and } 4" ]	[ "A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order." ]	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 6 }	6
omr-ws-5365fe5365cea67330e86630983186e2f42228fb0a72838a8c61f8bc421fd95f	A triangle has two altitudes $ h $ and $ k $ where $ h \neq k $. Determine the upper and lower bounds of the length of the third altitude in terms of $ h $ and $ k $.	[ "\\dfrac{hk}{\|h - k\|}" ]	[ "A triangle has two altitudes \$ h \$ and \$ k \$ where \$ h \\neq k \$. Determine the upper and lower bounds of the length of the third altitude in terms of \$ h \$ and \$ k \$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 6 }	6
omr-ws-60269221289049ceae1d728f082d363c57e7edf338274ac61a6cfda778c8241a	A triangle has two of its sides along the coordinate axes, and its third side touches the circle given by the equation $x^2 + y^2 = r^2$. Find the equation of the locus of the circumcenter of the triangle.	[ "\\dfrac{1}{x^2} + \\dfrac{1}{y^2} = \\dfrac{4}{r^2}" ]	[ "A triangle has two of its sides along the coordinate axes, and its third side touches the circle given by the equation \$x^2 + y^2 = r^2\$. Find the equation of the locus of the circumcenter of the triangle." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-ba2fb99d6833261aab8a3b5d7f59dabf07e449d6b4c857615a3925eb7b8ff7b6	A triangle has two of its vertices at $(0,1)$ and $(2,2)$ in a Cartesian coordinate system. Its third vertex is on the $x$-axis. If the area of the triangle is 2 square units, what are the possible coordinates for this third vertex?	[ "(-6, 0)" ]	[ "A triangle has two of its vertices at $(0,1)$ and $(2,2)$ in a Cartesian coordinate system. Its third vertex is on the $x$-axis. If the area of the triangle is 2 square units, what are the possible coordinates for this third vertex?" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 9 }	9
omr-ws-6c470e86438c6927e00a8b0fbf5dbdbe78f3bdea426bd4c534e9760c40002ff4	A triangle has two sides measuring 6 and 12 units in length. The median to the third side has length $ x $. Find the sum of all the possible integral values of $ x $.	[ "30" ]	[ "A triangle has two sides measuring 6 and 12 units in length. The median to the third side has length \$ x \$. Find the sum of all the possible integral values of \$ x \$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-b35fc0fbccdabb0eea06391ac066603a7c3e28133020d0318cd51ca0f1e89a8c	A triangle has two sides on the lines $x - 2y = 4$ and $x + 2y = 8$, and the third vertex at the point $(3, 4)$. What is the area of this triangle?	[ "6" ]	[ "A triangle has two sides on the lines \$x - 2y = 4\$ and \$x + 2y = 8\$, and the third vertex at the point \$(3, 4)\$. What is the area of this triangle?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 7, "genselect": 1 }	8
omr-ws-ce574681ac08654b4e50519a4f370b432a9b092deb83f34f0414f57615ad7b7c	A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.	[ "409" ]	[ "A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$." ]	[ "MATH_training_set", "aops_c5_contests_amp_programs" ]	[ "cot", "genselect", "tir" ]	{ "cot": 93, "genselect": 8, "tir": 17 }	118
omr-ws-3a339547c6bbb11599ff5bf67efb4abffedd97bf71908ee9f03c36fae8ed795e	A triangle has vertices $P=(-8,5)$, $Q=(-15,-19)$, and $R=(1,-7)$. The equation of the bisector of $\angle P$ can be written in the form $ax+2y+c=0$. Find $a+c$. [asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17); MP("P",P,N,f);M...	[ "89" ]	[ "A triangle has vertices $P=(-8,5)$, $Q=(-15,-19)$, and $R=(1,-7)$. The equation of the bisector of $\\angle P$ can be written in the form $ax+2y+c=0$. Find $a+c$.\n[asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17); MP(\"P\"...	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-56f7ffd742d0329a23fdfe851dd07c92af2b704935503aea0dc3cb57c02a1972	A triangle has vertices $P_{}^{}=(-8,5)$, $Q_{}^{}=(-15,-19)$, and $R_{}^{}=(1,-7)$. The equation of the bisector of $\angle P$ can be written in the form $ax+2y+c=0_{}^{}$. Find $a+c_{}^{}$. [asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-1...	[ "089" ]	[ "A triangle has vertices $P_{}^{}=(-8,5)$, $Q_{}^{}=(-15,-19)$, and $R_{}^{}=(1,-7)$. The equation of the bisector of $\\angle P$ can be written in the form $ax+2y+c=0_{}^{}$. Find $a+c_{}^{}$. \n[asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(-8,5),Q=(-15,-19),R=(...	[ "MATH_training_set" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 15 }	31
omr-ws-be89c7feff361f714a385efae892ca1f7a3991d31ee0bb776c234e65a7fbbe0b	A triangle has vertices A(0, 0), B(13, 0), and C(5, 7). The triangle is inscribed in a circle. What are the coordinates of the center of the circle? Express your answer in the form (x, y), where x and y are common fractions.	[ "\$\\left(\\frac{13}{2}, \\frac{9}{14}\\right)\$" ]	[ "A triangle has vertices A(0, 0), B(13, 0), and C(5, 7). The triangle is inscribed in a circle. What are the coordinates of the center of the circle? Express your answer in the form (x, y), where x and y are common fractions." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 25 }	41

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