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-661855b02deb4b24ec335e8261511c8b2b4c81154376339085de5c2657d463f5	$ x $, $ y $, and $ z $ are distinct non-zero integers such that $ -7 \le x, y, z \le 7 $. Compute the number of solutions $ (x, y, z) $ to the equation \[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{x+y+z}. \]	[ "504" ]	[ "\$ x \$, \$ y \$, and \$ z \$ are distinct non-zero integers such that \$ -7 \\le x, y, z \\le 7 \$. Compute the number of solutions \$ (x, y, z) \$ to the equation\n\\[\n\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} = \\frac{1}{x+y+z}.\n\\]" ]	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-98b1ff33698b194da5b83ad3897358cca84c811380bd629b0d9b37d5de5ed857	$(\cos 22.5^\circ)^2$ is a root of $x^2 + bx + c = 0$ where $b, c \in \mathbb{Q}$. Determine $(b, c)$.	[ "\$(b, c) = (-1, \\frac{1}{8})\$" ]	[ "\$(\\cos 22.5^\\circ)^2\$ is a root of \$x^2 + bx + c = 0\$ where \$b, c \\in \\mathbb{Q}\$. Determine \$(b, c)\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 31 }	47
omr-ws-6c06cbee81c8cccfc7e1d2bc3b3e9e54f38cdeec837ff755a30f94e3e1ad4e1b	$(a_n)_{n=0}^\infty$ is a sequence on integers. For every $n \ge 0$, $a_{n+1} = a_n^3 + a_n^2$. The number of distinct residues of $a_i$ in $\pmod{11}$ can be at most?	[ "3" ]	[ "\$(a_n)_{n=0}^\\infty\$ is a sequence on integers. For every \$n \\ge 0\$, \$a_{n+1} = a_n^3 + a_n^2\$. The number of distinct residues of \$a_i\$ in \$\\pmod{11}\$ can be at most?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 4 }	20
omr-ws-8f9880ba568bffba46b294c257b83a5122ec36ace4109fd6573f207e1b6d1e45	$1 \le n \le 455$ and $n^3 \equiv 1 \pmod{455}$. The number of solutions is?	[ "9" ]	[ "\$1 \\le n \\le 455\$ and \$n^3 \\equiv 1 \\pmod{455}\$. The number of solutions is?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-bdc66eff784ec01e5a770fac2ee5f681d3a438ccd2d0612b46e8f24eea41a466	$1989$ equal circles are arbitrarily placed on a table without overlap. What is the least number of colors needed such that all the circles can be painted with any two tangential circles colored differently?	[ "4" ]	[ "\$1989\$ equal circles are arbitrarily placed on a table without overlap. What is the least number of colors needed such that all the circles can be painted with any two tangential circles colored differently?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-c9bb15426d8e2dbd0ea9b08d75cceee0cd28e3c044d297198bcb5e24cdca65e8	$379$ is a prime. Let $\Omega(k)$ denote the smallest positive integer $n$ such that $379 \mid (k^n - 1)$. A fair $378$-sided die, with faces numbered $1, \ldots, 378$, is rolled, and the result is $r$. Let $p$ denote the expected value of $\Omega(r)$. If $p = \frac{m}{n}$ where $m$ and $n$ are ...	[ "647" ]	[ "\$379\$ is a prime. Let \$\\Omega(k)\$ denote the smallest positive integer \$n\$ such that \$379 \\mid (k^n - 1)\$. A fair \$378\$-sided die, with faces numbered \$1, \\ldots, 378\$, is rolled, and the result is \$r\$. Let \$p\$ denote the expected value of \$\\Omega(r)\$. If \\(p = \\frac{m}{n}...	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-edd0e4fb262e8a1921b3214662eec2f2ecb31141b5aec1b2602e0b73c34218c7	$5M$ moppers mop a messy mezzanine in 9 minutes. Meanwhile, 18 moppers mop a mat in $90/M$ minutes. Find the value of $M$.	[ "\$6\$" ]	[ "\$5M\$ moppers mop a messy mezzanine in 9 minutes. Meanwhile, 18 moppers mop a mat in \$90/M\$ minutes. Find the value of \$M\$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-9e84a38a80eb79a0a96b27d6c0c555a969450cd23985fdadf4e75b5bb7bd06dd	$A, B, C$ are playing a backgammon tournament. At first, $A$ plays with $B$. Then the winner plays with $C$. As the tournament goes on, the last winner plays with the player who did not play in the previous game. When a player wins two successive games, he wins the tournament. If each player has an equal chance...	[ "\$\\frac{2}{7}\$" ]	[ "\$A, B, C\$ are playing a backgammon tournament. At first, \$A\$ plays with \$B\$. Then the winner plays with \$C\$. As the tournament goes on, the last winner plays with the player who did not play in the previous game. When a player wins two successive games, he wins the tournament. If each player has an...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 32, "genselect": 2, "tir": 8 }	42
omr-ws-50e6cb089d5bafd1938c6d3391ebec4767887cb6b9c6a2732975ce8122666c39	$AB = AC$, $\angle BAC = 80^\circ$. Let $E$ be a point inside $\triangle ABC$ such that $AE = EC$ and $\angle EAC = 10^\circ$. What is the measure of $\angle EBC$?	[ "20^\\circ" ]	[ "\$AB = AC\$, \$\\angle BAC = 80^\\circ\$. Let \$E\$ be a point inside \$\\triangle ABC\$ such that \$AE = EC\$ and \$\\angle EAC = 10^\\circ\$. What is the measure of \$\\angle EBC\$?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 48, "tir": 19 }	67
omr-ws-32fde88dc02d9691de48b6c888cee9887ca7e38d586912422bd7968a01909da9	$ABCDEFGHIJ$ is a regular decagon. If sides $AB$ and $CD$ are extended to meet at point $K$, find the measure of $\angle K$.	[ "108" ]	[ "\$ABCDEFGHIJ\$ is a regular decagon. If sides \$AB\$ and \$CD\$ are extended to meet at point \$K\$, find the measure of \$\\angle K\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 5, "genselect": 3, "tir": 3 }	11
omr-ws-8e1ab3ef5d5b264cb47015e94d13b2b0a3347a743ae1d84022cdb1101e065400	$ABCDEFGH$ is an equiangular octagon with side lengths 2, $4\sqrt{2}$, 1, $3\sqrt{2}$, 2, $3\sqrt{2}$, 3, and $2\sqrt{2}$, in that order. Compute the area of the octagon.	[ "45" ]	[ "\$ABCDEFGH\$ is an equiangular octagon with side lengths 2, \$4\\sqrt{2}\$, 1, \$3\\sqrt{2}\$, 2, \$3\\sqrt{2}\$, 3, and \$2\\sqrt{2}\$, in that order. Compute the area of the octagon." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-39aeffd92910e4c115e4d1c393dd937955225dd610e30529662073f6a6f11a5e	$ABCDE$ is a convex pentagon with $m(\widehat{B}) = m(\widehat{D}) = 90^\circ$, $m(\widehat{C}) = 120^\circ$, $\|AB\| = 2$, $\|BC\| = \|CD\| = \sqrt{3}$, and $\|ED\| = 1$. Find $\|AE\|$.	[ "\$\\sqrt{3}\$" ]	[ "\$ABCDE\$ is a convex pentagon with \$m(\\widehat{B}) = m(\\widehat{D}) = 90^\\circ\$, \$m(\\widehat{C}) = 120^\\circ\$, \$\|AB\| = 2\$, \$\|BC\| = \|CD\| = \\sqrt{3}\$, and \$\|ED\| = 1\$. Find \$\|AE\|\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 33, "genselect": 5, "tir": 20 }	58
