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-47c8d67a8fb596cd0f88397aa53bf2d6909ec654feb4b4c366c1774255d6f6d3	Triangle $ABC$ is given. An arbitrary circle with center $J$, passing through $B$ and $C$, intersects the sides $AC$ and $AB$ at $E$ and $F$, respectively. Let $X$ be a point such that triangle $FXB$ is similar to triangle $EJC$ (with the same order) and the points $X$ and $C$ lie on the same ...	[ "1" ]	[ "Triangle \$ABC\$ is given. An arbitrary circle with center \$J\$, passing through \$B\$ and \$C\$, intersects the sides \$AC\$ and \$AB\$ at \$E\$ and \$F\$, respectively. Let \$X\$ be a point such that triangle \$FXB\$ is similar to triangle \$EJC\$ (with the same order) and the points \\(X\...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-77ef848c2ae868691dd0045355e7daf6f5bf6611271ba250c8900f16aa280d54	Triangle $ABC$ is given. Circle $\omega$ passes through $B$, touches $AC$ at $D$, and intersects sides $AB$ and $BC$ at $P$ and $Q$ respectively. Line $PQ$ intersects $BD$ and $AC$ at $M$ and $N$ respectively. Let $X$ be the intersection point of $\omega$, the circumcircle of \(\triangle...	[ "\\dfrac{2}{15}" ]	[ "Triangle \$ABC\$ is given. Circle \$\\omega\$ passes through \$B\$, touches \$AC\$ at \$D\$, and intersects sides \$AB\$ and \$BC\$ at \$P\$ and \$Q\$ respectively. Line \$PQ\$ intersects \$BD\$ and \$AC\$ at \$M\$ and \$N\$ respectively. Let \$X\$ be the intersection point of \\(\\om...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 2 }	2
omr-ws-56296602631e4706c399ef7385686fc581bb9b5d43c46d5775df6d1abedcf15b	Triangle $ABC$ is given. Circle $\omega$ with center $Q$ is tangent to side $BC$ and touches the circumcircle of triangle $ABC$ internally at point $A$. Let $M$ be the midpoint of $BC$ and $N$ be the midpoint of arc $BAC$ of the circumcircle of triangle $ABC$. Point $S$ is chosen on the segment ...	[ "\\dfrac{169}{225}" ]	[ "Triangle \$ABC\$ is given. Circle \$\\omega\$ with center \$Q\$ is tangent to side \$BC\$ and touches the circumcircle of triangle \$ABC\$ internally at point \$A\$. Let \$M\$ be the midpoint of \$BC\$ and \$N\$ be the midpoint of arc \$BAC\$ of the circumcircle of triangle \$ABC\$. Point \\(...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 5 }	5
omr-ws-702114e69dff2cf2693381a408d37c83ef8319315fc4e24c900c970652426fdb	Triangle $ABC$ is given. Find the locus of the intersection points of the medians of triangles $APB$, where $P$ is a point of the incircle of triangle $ABC$ not lying on $AB$.	[ "\\text{A circle centered at the centroid of } A, B, \\text{ and the incenter with radius one-third of the inradius}" ]	[ "Triangle \$ABC\$ is given. Find the locus of the intersection points of the medians of triangles \$APB\$, where \$P\$ is a point of the incircle of triangle \$ABC\$ not lying on \$AB\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-62ef17db29f9c33adac1aa30196329d1cc70583c980ad35f1cc2b66a59dd596f	Triangle $ABC$ is given. Point $M$ moves along the side $BA$, and point $N$ moves along the extension of the side $AC$ after point $C$ in such a way that $BM = CN$. Let $O$ be the circumcenter of triangle $AMN$. Find the equation of the line along which $O$ moves if $A = (0,0)$, $B = (2,0)$, and...	[ "x + y = 2" ]	[ "Triangle \$ABC\$ is given. Point \$M\$ moves along the side \$BA\$, and point \$N\$ moves along the extension of the side \$AC\$ after point \$C\$ in such a way that \$BM = CN\$. Let \$O\$ be the circumcenter of triangle \$AMN\$. Find the equation of the line along which \$O\$ moves if \\(A = (...