omr-ws-3a7767288e5b7b7c7a24085c1ea3ab921909bb8f611e67431e3af30b7b6aad5b	$ABCD$ is a cyclic quadrilateral that has an inscribed circle. The diagonals of $ABCD$ intersect at $P$. If $AB = 1$, $CD = 4$, and $BP : DP = 3 : 8$, then the area of the inscribed circle of $ABCD$ can be expressed as $\frac{p\pi}{q}$, where $p$ and $q$ are relatively prime positive integers. Deter...	[ "49" ]	[ "\$ABCD\$ is a cyclic quadrilateral that has an inscribed circle. The diagonals of \$ABCD\$ intersect at \$P\$. If \$AB = 1\$, \$CD = 4\$, and \$BP : DP = 3 : 8\$, then the area of the inscribed circle of \$ABCD\$ can be expressed as \$\\frac{p\\pi}{q}\$, where \$p\$ and \$q\$ are relatively pri...	[ "aops_c5_contests_amp_programs" ]	[ "cot", "genselect" ]	{ "cot": 14, "genselect": 8 }	22
omr-ws-9c29c7783b775957cf4bb79a9f06b4737e01fff52ef0582b909c09299557467a	$ABCD$ is a rectangle, $M$ is the midpoint of $BC$, and $MA \perp MB$. The perimeter $P_{ABCD} = 24$. Find the sides of the rectangle.	[ "8" ]	[ "\$ABCD\$ is a rectangle, \$M\$ is the midpoint of \$BC\$, and \$MA \\perp MB\$. The perimeter \$P_{ABCD} = 24\$. Find the sides of the rectangle." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 12 }	12
omr-ws-8294305c384c38cd40b495c75b7ffde9083e4d5824c1a948d3db0dbab5994afb	$ABCD$ is a square with $AB = 20$. Point $P$ is on $CD$ and point $Q$ is on $AD$ such that $AP \perp BQ$. Let $R$ be the intersection of $AP$ and $BQ$. Given $AR = 12$, find the area of quadrilateral $BCPR$.	[ "154" ]	[ "\$ABCD\$ is a square with \$AB = 20\$. Point \$P\$ is on \$CD\$ and point \$Q\$ is on \$AD\$ such that \$AP \\perp BQ\$. Let \$R\$ be the intersection of \$AP\$ and \$BQ\$. Given \$AR = 12\$, find the area of quadrilateral \$BCPR\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 6 }	22
omr-ws-1e3dd9c68d556a1c9ba68531ab72d6373bcbf6a8c2c241bb9f48002c5d1bfc95	$ABCD$ is a square. Describe the locus of points $P$, different from $A, B, C, D$, on the plane for which $\widehat{APB} + \widehat{CPD} = 180^\circ$.	[ "\\text{the two diagonals of the square}" ]	[ "\$ABCD\$ is a square. Describe the locus of points \$P\$, different from \$A, B, C, D\$, on the plane for which \$\\widehat{APB} + \\widehat{CPD} = 180^\\circ\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 10, "genselect": 7 }	17
omr-ws-6571e31238d6e50e07daa25f81c7b15235944329d7cd2bca4fb15d5c3e54ff32	$ABCD$ is a trapezoid, and the bases $AD$ and $BC$ are extended in both directions. The bisectors of the external angles $A$ and $B$ intersect at point $K$, and the bisectors of the external angles $C$ and $D$ intersect at point $E$. Find the perimeter of the trapezoid if $KE = 20$.	[ "40" ]	[ "\$ABCD\$ is a trapezoid, and the bases \$AD\$ and \$BC\$ are extended in both directions. The bisectors of the external angles \$A\$ and \$B\$ intersect at point \$K\$, and the bisectors of the external angles \$C\$ and \$D\$ intersect at point \$E\$. Find the perimeter of the trapezoid if \\(KE ...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-bae40dfbe73e0a97346462ec06256083e17ec41a4a0403f9dce8a075b2026435	$ABCD$ is an isosceles trapezium ($AD \parallel BC$). $N$ is the point of intersection of the perpendicular bisector to side $AB$ with line $BC$. It turned out that $AN \parallel CD$. Find the angles of the trapezoid.	[ "120^\\circ" ]	[ "\$ABCD\$ is an isosceles trapezium (\$AD \\parallel BC\$). \$N\$ is the point of intersection of the perpendicular bisector to side \$AB\$ with line \$BC\$. It turned out that \$AN \\parallel CD\$. Find the angles of the trapezoid." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 14 }	14
omr-ws-2a3c5c1b7b0721e56188c52a4482b8f61129b0ba927c01df2272d184d6076284	$ABCD$ is an isosceles trapezoid with $AB = 10$ and $CD = 8$. Point $E$ is the midpoint of $AB$ and point $F$ is the midpoint of $CD$, with $EF = 12$. Point $R$ is on segment $EF$ such that $\angle CRB = 90^\circ$. Find all possible lengths for $RE$.	[ "2, 10" ]	[ "\$ABCD\$ is an isosceles trapezoid with \$AB = 10\$ and \$CD = 8\$. Point \$E\$ is the midpoint of \$AB\$ and point \$F\$ is the midpoint of \$CD\$, with \$EF = 12\$. Point \$R\$ is on segment \$EF\$ such that \$\\angle CRB = 90^\\circ\$. Find all possible lengths for \$RE\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 5, "tir": 10 }	15
omr-ws-997d2f6736b7f3db823ac4ce46e868500ac092782c76732a65fb1e359a96e514	$ABC$ is a right triangle at $B$ with $AB = BC$. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $BM = BN$. The lines perpendicular to $AN$ drawn from $M$ and $B$ intersect side $AC$ at points $P$ and $Q$, respectively. If $AP = 6$ and $PQ = 5$, calculate \(...	[ "11" ]	[ "\$ABC\$ is a right triangle at \$B\$ with \$AB = BC\$. Let \$M\$ and \$N\$ be points on the sides \$AB\$ and \$BC\$, respectively, such that \$BM = BN\$. The lines perpendicular to \$AN\$ drawn from \$M\$ and \$B\$ intersect side \$AC\$ at points \$P\$ and \$Q\$, respectively. If \\(AP ...	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-73ec2c44d81c4c7215a18791b60f89fdfbba79484b6a68ff21d1c262316d12d7	$ABC$ is a triangle with $\overline{AB} = 20$, $\overline{AC} = 10$, and $\overline{BC} \ge 6$. Three points, $P$, $Q$, and $R$ are chosen from sides $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ respectively such that $\overline{AR} = 8$, $\overline{CQ} = 5$, and $\overline{BP} = 6$. ...	[ "25" ]	[ "\$ABC\$ is a triangle with \$\\overline{AB} = 20\$, \$\\overline{AC} = 10\$, and \$\\overline{BC} \\ge 6\$. Three points, \$P\$, \$Q\$, and \$R\$ are chosen from sides \$\\overline{BC}\$, \$\\overline{AC}\$, and \$\\overline{AB}\$ respectively such that \$\\overline{AR} = 8\$, \\(\\overline{C...	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-8b664b8d5f00ea794e532341a7474097d5965353739862337b5877829e668e8b	$ABC$ is a triangle. The points $A'$, $B'$, $C'$ are on sides $BC$, $CA$, $AB$ and $AA'$, $BB'$, $CC'$ meet at $O$. Also $\frac{AO}{A'O} + \frac{BO}{B'O} + \frac{CO}{C'O} = 92$. Find $\left(\frac{AO}{A'O}\right)\left(\frac{BO}{B'O}\right)\left(\frac{CO}{C'O}\right)$.	[ "94" ]	[ "\$ABC\$ is a triangle. The points \$A'\$, \$B'\$, \$C'\$ are on sides \$BC\$, \$CA\$, \$AB\$ and \$AA'\$, \$BB'\$, \$CC'\$ meet at \$O\$. Also \$\\frac{AO}{A'O} + \\frac{BO}{B'O} + \\frac{CO}{C'O} = 92\$. Find \\(\\left(\\frac{AO}{A'O}\\right)\\left(\\frac{BO}{B'O}\\right)\\left(\\frac{CO}{...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 15, "genselect": 8 }	23
omr-ws-29e5def6c92462f4d7f6421a3ac488f7446a2bd374c9b7e2eda079ceceb8e390	$ABC$ is an acute triangle with perimeter 60. $D$ is a point on $\overline{BC}$. The circumcircles of triangles $ABD$ and $ADC$ intersect $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$ respectively such that $DE = 8$ and $DF = 7$. If $\angle EBC \cong \angle BCF$, then the value of \(\frac{A...	[ "15" ]	[ "\$ABC\$ is an acute triangle with perimeter 60. \$D\$ is a point on \$\\overline{BC}\$. The circumcircles of triangles \$ABD\$ and \$ADC\$ intersect \$\\overline{AC}\$ and \$\\overline{AB}\$ at \$E\$ and \$F\$ respectively such that \$DE = 8\$ and \$DF = 7\$. If \\(\\angle EBC \\cong \\angle ...	[ "aops_c5_contests_amp_programs" ]	[ "cot", "genselect" ]	{ "cot": 11, "genselect": 4 }	15
omr-ws-c972f25c8b96ca6d58411be19f276ffaf672a7670b3165dfb2118f16c2432cf7	$ABC$ is an equilateral triangle. $D$ is a point inside $\triangle ABC$ such that $AD = 8$, $BD = 13$, and $\angle ADC = 120^\circ$. What is the length of $DC$?	[ "15" ]	[ "\$ABC\$ is an equilateral triangle. \$D\$ is a point inside \$\\triangle ABC\$ such that \$AD = 8\$, \$BD = 13\$, and \$\\angle ADC = 120^\\circ\$. What is the length of \$DC\$?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 23, "genselect": 8, "tir": 3 }	34
omr-ws-93ebe2b9bdd067437bbd96c8289ee63ebdfb15dc4b514b6e60e21d96c75a77db	$ABC$ is an isosceles triangle with base $\overline{AB}$. $D$ is a point on $\overline{AC}$ and $E$ is the point on the extension of $\overline{BD}$ past $D$ such that $\angle BAE$ is right. If $BD = 15$, $DE = 2$, and $BC = 16$, then $CD$ can be expressed as $\frac{m}{n}$, where $m$ and \(n...	[ "225" ]	[ "\$ABC\$ is an isosceles triangle with base \$\\overline{AB}\$. \$D\$ is a point on \$\\overline{AC}\$ and \$E\$ is the point on the extension of \$\\overline{BD}\$ past \$D\$ such that \$\\angle BAE\$ is right. If \$BD = 15\$, \$DE = 2\$, and \$BC = 16\$, then \$CD\$ can be expressed as \\(...	[ "aops_c5_contests_amp_programs" ]	[ "cot", "genselect" ]	{ "cot": 14, "genselect": 8 }	22
omr-ws-9dc5c8ff76b704acff42583304a2d53f139a4d0934197202b39ca32e02484ae9	$AB$ is a diameter of Circle $O$. $C$ is a point on Circle $O$. From $C$, draw a line perpendicular to $AB$, meeting $AB$ at $D$. Using $CD$ as the radius, draw Circle $C$. Circle $C$ intersects Circle $O$ at $P$ and $Q$. Connect $PQ$, which intersects $CD$ at $G$. If $GP = 12$ and \...	[ "24" ]	[ "\$AB\$ is a diameter of Circle \$O\$. \$C\$ is a point on Circle \$O\$. From \$C\$, draw a line perpendicular to \$AB\$, meeting \$AB\$ at \$D\$. Using \$CD\$ as the radius, draw Circle \$C\$. Circle \$C\$ intersects Circle \$O\$ at \$P\$ and \$Q\$. Connect \$PQ\$, which intersects \\...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 14, "genselect": 5 }	19
omr-ws-c445c4a78ed4f9dc758f438371dc49981f9d4e6ac99e45dea5c719958483dfda	$AB$ is tangent to a circle at $A$, $BC$ passes through the center of the circle, and $CD$ is a chord of the circle that is parallel to $AB$. If $AB=6$ and $BC=12$, what is the length of $CD$?	[ "\$\\frac{36}{5}\$" ]	[ "\$AB\$ is tangent to a circle at \$A\$, \$BC\$ passes through the center of the circle, and \$CD\$ is a chord of the circle that is parallel to \$AB\$. If \$AB=6\$ and \$BC=12\$, what is the length of \$CD\$?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 12 }	27
omr-ws-eadc304b86e94c68da596d0c2510a14275d18eb4b7c1b3420813b88c4dd9d7c4	$AH$ is the height of the isosceles triangle $ABC$ ($AB = BC$), and $AK$ is the bisector of the angle $BAH$. Find the angle $\angle KAC$.	[ "\$22.5^\\circ\$" ]	[ "\$AH\$ is the height of the isosceles triangle \$ABC\$ (\$AB = BC\$), and \$AK\$ is the bisector of the angle \$BAH\$. Find the angle \$\\angle KAC\$." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-80b3063cc2bac0465a6b8c9af75dcedaefa6244726bacc353edae441a7707d26	$A$ and $B$ are fixed points on a plane, and $L$ is a line passing through $A$ but not through $B$. $C$ is a variable point moving from $A$ toward infinity along a half-line of $L$. The incircle of $\triangle ABC$ touches $BC$ at $D$ and $AC$ at $E$. Let $P$ be the fixed point through which ...	[ "(2+\\sqrt{2}, \\sqrt{2})" ]	[ "\$A\$ and \$B\$ are fixed points on a plane, and \$L\$ is a line passing through \$A\$ but not through \$B\$. \$C\$ is a variable point moving from \$A\$ toward infinity along a half-line of \$L\$. The incircle of \$\\triangle ABC\$ touches \$BC\$ at \$D\$ and \$AC\$ at \$E\$. Let \$P\$...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-5242075dc731fdd542b2803034200dfa7e879ee67938487e910da05325f0abd2	$A_1A_2A_3A_4A_5A_6A_7$ is a regular heptagon (7-sided figure) centered at the origin where $A_1 = (\sqrt[91]{6}, 0)$. $B_1B_2B_3 \ldots B_{13}$ is a regular triskaidecagon (13-sided figure) centered at the origin where $B_1 = (0, \sqrt[91]{41})$. Compute the product of all lengths $A_iB_j$, where $i$ range...	[ "\\sqrt{1717}" ]	[ "\$A_1A_2A_3A_4A_5A_6A_7\$ is a regular heptagon (7-sided figure) centered at the origin where \$A_1 = (\\sqrt[91]{6}, 0)\$. \$B_1B_2B_3 \\ldots B_{13}\$ is a regular triskaidecagon (13-sided figure) centered at the origin where \$B_1 = (0, \\sqrt[91]{41})\$. Compute the product of all lengths \$A_iB_j\$,...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-5f82ab029d5b9626ed76de536ba49010b5794910e2d82773edec58f85229f44f	$C_{1}$ and $C_{2}$ are two non-intersecting circles whose radii are in the ratio $1:2$. A third circle cuts the smaller circle at $A$ and $B$ and the bigger one at $C$ and $D$. $AB$ and $CD$ intersect at $P$. The ratio of the lengths of the tangents from $P$ to the circles $C_{1}$ and $C_{2}$...	[ "1:1" ]	[ "\$C_{1}\$ and \$C_{2}\$ are two non-intersecting circles whose radii are in the ratio \$1:2\$. A third circle cuts the smaller circle at \$A\$ and \$B\$ and the bigger one at \$C\$ and \$D\$. \$AB\$ and \$CD\$ intersect at \$P\$. The ratio of the lengths of the tangents from \$P\$ to the circ...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-47c42e5854da6416e716fe9406d91301f6878f82873f1a0cf2e71c0b089a7183	$M, N$, and $P$ are three collinear points, with $N$ between $M$ and $P$. Let $r$ be the perpendicular bisector of $NP$. A point $O$ is taken over $r$. $\omega$ is the circle with center $O$ passing through $N$. The tangents through $M$ to $\omega$ intersect $\omega$ at $T$ and $T'$. F...	