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-c16b2b4aaa7e4fb726af1b4b89877e138fa28c595e123651b6bedb9f7db26048	Triangle $ABC$ is given. Points $D$ and $E$ are chosen on the line $AB$ such that $D - A - B - E$, $AD = AC$, and $BE = BC$. The bisectors of the inner angles at points $A$ and $B$ meet the opposite sides at points $P$ and $Q$, respectively, and the circumcircle of triangle $ABC$ at points $M$...	[ "1" ]	[ "Triangle \$ABC\$ is given. Points \$D\$ and \$E\$ are chosen on the line \$AB\$ such that \$D - A - B - E\$, \$AD = AC\$, and \$BE = BC\$. The bisectors of the inner angles at points \$A\$ and \$B\$ meet the opposite sides at points \$P\$ and \$Q\$, respectively, and the circumcircle of trian...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 2 }	7
omr-ws-8879b67c5744d3f3dae322335636731120abb5e97c972158b4df20c029a9635f	Triangle $ABC$ is inscribed in $\Gamma$, with orthocenter $H$ and circumcenter $O$. $H_B$ is the reflection of $H$ over $AC$, and $H_C$ is the reflection of $H$ over $AB$. Let $\omega_B$ be the circumcircle of $\triangle BHH_C$ and $\omega_C$ be the circumcircle of $\triangle CHH_B$. Let \(P...	[ "R" ]	[ "Triangle \$ABC\$ is inscribed in \$\\Gamma\$, with orthocenter \$H\$ and circumcenter \$O\$. \$H_B\$ is the reflection of \$H\$ over \$AC\$, and \$H_C\$ is the reflection of \$H\$ over \$AB\$. Let \$\\omega_B\$ be the circumcircle of \$\\triangle BHH_C\$ and \$\\omega_C\$ be the circumcir...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 5 }	5
omr-ws-c4c19837c57b8661e98b914db154aa32f6259004446106a45a7fd15c8c12b27b	Triangle $ABC$ is inscribed in a circle $(O)$ with incenter $I$. Given that $AI$ is perpendicular to $OI$, find the maximum possible value of $\frac{S_{ABC}}{AI^2}$.	[ "\\dfrac{3\\sqrt{3}}{4}" ]	[ "Triangle \$ABC\$ is inscribed in a circle \$(O)\$ with incenter \$I\$. Given that \$AI\$ is perpendicular to \$OI\$, find the maximum possible value of \$\\frac{S_{ABC}}{AI^2}\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-0052e5fa9794ff6faf8655680d530623dce97d6f312861b7d97f20f48bb28898	Triangle $ABC$ is inscribed in a circle $(O)$ with incenter $I$. The lines $BI$ and $CI$ intersect $AC$ and $AB$ at points $E$ and $F$ respectively. The line $EF$ intersects the circumcircle of $ABC$ at points $M$ and $N$. The lines $IM$ and $IN$ intersect the circumcircle $(O)$ again at...	[ "\\dfrac{1}{2}" ]	[ "Triangle \$ABC\$ is inscribed in a circle \$(O)\$ with incenter \$I\$. The lines \$BI\$ and \$CI\$ intersect \$AC\$ and \$AB\$ at points \$E\$ and \$F\$ respectively. The line \$EF\$ intersects the circumcircle of \$ABC\$ at points \$M\$ and \$N\$. The lines \$IM\$ and \$IN\$ intersec...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 2 }	2
omr-ws-d2301a16f93694ee1c1f651310c928489a453ec162ade1a20a9a58bc68a250b5	Triangle $ABC$ is inscribed in a circle $(O)$. Point $M$ moves on the arc $BC$ that does not contain $A$. Point $K$ is such that $BMCK$ is a parallelogram. $H$ is the orthocenter of triangle $ABK$. Find the locus of $H$.	[ "\\text{The circle obtained by reflecting } (O) \\text{ over the midpoint of } AB" ]	[ "Triangle \$ABC\$ is inscribed in a circle \$(O)\$. Point \$M\$ moves on the arc \$BC\$ that does not contain \$A\$. Point \$K\$ is such that \$BMCK\$ is a parallelogram. \$H\$ is the orthocenter of triangle \$ABK\$. Find the locus of \$H\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-c89c10dd8f0a83752dea055497ea3e014f05716aa045ed257893835c291be09f	Triangle $ABC$ is inscribed in a circle $(O)$. Point $P$ is located inside the triangle such that $AP$ is perpendicular to $BC$. The circle with diameter $AP$ intersects sides $AC$ and $AB$ at points $E$ and $F$ respectively, and intersects the circumcircle $(O)$ at points $G$ and $A$. If \(BE...	[ "\\dfrac{1}{4}" ]	[ "Triangle \$ABC\$ is inscribed in a circle \$(O)\$. Point \$P\$ is located inside the triangle such that \$AP\$ is perpendicular to \$BC\$. The circle with diameter \$AP\$ intersects sides \$AC\$ and \$AB\$ at points \$E\$ and \$F\$ respectively, and intersects the circumcircle \$(O)\$ at poin...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 4 }	4
omr-ws-c4fa92af099b1916d54009264968819f21bc217c2a570a7ce0ea6f3574e4b74e	Triangle $ABC$ is inscribed in a circle $(O)$. Point $P$ is on the angle bisector of $\angle BAC$. Circles $(PAB)$ and $(PAC)$ intersect $CA$ and $BA$ at points $E$ and $F$ respectively. If $OP$ and $EF$ intersect at point $Q$, and the length of $OP$ is 10 units, calculate the length of \(OQ...	[ "5" ]	[ "Triangle \$ABC\$ is inscribed in a circle \$(O)\$. Point \$P\$ is on the angle bisector of \$\\angle BAC\$. Circles \$(PAB)\$ and \$(PAC)\$ intersect \$CA\$ and \$BA\$ at points \$E\$ and \$F\$ respectively. If \$OP\$ and \$EF\$ intersect at point \$Q\$, and the length of \$OP\$ is 10 u...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-fbfc9ab0970a2d883ffead89d08bb36599785f76de5566e7799f268bf0388066	Triangle $ABC$ is inscribed in a circle $(O)$. Point $P$ lies on the arc $BC$ that does not contain $A$. Point $M$ lies on $AB$ such that $\frac{MA}{MP} = k$ (a constant). The circumcircle of $\triangle MBP$ intersects the circumcircle of $\triangle MAC$ at point $Q$. Let $R$ be the radius of th...	[ "100" ]	[ "Triangle \$ABC\$ is inscribed in a circle \$(O)\$. Point \$P\$ lies on the arc \$BC\$ that does not contain \$A\$. Point \$M\$ lies on \$AB\$ such that \$\\frac{MA}{MP} = k\$ (a constant). The circumcircle of \$\\triangle MBP\$ intersects the circumcircle of \$\\triangle MAC\$ at point \$Q\$....	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-f33b0c8b3995a256900a364038f4d87f9d308efeacec4fc130af0204b576b4c5	Triangle $ABC$ is inscribed in a circle $(O)$. $AD$ is the angle bisector of $\angle BAC$, and $AM$ is the median from $A$ to $BC$. The line passing through $D$ and perpendicular to $BC$ intersects $AM$ at $J$. Let $r$ be the radius of the circle centered at $J$ with radius $JD$. If $A$ mo...	[ "\\dfrac{R}{2}" ]	[ "Triangle \$ABC\$ is inscribed in a circle \$(O)\$. \$AD\$ is the angle bisector of \$\\angle BAC\$, and \$AM\$ is the median from \$A\$ to \$BC\$. The line passing through \$D\$ and perpendicular to \$BC\$ intersects \$AM\$ at \$J\$. Let \$r\$ be the radius of the circle centered at \$J\$...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 4, "genselect": 2 }	6
omr-ws-58b3c361663f96f0239bf623f125de407ee60a5f3f3513a7404c951513dd9d0c	Triangle $ABC$ is inscribed in a circle $\Gamma$. A chord $MN = 1$ of $\Gamma$ intersects sides $AB$ and $AC$ at points $X$ and $Y$ respectively, with $M$, $X$, $Y$, $N$ in that order on $MN$. Let $UV$ be the diameter of $\Gamma$ perpendicular to $MN$ with $U$ and $A$ in the same sem...	[ "\$\\frac{1}{11}\$" ]	[ "Triangle \$ABC\$ is inscribed in a circle \$\\Gamma\$. A chord \$MN = 1\$ of \$\\Gamma\$ intersects sides \$AB\$ and \$AC\$ at points \$X\$ and \$Y\$ respectively, with \$M\$, \$X\$, \$Y\$, \$N\$ in that order on \$MN\$. Let \$UV\$ be the diameter of \$\\Gamma\$ perpendicular to \\(MN...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 29, "tir": 8 }	37