[ "\$3(m+p)(x^{2}+y^{2})+2m(2m+p)x+m^{2}(m-p)=0\$" ]	[ "\$M, N\$, and \$P\$ are three collinear points, with \$N\$ between \$M\$ and \$P\$. Let \$r\$ be the perpendicular bisector of \$NP\$. A point \$O\$ is taken over \$r\$. \$\\omega\$ is the circle with center \$O\$ passing through \$N\$. The tangents through \$M\$ to \$\\omega\$ intersec...	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-46715958c134f95d8941663d31fc7a0a372f5f9573602e05834c0c0040f38689	$N$ consecutive integers sum to 2003, where $N$ is an odd number greater than three. Determine the smallest of these integers.	[ "-1000" ]	[ "\$N\$ consecutive integers sum to 2003, where \$N\$ is an odd number greater than three. Determine the smallest of these integers." ]	[ "aops_c5_contests_amp_programs" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 3 }	19
omr-ws-d3ed85e09452cf52966988529254f3f597469015a8dc48027f72b62179b1215e	$N$ is a 50-digit number (in the decimal scale). All digits except the 26th digit (from the left) are 1. If $N$ is divisible by 13, what is the 26th digit?	[ "3" ]	[ "\$N\$ is a 50-digit number (in the decimal scale). All digits except the 26th digit (from the left) are 1. If \$N\$ is divisible by 13, what is the 26th digit?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 47, "genselect": 8 }	55
omr-ws-8ebcac5c85be81c455043462669c483c3db5348d9637583cb631ca4192d5c572	$O$ is the center of a semicircle with radius $OA$ on the base. $C$ is a point on the base between $O$ and $A$, and $B$ is a point on the semicircle such that $\overline{BC}$ is perpendicular to $\overline{OA}$. $BO$ is extended through $O$ to $E$ and $D$ is the intersection of the lines $AE$ ...	[ "\\sqrt{2 - \\sqrt{2}}" ]	[ "\$O\$ is the center of a semicircle with radius \$OA\$ on the base. \$C\$ is a point on the base between \$O\$ and \$A\$, and \$B\$ is a point on the semicircle such that \$\\overline{BC}\$ is perpendicular to \$\\overline{OA}\$. \$BO\$ is extended through \$O\$ to \$E\$ and \$D\$ is the in...	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-f9ae1d62df1577da245059298be3dccad29bf99e3b2fa2674dec9796dfa6bb55	$PA$ and $PB$ are tangents to a circle, and a secant passing through $P$ intersects the circle at $C$ and $D$ respectively, with $D-C-P$. Chords $AC$, $BC$, $BD$, and $DA$ are drawn. If $AC = 9$, $AD = 12$, and $BD = 10$, find $BC$.	[ "7.5" ]	[ "\$PA\$ and \$PB\$ are tangents to a circle, and a secant passing through \$P\$ intersects the circle at \$C\$ and \$D\$ respectively, with \$D-C-P\$. Chords \$AC\$, \$BC\$, \$BD\$, and \$DA\$ are drawn. If \$AC = 9\$, \$AD = 12\$, and \$BD = 10\$, find \$BC\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 5, "genselect": 5, "tir": 5 }	15
omr-ws-083a21222b237c0ca059e3488f005cfb351c248c04ccbcf6b0c05ee314243a25	$P$ and $Q$ are points on the bisector of the exterior angle $A$ of triangle $ABC$ with $A$ between $P$ and $Q$, and $PB$ is parallel to $QC$. Point $D$ is on $BC$ such that $DP = DQ$. If the length of $AB$ is 10 units, find the length of $DQ$.	[ "10" ]	[ "\$P\$ and \$Q\$ are points on the bisector of the exterior angle \$A\$ of triangle \$ABC\$ with \$A\$ between \$P\$ and \$Q\$, and \$PB\$ is parallel to \$QC\$. Point \$D\$ is on \$BC\$ such that \$DP = DQ\$. If the length of \$AB\$ is 10 units, find the length of \$DQ\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 4, "genselect": 2 }	6
omr-ws-4832bb58d1c2e5920d342a7b2d4e99b8a4b5961a47baecf9b2215bb9b17c9767	$P$ is a point in the interior of $\triangle ABC$, and $\angle ABP = \angle PCB = 10^\circ$. (a) If $\angle PBC = 10^\circ$ and $\angle ACP = 20^\circ$, what is the value of $\angle BAP$? (b) If $\angle PBC = 20^\circ$ and $\angle ACP = 10^\circ$, what is the value of $\angle BAP$?	[ "30^\\circ" ]	[ "\$P\$ is a point in the interior of \$\\triangle ABC\$, and \$\\angle ABP = \\angle PCB = 10^\\circ\$. \n(a) If \$\\angle PBC = 10^\\circ\$ and \$\\angle ACP = 20^\\circ\$, what is the value of \$\\angle BAP\$?\n(b) If \$\\angle PBC = 20^\\circ\$ and \$\\angle ACP = 10^\\circ\$, what is the value o...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-4c8541efa6d5a47fdf8e00b2a456a867cdedb4a884d08dd16819c91e66d8c6c4	$P$ is a point outside circle $C$. Draw tangents $PA$ and $PB$ to $C$. Choose a point $K$ on $AB$. The circumcircle of triangle $PBK$ intersects $C$ again at $T$. $P'$ is the reflection of $P$ with respect to $A$. If $\angle PBT = \theta$ and $\angle P'KA = \phi$, calculate \(\theta + \phi...	[ "90" ]	[ "\$P\$ is a point outside circle \$C\$. Draw tangents \$PA\$ and \$PB\$ to \$C\$. Choose a point \$K\$ on \$AB\$. The circumcircle of triangle \$PBK\$ intersects \$C\$ again at \$T\$. \$P'\$ is the reflection of \$P\$ with respect to \$A\$. If \$\\angle PBT = \\theta\$ and \\(\\angle P'K...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 6 }	6
omr-ws-2cf7481e9e23c71ebd0d14ecd3c9ad074a482f7efc24640fd1f2a068ad2e2d47	$P$ is a variable point on a given straight line, and $A$ is a fixed point. $Q$ is a point on $AP$ or its extension such that the ratio $AQ : AP$ is constant. Find the locus of $Q$.	[ "The locus of \$Q\$ is 2 lines." ]	[ "\$P\$ is a variable point on a given straight line, and \$A\$ is a fixed point. \$Q\$ is a point on \$AP\$ or its extension such that the ratio \$AQ : AP\$ is constant. Find the locus of \$Q\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "tir" ]	{ "tir": 1 }	1
omr-ws-ecdf4abe71e903468ba55ca64bb62992a2b1cfaf73e73b8a68d567738b167ec0	$P$ is the intersection point of diagonals of cyclic $ABCD$. The circumcenters of $\triangle APB$ and $\triangle CPD$ lie on the circumcircle of $ABCD$. If $AC + BD = 18$, then the area of $ABCD$ is?	[ "\$\\frac{81\\sqrt{3}}{4}\$" ]	[ "\$P\$ is the intersection point of diagonals of cyclic \$ABCD\$. The circumcenters of \$\\triangle APB\$ and \$\\triangle CPD\$ lie on the circumcircle of \$ABCD\$. If \$AC + BD = 18\$, then the area of \$ABCD\$ is?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 4, "genselect": 4, "tir": 2 }	10