omr-ws-13f14360200e51560251a5a21761ddd021b58e74eb8d29eb4168782b0c1d5c22	Triangle $ABC$ is inscribed in a circle with $AB = 15$ and $BC = 25$. A tangent to the circle is drawn through $B$, and a line through $A$ parallel to this tangent intersects $BC$ at $D$. Find the length of $DC$.	[ "16" ]	[ "Triangle \$ABC\$ is inscribed in a circle with \$AB = 15\$ and \$BC = 25\$. A tangent to the circle is drawn through \$B\$, and a line through \$A\$ parallel to this tangent intersects \$BC\$ at \$D\$. Find the length of \$DC\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 1 }	17
omr-ws-fd48914b0eea29f82491d2ee308d3d0404599d9746f54881d1a058b2ebfedb63	Triangle $ABC$ is inscribed in a circle. The measures of the non-overlapping minor arcs $AB$, $BC$, and $CA$ are $x+75^\circ$, $2x+25^\circ$, and $3x-22^\circ$ respectively. Determine one interior angle of the triangle in degrees.	[ "61" ]	[ "Triangle \$ABC\$ is inscribed in a circle. The measures of the non-overlapping minor arcs \$AB\$, \$BC\$, and \$CA\$ are \$x+75^\\circ\$, \$2x+25^\\circ\$, and \$3x-22^\\circ\$ respectively. Determine one interior angle of the triangle in degrees." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 30 }	46
omr-ws-23ef898e806c0d43eeaf556f42c86bf71f9e760ca45d66a83ae0503fc86738b8	Triangle $ABC$ is inscribed in a circumcircle $O$. The angle bisectors of $\angle ABC$ and $\angle ACB$ intersect $AC$ and $AB$ at points $D$ and $E$ respectively. The line $DE$ intersects the circumcircle $O$ at points $M$ and $N$ (with $M$ to the left and $N$ to the right). If $I$ is the...	[ "0" ]	[ "Triangle \$ABC\$ is inscribed in a circumcircle \$O\$. The angle bisectors of \$\\angle ABC\$ and \$\\angle ACB\$ intersect \$AC\$ and \$AB\$ at points \$D\$ and \$E\$ respectively. The line \$DE\$ intersects the circumcircle \$O\$ at points \$M\$ and \$N\$ (with \$M\$ to the left and \\(...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 6 }	6
omr-ws-c6919746d553e49f94e0b095b55a53d0a91f44b1116afde54e6fa0cab1d50619	Triangle $ABC$ is inscribed in a unit circle. The angle bisectors of $A$, $B$, and $C$ are extended to intersect the circle at points $A_1$, $B_1$, and $C_1$ respectively. Calculate the maximum possible value of the expression \( AA_1 \cdot \cos \frac{A}{2} + BB_1 \cdot \cos \frac{B}{2} + CC_1 \cdot \cos ...	[ "3\\sqrt{3}" ]	[ "Triangle \$ABC\$ is inscribed in a unit circle. The angle bisectors of \$A\$, \$B\$, and \$C\$ are extended to intersect the circle at points \$A_1\$, \$B_1\$, and \$C_1\$ respectively. Calculate the maximum possible value of the expression \\( AA_1 \\cdot \\cos \\frac{A}{2} + BB_1 \\cdot \\cos \\fra...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 6, "genselect": 6 }	12
omr-ws-eb8ff568e474c946cdedc3998cb7045b825d1ce32ce06ce02c56d8434afcaedc	Triangle $ABC$ is inscribed in a unit circle. The angle bisectors of $A$, $B$, and $C$ are extended to intersect the circle at points $A_1$, $B_1$, and $C_1$ respectively. Calculate the maximum possible value of the expression \[ AA_1 \cdot \cos \frac{A}{2} \cdot \sin A + BB_1 \cdot \cos \frac{B}{2} \cdot...	[ "\\dfrac{9}{2}" ]	[ "Triangle \$ABC\$ is inscribed in a unit circle. The angle bisectors of \$A\$, \$B\$, and \$C\$ are extended to intersect the circle at points \$A_1\$, \$B_1\$, and \$C_1\$ respectively. Calculate the maximum possible value of the expression\n\n\\[ AA_1 \\cdot \\cos \\frac{A}{2} \\cdot \\sin A + BB_1 ...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 4, "genselect": 4 }	8
omr-ws-073b82df77be0bbfb3e1679825af70697b6e22f0dfe97e1aa40fdd7fc24e30a6	Triangle $ABC$ is inscribed in circle $(I)$. The circle $(I)$ touches $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$ respectively. $DI$ intersects $EF$ at $T$. Let $L$ be a point such that $A$ is the midpoint of $LI$. If $\angle ABC = 2\theta$, find the value of $\angle LBT$ in terms...	[ "\\theta" ]	[ "Triangle \$ABC\$ is inscribed in circle \$(I)\$. The circle \$(I)\$ touches \$BC\$, \$CA\$, and \$AB\$ at points \$D\$, \$E\$, and \$F\$ respectively. \$DI\$ intersects \$EF\$ at \$T\$. Let \$L\$ be a point such that \$A\$ is the midpoint of \$LI\$. If \$\\angle ABC = 2\\theta\$, fi...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 6 }	6
omr-ws-ddd038877160fcf0593335da2eb277cf201dea7a3439ee3610594cbe2ef378e7	Triangle $ABC$ is inscribed in circle $(O)$. Another circle $(O')$ touches $AB$ and $AC$ at points $P$ and $Q$ respectively and touches circle $(O)$ internally at point $S$. The lines $SP$ and $SQ$ intersect circle $(O)$ again at points $M$ and $N$ respectively. Points $E$, $D$, and \(F\...	[ "1" ]	[ "Triangle \$ABC\$ is inscribed in circle \$(O)\$. Another circle \$(O')\$ touches \$AB\$ and \$AC\$ at points \$P\$ and \$Q\$ respectively and touches circle \$(O)\$ internally at point \$S\$. The lines \$SP\$ and \$SQ\$ intersect circle \$(O)\$ again at points \$M\$ and \$N\$ respective...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-d4d8af8b0e38dfb424ee8664bd94efccb10fc26da2de63c92b33bbd3ca4264c1	Triangle $ABC$ is inscribed in circle $(O)$. Point $P$ is an arbitrary point inside triangle $ABC$. The lines $PA$, $PB$, and $PC$ intersect the circle $(O)$ again at points $A_1$, $B_1$, and $C_1$ respectively. Points $A_0$, $B_0$, and $C_0$ are the midpoints of $BC$, $AC$, and $AB$ r...	[ "15" ]	[ "Triangle \$ABC\$ is inscribed in circle \$(O)\$. Point \$P\$ is an arbitrary point inside triangle \$ABC\$. The lines \$PA\$, \$PB\$, and \$PC\$ intersect the circle \$(O)\$ again at points \$A_1\$, \$B_1\$, and \$C_1\$ respectively. Points \$A_0\$, \$B_0\$, and \$C_0\$ are the midpoint...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-1f7e9ebab30d50a1c2616572bb4010ac1e916b53b9755721251e498f8a98bf14	Triangle $ABC$ is inscribed in circle $C$. A circle with center $O$ is drawn, tangent to side $BC$ at point $P$, and internally tangent to the arc $BC$ of $C$ not containing $A$ at point $Q$. If $\angle BAO = \angle CAO$, calculate the value of $\frac{\angle PAO}{\angle QAO}$.	[ "\\dfrac{1}{2}" ]	[ "Triangle \$ABC\$ is inscribed in circle \$C\$. A circle with center \$O\$ is drawn, tangent to side \$BC\$ at point \$P\$, and internally tangent to the arc \$BC\$ of \$C\$ not containing \$A\$ at point \$Q\$. If \$\\angle BAO = \\angle CAO\$, calculate the value of \\(\\frac{\\angle PAO}{\\ang...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 5 }	10
omr-ws-fd2d13da82e413866a4e65ec140835e585980f4fd95ca334d9d94c62cd78bc00	Triangle $ABC$ is inscribed in circle $C$. The bisectors of the angles $A$, $B$, and $C$ meet the circle $C$ again at the points $A'$, $B'$, and $C'$. Let $I$ be the incenter of $ABC$. Calculate the minimum possible value of \[ \frac{IA'}{IA} + \frac{IB'}{IB} + \frac{IC'}{IC}. \]	[ "3" ]	[ "Triangle \$ABC\$ is inscribed in circle \$C\$. The bisectors of the angles \$A\$, \$B\$, and \$C\$ meet the circle \$C\$ again at the points \$A'\$, \$B'\$, and \$C'\$. Let \$I\$ be the incenter of \$ABC\$. Calculate the minimum possible value of\n\\[\n\\frac{IA'}{IA} + \\frac{IB'}{IB} + \\fr...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-f4511e5e8a19a1c586f83f6d9ac282c63699f29d18d58f34898b1d7d230bb943	Triangle $ABC$ is inscribed in circle $O$. The tangent lines to circle $O$ at $B$ and $C$ meet at $T$. Point $S$ lies on ray $BC$ such that $AS$ is perpendicular to $AT$. Points $B_1$ and $C_1$ lie on ray $ST$ (with $C_1$ in between $B_1$ and $S$) such that $B_1T = BT = C_1T$. Let \(k\...	[ "\\dfrac{3}{5}" ]	[ "Triangle \$ABC\$ is inscribed in circle \$O\$. The tangent lines to circle \$O\$ at \$B\$ and \$C\$ meet at \$T\$. Point \$S\$ lies on ray \$BC\$ such that \$AS\$ is perpendicular to \$AT\$. Points \$B_1\$ and \$C_1\$ lie on ray \$ST\$ (with \$C_1\$ in between \$B_1\$ and \$S\$) suc...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 4 }	4