omr-ws-7754b9d0e5038d54c22a95a7bb4e525bcd29b15e32e4ecf02be20be369090831	$S(x)$ denotes the sum of the digits of $x$ in decimal representation. If $m \in \mathbb{N}$ satisfies $m \equiv S(m) \equiv 0 \pmod{6}$, we call it a good number. How many good numbers are there less than 2012?	[ "167" ]	[ "\$S(x)\$ denotes the sum of the digits of \$x\$ in decimal representation. If \$m \\in \\mathbb{N}\$ satisfies \$m \\equiv S(m) \\equiv 0 \\pmod{6}\$, we call it a good number. How many good numbers are there less than 2012?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 8, "genselect": 7 }	15
omr-ws-70c7e91a8a3742433dd3eed0491c0be8b63ccab5ad8e934248e2548d8748e282	$S$ is a set of positive integers containing 1 and 2002. No elements are larger than 2002. For every $n$ in $S$, the arithmetic mean of the other elements of $S$ is an integer. What is the largest possible number of elements of $S$?	[ "30" ]	[ "\$S\$ is a set of positive integers containing 1 and 2002. No elements are larger than 2002. For every \$n\$ in \$S\$, the arithmetic mean of the other elements of \$S\$ is an integer. What is the largest possible number of elements of \$S\$?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 2, "tir": 7 }	9
omr-ws-7ce924e8f52c2e1097929bcbed77356debde69832fc9b85fed51aa032e4d6acf	$\cot 10^\circ + \tan 5^\circ =$	[ "\\csc 10^\\circ" ]	[ "\$\\cot 10^\\circ + \\tan 5^\\circ =\$" ]	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-cdf243a94496b52cf4d53d4bef937ac7115c686b9b9e7f16d6a5bac126272031	$\mathbb{Q}$ is the set of all rational numbers. Find all functions $f: \mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{Q}$ such that for all $x, y, z \in \mathbb{Q}$, the equation $f(x, y) + f(y, z) + f(z, x) = f(0, x + y + z)$ holds.	[ "f(x, y) = c y" ]	[ "\$\\mathbb{Q}\$ is the set of all rational numbers. Find all functions \$f: \\mathbb{Q} \\times \\mathbb{Q} \\rightarrow \\mathbb{Q}\$ such that for all \$x, y, z \\in \\mathbb{Q}\$, the equation \$f(x, y) + f(y, z) + f(z, x) = f(0, x + y + z)\$ holds." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 8 }	8
omr-ws-0b356253e6f7e852b570643ccfdc34efeeaa249bb01474dd266b7976f9d7c3ad	$\prod_{k=1}^{45} \sin(2k-1)^\circ = \frac{1}{2^n}$. Determine the rational number $n$.	[ "\$\\frac{89}{2}\$" ]	[ "\$\\prod_{k=1}^{45} \\sin(2k-1)^\\circ = \\frac{1}{2^n}\$. Determine the rational number \$n\$." ]	[ "aops_c6_high_school_olympiads", "c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 38, "genselect": 4, "tir": 29 }	71
omr-ws-2c1b49749729e23ea9e54bace7d085a14614ff167f833ec640c00aa8d7122461	$\tan(11x) = \tan(34^\circ)$ and $\tan(19x) = \tan(21^\circ)$. Compute $\tan(5x)$.	[ "\\tan 13^\\circ" ]	[ "\$\\tan(11x) = \\tan(34^\\circ)\$ and \$\\tan(19x) = \\tan(21^\\circ)\$. Compute \$\\tan(5x)\$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 5 }	5
omr-ws-82dced8a82d5310087ab0eaefe02116373d6a67f61e0b71a908b0bd1a184227c	$\triangle ABC$ and $\triangle ADC$ are both right triangles with $m\angle B = m\angle D = 90^\circ$. If $BC = 5$, $AB = 12$, and $\overleftrightarrow{AB}$ is the angle bisector of $\angle DAC$, then find $AD$.	[ "\$\\frac{119}{13}\$" ]	[ "\$\\triangle ABC\$ and \$\\triangle ADC\$ are both right triangles with \$m\\angle B = m\\angle D = 90^\\circ\$. If \$BC = 5\$, \$AB = 12\$, and \$\\overleftrightarrow{AB}\$ is the angle bisector of \$\\angle DAC\$, then find \$AD\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 14, "genselect": 7, "tir": 21 }	42
omr-ws-5d7f1c8bf8525b848a6821c61d5333a71f41bf5783e6b5f143dda73cb3b350eb	$\triangle ABC$ and $\triangle ADE$ are two secants of a circle of radius 3 cm. $A$ is at a distance of 5 cm from the center of the circle. The secants include an angle of $30^\circ$. Then the area of $\triangle ACE$ is $10 \text{ cm}^2$. Then the area of $\triangle ADB$ (in $\text{cm}^2$) is ...	[ "\\dfrac{8}{5}" ]	[ "\$\\triangle ABC\$ and \$\\triangle ADE\$ are two secants of a circle of radius 3 cm. \$A\$ is at a distance of 5 cm from the center of the circle. The secants include an angle of \$30^\\circ\$. Then the area of \$\\triangle ACE\$ is \$10 \\text{ cm}^2\$. Then the area of \$\\triangle ADB\$ (in \\(\\...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-02360b98c7c14321b4393b2ba2b2c148bb77b3d51c29b565cf44a742a1b54efb	$\triangle ABC$ has all side lengths distinct. Two altitudes of this triangle have lengths 4 and 12. If the length of the third altitude is also an integer, find its maximum value.	[ "5" ]	[ "\$\\triangle ABC\$ has all side lengths distinct. Two altitudes of this triangle have lengths 4 and 12. If the length of the third altitude is also an integer, find its maximum value." ]	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-3361927b607ce77092d9ddd6c1bc823f86bd394514744cf9eb40c7de43140571	$\triangle ABC$ is an isosceles triangle with the length of sides $ AB $ and $ AC $ equal. Suppose $ D $ is a point on side $ AB $ and $ E $ is a point on side $ AC $ such that $ DE $ and $ BC $ are parallel. Further, suppose triangle $ ADE $ and trapezoid $ BCED $ have equal area and equal perime...	[ "\$\\sqrt{2} - 1\$" ]	[ "\$\\triangle ABC\$ is an isosceles triangle with the length of sides \$ AB \$ and \$ AC \$ equal. Suppose \$ D \$ is a point on side \$ AB \$ and \$ E \$ is a point on side \$ AC \$ such that \$ DE \$ and \$ BC \$ are parallel. Further, suppose triangle \$ ADE \$ and trapezoid \$ BCED \$ have...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 14, "genselect": 8, "tir": 21 }	43
omr-ws-1bc6cee9b56cd55af620df544d99cf33633b0a7e3979c18fac41460bec3f782c	$\triangle ABC$ satisfies $\overline{AB} = 13$, $\overline{BC} = 15$, $\overline{CA} = 14$. There exist three circles of equal radii that are each tangent to the interior of two sides of $\triangle ABC$, and no two circles coincide. These circles intersect at a common point $P$. Given that $O$ is the circ...	[ "73" ]	[ "\$\\triangle ABC\$ satisfies \$\\overline{AB} = 13\$, \$\\overline{BC} = 15\$, \$\\overline{CA} = 14\$. There exist three circles of equal radii that are each tangent to the interior of two sides of \$\\triangle ABC\$, and no two circles coincide. These circles intersect at a common point \$P\$. Given ...	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 6, "tir": 2 }	8
omr-ws-d6a4e7941554ff8c7dc8bc0d158ce60bd7961b764d78f882be8b1e51b13993ac	$\zeta_1, \zeta_2$, and $\zeta_3$ are complex numbers such that \[ \zeta_1 + \zeta_2 + \zeta_3 = 1, \] \[ \zeta_1^2 + \zeta_2^2 + \zeta_3^2 = 3, \] \[ \zeta_1^3 + \zeta_2^3 + \zeta_3^3 = 7. \] Compute $\zeta_1^7 + \zeta_2^7 + \zeta_3^7$.	[ "71" ]	[ "\$\\zeta_1, \\zeta_2\$, and \$\\zeta_3\$ are complex numbers such that\n\\[\n\\zeta_1 + \\zeta_2 + \\zeta_3 = 1,\n\\]\n\\[\n\\zeta_1^2 + \\zeta_2^2 + \\zeta_3^2 = 3,\n\\]\n\\[\n\\zeta_1^3 + \\zeta_2^3 + \\zeta_3^3 = 7.\n\\]\nCompute \$\\zeta_1^7 + \\zeta_2^7 + \\zeta_3^7\$." ]	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-2fdb5a683a5d2e657b4b44ddd91fae5bdec42bc1572de69a9b8c43884d4231ee	$\{A_n\}_{n \ge 1}$ is a sequence of positive integers such that \[ a_n = 2a_{n-1} + n^2 \] for all integers $n > 1$. Compute the remainder obtained when $a_{2004}$ is divided by 1000 if $a_1 = 1$.	[ "58" ]	[ "\$\\{A_n\\}_{n \\ge 1}\$ is a sequence of positive integers such that\n\\[\na_n = 2a_{n-1} + n^2\n\\]\nfor all integers \$n > 1\$. Compute the remainder obtained when \$a_{2004}\$ is divided by 1000 if \$a_1 = 1\$." ]	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-649fd1eb5f92cd40cdbdd6b4efa0b72772e1fdbd9040cf5217ba81374b00be4b	$a, b, c, d$ satisfy the following system of equations: \[ ab + c + d = 13 \] \[ bc + d + a = 27 \] \[ cd + a + b = 30 \] \[ da + b + c = 17. \] Compute the value of $a + b + c + d$.	[ "210" ]	[ "\$a, b, c, d\$ satisfy the following system of equations:\n\\[\nab + c + d = 13\n\\]\n\\[\nbc + d + a = 27\n\\]\n\\[\ncd + a + b = 30\n\\]\n\\[\nda + b + c = 17.\n\\]\nCompute the value of \$a + b + c + d\$." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-3ea16373e4bdeac07eadb46a9dd3f3bf8b3600e272dbe61319fe7f05b7bd46ca	$a, b, c$ are positive integers such that $a < b < c$. The system of equations $2x + y = 2003$ and $y = \|x - a\| + \|x - b\| + \|x - c\|$ has exactly one solution. Find the minimum value of $c$.	[ "1002" ]	[ "\$a, b, c\$ are positive integers such that \$a < b < c\$. The system of equations \$2x + y = 2003\$ and \$y = \|x - a\| + \|x - b\| + \|x - c\|\$ has exactly one solution. Find the minimum value of \$c\$." ]	[ "aops_c5_contests_amp_programs" ]	[ "cot", "genselect" ]	{ "cot": 9, "genselect": 5 }	14
omr-ws-75636515026c84032e5a11bfa98851a41b62b38b54ab00ffb11547f214ba1121	$a$ is a real number. $x_1$ and $x_2$ are the distinct roots of $x^2 + ax + 2 = x$. $x_3$ and $x_4$ are the distinct roots of $(x - a)^2 + a(x - a) + 2 = x$. If $x_3 - x_1 = 3(x_4 - x_2)$, then $x_4 - x_2$ will be?	[ "\$\\frac{a}{2}\$" ]	[ "\$a\$ is a real number. \$x_1\$ and \$x_2\$ are the distinct roots of \$x^2 + ax + 2 = x\$. \$x_3\$ and \$x_4\$ are the distinct roots of \$(x - a)^2 + a(x - a) + 2 = x\$. If \$x_3 - x_1 = 3(x_4 - x_2)\$, then \$x_4 - x_2\$ will be?" ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-51dc30c4450f0f870d1861a366ade088dab12ebd19e461ed2f884bb66bada5ee	$a$, $b$, $c$, $d$ are real numbers such that $a \neq b$, $b \neq c$, $c \neq d$, $d \neq a$, and $\frac{1}{(a-b)^2} + \frac{1}{(b-c)^2} + \frac{1}{(c-d)^2} + \frac{1}{(d-a)^2} = 1$. Find the minimum value of $a^2 + b^2 + c^2 + d^2$.	[ "4" ]	[ "\$a\$, \$b\$, \$c\$, \$d\$ are real numbers such that \$a \\neq b\$, \$b \\neq c\$, \$c \\neq d\$, \$d \\neq a\$, and \$\\frac{1}{(a-b)^2} + \\frac{1}{(b-c)^2} + \\frac{1}{(c-d)^2} + \\frac{1}{(d-a)^2} = 1\$. Find the minimum value of \$a^2 + b^2 + c^2 + d^2\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 14, "genselect": 6 }	20
omr-ws-5d66df73b6938d5dac2ff0df6b3f2b57d8198643d877fda82aaa1bb399cac072	$a_1, a_2, \ldots, a_{2014}$ is a permutation of $1, 2, 3, \ldots, 2014$. What is the greatest number of perfect squares that can be in the set $\{a_1^2 + a_2, a_2^2 + a_3, a_3^2 + a_4, \ldots, a_{2013}^2 + a_{2014}, a_{2014}^2 + a_1\}$?	[ "1007" ]	[ "\$a_1, a_2, \\ldots, a_{2014}\$ is a permutation of \$1, 2, 3, \\ldots, 2014\$. What is the greatest number of perfect squares that can be in the set \$\\{a_1^2 + a_2, a_2^2 + a_3, a_3^2 + a_4, \\ldots, a_{2013}^2 + a_{2014}, a_{2014}^2 + a_1\\}\$?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 12, "genselect": 6 }	18
omr-ws-574e791a8276ca29c2f1df13ef2df9abe4e4ba59d1ad6f2a933404ab44fa6c32	$f(x) = \frac{x^5}{5x^4 - 10x^3 + 10x^2 - 5x + 1}$. $\sum_{i=1}^{2009} f\left(\frac{i}{2009}\right) = ?$	[ "1005" ]	[ "\$f(x) = \\frac{x^5}{5x^4 - 10x^3 + 10x^2 - 5x + 1}\$. \$\\sum_{i=1}^{2009} f\\left(\\frac{i}{2009}\\right) = ?\$" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-84edc10f6b59f5425463dc656c3eaedbdb51cc33e2b1246718e6eac5f254c91a	$f(x) = x^3 - 6x^2 + 17x$. If $f(a) = 16$ and $f(b) = 20$, find $a + b$.	[ "4" ]	[ "\$f(x) = x^3 - 6x^2 + 17x\$. If \$f(a) = 16\$ and \$f(b) = 20\$, find \$a + b\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-2578166777cfbdedfc907063757edb1bfcc1feae2739a1cf9b3ae48875f891f2	$i$ integers are written out and then added together. Carl adds 100 to the sum of the first $i$ integers and gets the sum of the next $i$ integers. Find $i$.	[ "10" ]	[ "\$i\$ integers are written out and then added together. Carl adds 100 to the sum of the first \$i\$ integers and gets the sum of the next \$i\$ integers. Find \$i\$." ]	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-061f8ab4342a90ebe4a03846c0890cb520794b7c9f26d9724f7f4eba56d0481e	$k$ is a unique integer such that: \[\lfloor \log_2 1 \rfloor + \lfloor \log_2 2 \rfloor + \lfloor \log_2 3 \rfloor + \cdots + \lfloor \log_2 k \rfloor = 1994\] Find $k$.	[ "312" ]	[ "\$k\$ is a unique integer such that:\n\\[\\lfloor \\log_2 1 \\rfloor + \\lfloor \\log_2 2 \\rfloor + \\lfloor \\log_2 3 \\rfloor + \\cdots + \\lfloor \\log_2 k \\rfloor = 1994\\]\nFind \$k\$." ]	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-fa647972902f3976a00e43744c8d9cd2c1d178deb4f756864820f11f8e0b255e	$n \ge 2$ kids are trick-or-treating. They enter a haunted house in a single-file line such that each kid is friends with precisely the kids (or kid) adjacent to him. Inside the haunted house, they get mixed up and out of order. They meet up again at the exit, and leave in single file. After leaving, they realize tha...	[ "2^{n-1}" ]	[ "\$n \\ge 2\$ kids are trick-or-treating. They enter a haunted house in a single-file line such that each kid is friends with precisely the kids (or kid) adjacent to him. Inside the haunted house, they get mixed up and out of order. They meet up again at the exit, and leave in single file. After leaving, they rea...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-60e5c0c2d0d874984c0adffbc6b1fec70a12ce9ca30acbc245696d8162499300	$n$ buckets are arranged in a circle, each containing 1 token. Starting with one of the buckets, you take out a token and move it into the next bucket counterclockwise. You then take two tokens out of this bucket and move them into the next one. This process is repeated over and over, alternating moving one token and...	[ "999" ]	[ "\$n\$ buckets are arranged in a circle, each containing 1 token. Starting with one of the buckets, you take out a token and move it into the next bucket counterclockwise. You then take two tokens out of this bucket and move them into the next one. This process is repeated over and over, alternating moving one to...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 3, "genselect": 1 }	4