omr-ws-6817cccd0039e20d632510051cdc2bd4c20ba87824d5c6fe7937da156da257fa	Triangle $ABC$ is inscribed in circle $S$. Circle which contains $B$ and $C$ cuts $AB$ and $AC$ again at $P$ and $R$, respectively. Intersection point of $BR$ and $CP$ is $L$, and $LR$ and $LP$ intersect $S$ in $D$ and $E$ respectively. Angle bisector of $\angle BDE$ cuts $ER$ in \(K...	[ "\\dfrac{1}{2}" ]	[ "Triangle \$ABC\$ is inscribed in circle \$S\$. Circle which contains \$B\$ and \$C\$ cuts \$AB\$ and \$AC\$ again at \$P\$ and \$R\$, respectively. Intersection point of \$BR\$ and \$CP\$ is \$L\$, and \$LR\$ and \$LP\$ intersect \$S\$ in \$D\$ and \$E\$ respectively. Angle bisector...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-a3940c56c19c7db2786e3ac799c3b527284d20d5b97596eb6ccb9c940d49c29b	Triangle $ABC$ is inscribed in circle $W$ with $AB = 5$, $BC = 7$, and $AC = 3$. The angle bisector of $\angle A$ meets side $BC$ at $D$ and circle $W$ at a second point $E$. Let $C$ be the circle with diameter $DE$. Circles $W$ and $C$ meet at $E$ and at a second point $F$. Then \(AF^2 ...	[ "919" ]	[ "Triangle \$ABC\$ is inscribed in circle \$W\$ with \$AB = 5\$, \$BC = 7\$, and \$AC = 3\$. The angle bisector of \$\\angle A\$ meets side \$BC\$ at \$D\$ and circle \$W\$ at a second point \$E\$. Let \$C\$ be the circle with diameter \$DE\$. Circles \$W\$ and \$C\$ meet at \$E\$ and a...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 15, "genselect": 5 }	20
omr-ws-c90649e103d306211c21414b55ed6334cfc864dfd281a8d6c8c619611773d107	Triangle $ABC$ is inscribed in circle $\Gamma$ with incenter $I$. The line $AI$ intersects $\Gamma$ at $A' \neq A$, and $l$ is the tangent to $\Gamma$ at $A$. The line through $A'$ parallel to $l$ intersects $AB$ and $AC$ at $P$ and $Q$, respectively. Let $X \neq B$ be the intersection o...	[ "no_valid_answer_found" ]	[ "Triangle \$ABC\$ is inscribed in circle \$\\Gamma\$ with incenter \$I\$. The line \$AI\$ intersects \$\\Gamma\$ at \$A' \\neq A\$, and \$l\$ is the tangent to \$\\Gamma\$ at \$A\$. The line through \$A'\$ parallel to \$l\$ intersects \$AB\$ and \$AC\$ at \$P\$ and \$Q\$, respectively....	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-1a27dee12a1136666f4e13a7d0654639a95c0edecb0be17128f5686ad292dfae	Triangle $ABC$ is inscribed in circle $\Gamma$. Let $H$ and $O$ denote its orthocenter and circumcenter, respectively. Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Rays $MH$ and $NH$ meet $\Gamma$ at $P$ and $Q$, respectively. Lines $MN$ and $PQ$ meet at $R$. If...	[ "90" ]	[ "Triangle \$ABC\$ is inscribed in circle \$\\Gamma\$. Let \$H\$ and \$O\$ denote its orthocenter and circumcenter, respectively. Let \$M\$ and \$N\$ be the midpoints of sides \$AB\$ and \$AC\$, respectively. Rays \$MH\$ and \$NH\$ meet \$\\Gamma\$ at \$P\$ and \$Q\$, respectively. Lines \\...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 5 }	5