omr-ws-d8ce05b85bb5c758c4ec990be0db571a8b25c820706a6fb91104048880299282	$n$ integers are arranged along a circle in such a way that each number is equal to the absolute value of the difference of the two numbers following that number in the clockwise direction. If the sum of all numbers is 278, how many different values can $n$ take?	[ "2" ]	[ "\$n\$ integers are arranged along a circle in such a way that each number is equal to the absolute value of the difference of the two numbers following that number in the clockwise direction. If the sum of all numbers is 278, how many different values can \$n\$ take?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 38, "genselect": 8 }	46
omr-ws-6d140e7e05fa328e4243b1f7cb5133580d46b42bfef3a627f8865da010872748	$n$ teams take part in a tournament. For any two teams $A$ and $B$, if the game is played at $A$'s town, then call this game $A$'s "home game", and $B$'s "away game". Any two teams play against each other just twice, each having a "home game" respectively. Knowing that every team can be arranged more than o...	[ "8" ]	[ "\$n\$ teams take part in a tournament. For any two teams \$A\$ and \$B\$, if the game is played at \$A\$'s town, then call this game \$A\$'s \"home game\", and \$B\$'s \"away game\". Any two teams play against each other just twice, each having a \"home game\" respectively. Knowing that every team can ...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 7, "genselect": 5 }	12
omr-ws-b5e4d1a134206933129dc1f9fadc263c4d9324bbe62c66e690a53c2c5ab8e24d	$x + 19y \equiv 0 \pmod{23}$ and $x + y < 69$. How many pairs of $(x, y)$ are there in positive integers?	[ "100" ]	[ "\$x + 19y \\equiv 0 \\pmod{23}\$ and \$x + y < 69\$. How many pairs of \$(x, y)\$ are there in positive integers?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 17, "genselect": 8, "tir": 10 }	35
omr-ws-61b8f357331b654e379336c4d107b869f3172114378b252b35a88129db4e4232	$x$ and $y$ are non-negative integers such that $3^{x+6}6^{y+2}$ is a perfect cube. Find the least possible product of $xy$.	[ "0" ]	[ "\$x\$ and \$y\$ are non-negative integers such that \$3^{x+6}6^{y+2}\$ is a perfect cube. Find the least possible product of \$xy\$." ]	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-0d9a0e06fac81a12ef1320f9e60cab4ad85dae511ed583976d9f4f550c6e90d1	$x$ satisfies the equation $(1 + i)x^3 + 8ix^2 + (-8 + 8i)x + 36 = 0$. Compute the largest possible value of $\|x\|$.	[ "3\\sqrt{2}" ]	[ "\$x\$ satisfies the equation \$(1 + i)x^3 + 8ix^2 + (-8 + 8i)x + 36 = 0\$. Compute the largest possible value of \$\|x\|\$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-ca4852a3b126e9871bff86bb6fdfc339a771d59c69ed1ffa44c6c74d12773afa	\[ \left(\frac{66......64}{16....6}\right)^2 + \left(\frac{16....6}{66......64}\right)^2 = m \] Find $ \lfloor 100m \rfloor $. (In both the denominator and numerator, there is the same number of 6's.)	[ "1606" ]	[ "\\[ \\left(\\frac{66......64}{16....6}\\right)^2 + \\left(\\frac{16....6}{66......64}\\right)^2 = m \\]\n\nFind \$ \\lfloor 100m \\rfloor \$. (In both the denominator and numerator, there is the same number of 6's.)" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-b36b43eae055f2a5c0e8bb9fcb54c50ef16703a2ab11ea795229dfb7bd9e38e8	\[ \textbf{I}_{0}^{\infty}(m) = \int_{0}^{\infty} \frac{\ln^{m}(1+x^{2})}{1+x^{2}} \, \textbf{d}x. \]	[ " \\frac{\\pi}{2} \\cdot (-2)^m \\cdot \\left. \\frac{d^m}{ds^m} \\frac{\\Gamma\\left(\\frac{1}{2} - \\frac{s}{2}\\right)}{\\Gamma(1 - \\frac{s}{2})} \\right\|_{s=0} " ]	[ "\\[ \\textbf{I}_{0}^{\\infty}(m) = \\int_{0}^{\\infty} \\frac{\\ln^{m}(1+x^{2})}{1+x^{2}} \\, \\textbf{d}x. \\]" ]	[ "aops_c7_college_math" ]	[ "additional_problems", "genselect" ]	{ "additional_problems": 1, "genselect": 4 }	5
omr-ws-21a391174cfc9ecdc799f2e4d4259adbf26317fb1163d9851f51610732b73084	\begin{align} 2a + 3b + 5c + 7d &= 34 \\ 3(d+c) &= b \\ 3b + c &= a \\ c - 1 &= d \\ \end{align} Given the above system of equations, find $a \cdot b \cdot c \cdot d$.	[ "0" ]	[ "\\begin{align}\n2a + 3b + 5c + 7d &= 34 \\\\\n3(d+c) &= b \\\\\n3b + c &= a \\\\\nc - 1 &= d \\\\\n\\end{align} Given the above system of equations, find $a \\cdot b \\cdot c \\cdot d$." ]	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-76550f01aafbfcf1980e9861cf5b69cfc18bd879f18d3a01d2a08362faded15b	a) How many four-digit whole numbers have an odd left-most digit, an even second digit, and all four digits are different? b) How many four-digit whole numbers have an even left-most digit, an odd second digit, and four different digits?	[ "a) 1400, b) 1120" ]	[ "a) How many four-digit whole numbers have an odd left-most digit, an even second digit, and all four digits are different?\nb) How many four-digit whole numbers have an even left-most digit, an odd second digit, and four different digits?" ]	[ "aops_c4_high_school_math" ]	[ "tir" ]	{ "tir": 23 }	23
omr-ws-9934caccf1594fe1c2518989a59a6a7d69c1012847183ccdeeb3c8c7c57477b9	a) In a convex quadrilateral $ABCD$, the areas of triangles $ABC$, $BCD$, $CDA$, and $DAB$ are in the ratio $1:1:1:x$. Find all possible values of $x$ and provide an example for each value of $x$.	[ "1" ]	[ "a) In a convex quadrilateral $ABCD$, the areas of triangles $ABC$, $BCD$, $CDA$, and $DAB$ are in the ratio $1:1:1:x$. Find all possible values of $x$ and provide an example for each value of $x$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 14 }	14
omr-ws-02f18dc0ffb24d702329ef9ecbabab54e3c3c3da9e057c4bbdc54175e13876b7	a) In the central cell of a $7 \times 7$ square composed of matches and divided into $1 \times 1$ cells, there is a prize. Two people take turns removing one match. The prize is taken by the one, after whose turn it will be possible to get to the prize from the area external to the square. Who can secure the victory - ...	[ "\\text{Second player}" ]	[ "a) In the central cell of a $7 \\times 7$ square composed of matches and divided into $1 \\times 1$ cells, there is a prize. Two people take turns removing one match. The prize is taken by the one, after whose turn it will be possible to get to the prize from the area external to the square. Who can secure the vic...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 8 }	8
omr-ws-10c4f43a60d679bf521918118e773fc008db473d5db04e6899234cf10824cd08	a) List three different natural numbers $ m $, $ n $, and $ k $ for which $ m! = n! \cdot k! $. b) Is it possible to come up with 1999 such triplets?	[ "\$ m = 3, n = 2, k = 1 \$ and Yes" ]	[ "a) List three different natural numbers \$ m \$, \$ n \$, and \$ k \$ for which \$ m! = n! \\cdot k! \$.\nb) Is it possible to come up with 1999 such triplets?" ]	[ "aops_c4_high_school_math" ]	[ "tir" ]	{ "tir": 9 }	9
omr-ws-20a65a62e7db2573dfa3942559236636f2002105092ad5a76c123256887119a2	a) What is the smallest number of points that can be marked inside a regular pentagon $ A_1A_2A_3A_4A_5 $ such that each triangle $ A_iA_jA_k $ (where $ 1 \le i < j < k \le 5 $) has at least one marked point inside it? b) The same problem for a regular $ n $-gon.	[ "\\left\\lfloor \\dfrac{n}{2} \\right\\rfloor" ]	[ "a) What is the smallest number of points that can be marked inside a regular pentagon \$ A_1A_2A_3A_4A_5 \$ such that each triangle \$ A_iA_jA_k \$ (where \$ 1 \\le i < j < k \\le 5 \$) has at least one marked point inside it?\nb) The same problem for a regular \$ n \$-gon." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 8 }	8
omr-ws-9b514c09a413215c4058778afd163dc409ad5823f5e7f3b614578c7e54a4f408	b) In a convex quadrilateral $ABCD$, the areas of triangles $ABC$, $BCD$, $CDA$, and $DAB$ are in the ratio $1:2:3:x$. Find all possible values of $x$ and provide an example for each value of $x$.	[ "2" ]	[ "b) In a convex quadrilateral $ABCD$, the areas of triangles $ABC$, $BCD$, $CDA$, and $DAB$ are in the ratio $1:2:3:x$. Find all possible values of $x$ and provide an example for each value of $x$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-a3ba194e80a46fc3ebc3bdf0e8c64745d94f4f7ee03aa270e1addfe9318855c6	i) Find a point $D$ in the interior of triangle $ABC$ such that the areas of triangles $ABD$, $BCD$, and $CAD$ are equal. ii) Find a point $D$ outside triangle $ABC$ such that the areas of triangles $ABD$, $BCD$, and $CAD$ are equal.	[ "D" ]	[ "i) Find a point $D$ in the interior of triangle $ABC$ such that the areas of triangles $ABD$, $BCD$, and $CAD$ are equal.\n \n ii) Find a point $D$ outside triangle $ABC$ such that the areas of triangles $ABD$, $BCD$, and $CAD$ are equal." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 3 }	3
omr-ws-67ee465f572865ea90debe55f0c27bda7f613417fbc0e4f0430564a6ff824013	makes. It takes the second computer, working alone, 2 min more to do the job than the first computer working alone. It takes the third computer, working alone, twice as much time as the first computer. The parts of the work being equivalent, the job can be divided between the three computers. In that case, working toge...	[ "6 min, 8 min, 12 min" ]	[ "makes. It takes the second computer, working alone, 2 min more to do the job than the first computer working alone. It takes the third computer, working alone, twice as much time as the first computer. The parts of the work being equivalent, the job can be divided between the three computers. In that case, working...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-8f9002a39fd4be31f228f28e7421b154c66308378edd61b8812a58702780375e	mathnerd314 is at the center of a circular building of radius 127. He walks north halfway to the wall, then east halfway to the wall, then south halfway to the wall, and finally west halfway to the wall. How far is he from the center after these movements?	[ "26.49" ]	[ "mathnerd314 is at the center of a circular building of radius 127. He walks north halfway to the wall, then east halfway to the wall, then south halfway to the wall, and finally west halfway to the wall. How far is he from the center after these movements?" ]	[ "aops_c4_high_school_math" ]	[ "tir" ]	{ "tir": 5 }	5
omr-ws-fd009f3c995842d70128241934dbfe85b8670cfac1eebb620fb857ffe2cd7d4e	{\it Buffon's needle problem.} A plane is ruled with parallel straight lines at distance of $2a$ apart. A needle $2l$ long $(l < a)$ is thrown on the plane at random. Find the probability that the needle will hit any of the lines.	[ "$\\frac{2l}{\\pi a}$" ]	[ "{\\it Buffon's needle problem.} A plane is ruled with parallel straight lines at distance of $2a$ apart. A needle $2l$ long $(l < a)$ is thrown on the plane at random. Find the probability that the needle will hit any of the lines." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-053af25dfa7dc681f4cbb7203084ea43e137a15643943d9f3279dc3ff43de999	£1000 is placed in a bank account at the start of each year, beginning in the year 2000. The account pays 5% interest per year at the end of each year. How much is the account after 15 years?	[ "22657.49" ]	[ "£1000 is placed in a bank account at the start of each year, beginning in the year 2000. The account pays 5% interest per year at the end of each year. How much is the account after 15 years?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 29 }	45
omr-ws-a46ac5819408215e3c42b53efb9049925f245e220c8cf6415868c0f63513bad0	ΔABC with ∠A = 90° has equal area and perimeter. I is the incenter of ΔABC and IB = √13. ΔBCD with ∠BCD = 90° has 18.2 times the area of ΔABC and has 91/30 the incenter of ΔABC. Compute the length of BD.	[ "\\dfrac{1259}{15}" ]	[ "ΔABC with ∠A = 90° has equal area and perimeter. I is the incenter of ΔABC and IB = √13. ΔBCD with ∠BCD = 90° has 18.2 times the area of ΔABC and has 91/30 the incenter of ΔABC. Compute the length of BD." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 10, "genselect": 2 }	12

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