omr-ws-0da7e13e0844224ebdbe5355ea769c893f82906cfb901d418fbcd4456a5792e6	Triangle $ABC$ is inscribed in circle $\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $\overline{DE}$. Circles $\omega$ and $\gamma$ meet at $E$ and a s...	[ "253" ]	[ "Triangle \$ABC\$ is inscribed in circle \$\\omega\$ with \$AB=5\$, \$BC=7\$, and \$AC=3\$. The bisector of angle \$A\$ meets side \$\\overline{BC}\$ at \$D\$ and circle \$\\omega\$ at a second point \$E\$. Let \$\\gamma\$ be the circle with diameter \$\\overline{DE}\$. Circles \$\\omega\$...	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-6a5d1b5d09db7b3ee8b1115ade6973d052ebf0660eca6edd6e6057cc0098d438	Triangle $ABC$ is inscribed in circle $\omega$. Circle $\gamma$ is tangent to $AB$ and $AC$ at points $P$ and $Q$ respectively, and is also tangent to circle $\omega$ at point $S$. Let the intersection of $AS$ and $PQ$ be $T$. Calculate the value of $\frac{BT}{CT}$ given that $AB = 13$, \(AC...	[ "\\dfrac{13}{16}" ]	[ "Triangle \$ABC\$ is inscribed in circle \$\\omega\$. Circle \$\\gamma\$ is tangent to \$AB\$ and \$AC\$ at points \$P\$ and \$Q\$ respectively, and is also tangent to circle \$\\omega\$ at point \$S\$. Let the intersection of \$AS\$ and \$PQ\$ be \$T\$. Calculate the value of \\(\\frac{BT}{...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 2 }	2
omr-ws-f078cada03c8acf069dc6f04674adac5bbfa930d62f9f46420d5f43496c08280	Triangle $ABC$ is inscribed in circle $\omega$. Circle $\omega_1$ is internally tangent to $\omega$ and touches sides $AB$ and $AC$ at points $M$ and $N$, respectively. Circle $\omega_2$ is also internally tangent to $\omega$ and touches sides $AB$ and $BC$ at points $P$ and $K$, respectivel...	[ "\\dfrac{40}{9}" ]	[ "Triangle \$ABC\$ is inscribed in circle \$\\omega\$. Circle \$\\omega_1\$ is internally tangent to \$\\omega\$ and touches sides \$AB\$ and \$AC\$ at points \$M\$ and \$N\$, respectively. Circle \$\\omega_2\$ is also internally tangent to \$\\omega\$ and touches sides \$AB\$ and \$BC\$ at p...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 6 }	6
omr-ws-2f18271f35fff832bab19d060e8852949dddfb5e45956a3af3146e0b88bd6ed3	Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies on line $BC$ such that line $PA$ is tangent to $\omega$. The bisector of $\angle APB$ meets segments $AB$ and $AC$ at $D$ and $E$ respectively. Segments $BE$ and $CD$ meet at $Q$. Given that line $PQ$ passes through the center ...	[ "90^\\circ", "\$60^\\circ\$" ]	[ "Triangle \$ABC\$ is inscribed in circle \$\\omega\$. Point \$P\$ lies on line \$BC\$ such that line \$PA\$ is tangent to \$\\omega\$. The bisector of \$\\angle APB\$ meets segments \$AB\$ and \$AC\$ at \$D\$ and \$E\$ respectively. Segments \$BE\$ and \$CD\$ meet at \$Q\$. Given that li...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 13, "genselect": 6, "tir": 4 }	23
omr-ws-fa0410e7af686c8d9215786def18608004d12c58d05f83fda71131f6fadc6a12	Triangle $ABC$ is inscribed in circle $\omega$. The chord $AD$ of this circle is the angle bisector of triangle $ABC$ and intersects $BC$ at point $L$. The chord $DE$ of the circle $\omega$ is perpendicular to the side $AC$ and intersects it at point $K$. Find $\frac{AK}{KC}$ if \(\frac{BL}{LC} = ...	[ "3" ]	[ "Triangle \$ABC\$ is inscribed in circle \$\\omega\$. The chord \$AD\$ of this circle is the angle bisector of triangle \$ABC\$ and intersects \$BC\$ at point \$L\$. The chord \$DE\$ of the circle \$\\omega\$ is perpendicular to the side \$AC\$ and intersects it at point \$K\$. Find \\(\\frac{AK...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-f63c4e73413d569d98390a816e340aff8364163880e3825644b3a303d07fe522	Triangle $ABC$ is inscribed in circle $\omega$. The tangent lines to $\omega$ at $B$ and $C$ meet at $T$. Point $H$ lies on line $AT$. $BH$ and $CH$ meet $CA$ and $AB$ at $E$ and $F$, respectively. The circumcircle of triangle $AEF$ meets $\omega$ at $G$. $AM$ meets $\omega$ at \(N...	[ "10" ]	[ "Triangle \$ABC\$ is inscribed in circle \$\\omega\$. The tangent lines to \$\\omega\$ at \$B\$ and \$C\$ meet at \$T\$. Point \$H\$ lies on line \$AT\$. \$BH\$ and \$CH\$ meet \$CA\$ and \$AB\$ at \$E\$ and \$F\$, respectively. The circumcircle of triangle \$AEF\$ meets \$\\omega\$ ...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 6 }	6
omr-ws-145d414ffd5b50b23de75879ae456d49e8d6c8f3f422f9fffce49153def5c45b	Triangle $ABC$ is inscribed in circle $\omega$. The tangent lines to $\omega$ at $B$ and $C$ meet at $T$. Point $S$ lies on ray $BC$ such that $AS \perp AT$. Points $B_1$ and $C_1$ lie on ray $ST$ (with $C_1$ in between $B_1$ and $S$) such that $B_1T = BT = C_1T$. If \(\angle BAC = 30^\c...	[ "30" ]	[ "Triangle \$ABC\$ is inscribed in circle \$\\omega\$. The tangent lines to \$\\omega\$ at \$B\$ and \$C\$ meet at \$T\$. Point \$S\$ lies on ray \$BC\$ such that \$AS \\perp AT\$. Points \$B_1\$ and \$C_1\$ lie on ray \$ST\$ (with \$C_1\$ in between \$B_1\$ and \$S\$) such that \\(B_1T...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 8, "genselect": 4 }	12
omr-ws-cbb0b178c2ec01a09a37490fad24240eb6aad038dc768fa692e2aff79beaa615	Triangle $ABC$ is inscribed in circle $w$. Line $l_{1}$ bisects $\angle BAC$ and meets segments $BC$ and $w$ at points $D$ and $M$, respectively. Let $y$ denote the circle centered at $M$ with radius $BM$. Line $l_{2}$ passes through $D$ and meets circle $y$ at points $X$ and $Y$. Calcul...	[ "60^\\circ" ]	[ "Triangle \$ABC\$ is inscribed in circle \$w\$. Line \$l_{1}\$ bisects \$\\angle BAC\$ and meets segments \$BC\$ and \$w\$ at points \$D\$ and \$M\$, respectively. Let \$y\$ denote the circle centered at \$M\$ with radius \$BM\$. Line \$l_{2}\$ passes through \$D\$ and meets circle \$y\$...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 4 }	4
omr-ws-a060dc69e85b2056d4fd40daebe5681dab037583478eda9818cd39770fd56a0a	Triangle $ABC$ is inscribed in the circle $(O)$. Point $M$ (distinct from $B$ and $C$) is chosen on the side $BC$. Let $E$ and $F$ be the centers of the circumcircles of triangles $ABM$ and $ACM$, respectively. The lines $BE$ and $CF$ intersect at $D$. If the radius of circle $(O)$ is $R$ ...	[ "\\dfrac{a^2}{4}" ]	[ "Triangle \$ABC\$ is inscribed in the circle \$(O)\$. Point \$M\$ (distinct from \$B\$ and \$C\$) is chosen on the side \$BC\$. Let \$E\$ and \$F\$ be the centers of the circumcircles of triangles \$ABM\$ and \$ACM\$, respectively. The lines \$BE\$ and \$CF\$ intersect at \$D\$. If the rad...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 4 }	4
omr-ws-6d779b495505b3d319244479116d8dc522c578ca7fee9e55a258459afdcffcf4	Triangle $ABC$ is inscribed in the circle $\Gamma$. Let $\Gamma_a$ denote the circle internally tangent to $\Gamma$ and also tangent to sides $AB$ and $AC$. Let $A'$ denote the point of tangency of $\Gamma$ and $\Gamma_a$. Define $B'$ and $C'$ similarly. Let $P$ be the point of concurrency of \(...	[ "1 - \\dfrac{2r}{R}" ]	[ "Triangle \$ABC\$ is inscribed in the circle \$\\Gamma\$. Let \$\\Gamma_a\$ denote the circle internally tangent to \$\\Gamma\$ and also tangent to sides \$AB\$ and \$AC\$. Let \$A'\$ denote the point of tangency of \$\\Gamma\$ and \$\\Gamma_a\$. Define \$B'\$ and \$C'\$ similarly. Let \$P\$...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-d887155c406cf58e79110aedf41ee49494642416a756f50f6a57d230a1dbee94	Triangle $ABC$ is isosceles with $AB = AC$ and altitude $AM = 11$. There is a point $D$ on $\overline{AM}$ such that $AD = 10$ and $\angle BDC = 3 \angle BAC$. The perimeter of $\triangle ABC$ can be written in the form $a + \sqrt{b}$, where $a$ and $b$ are integers. Find $a + b$.	[ "616" ]	[ "Triangle \$ABC\$ is isosceles with \$AB = AC\$ and altitude \$AM = 11\$. There is a point \$D\$ on \$\\overline{AM}\$ such that \$AD = 10\$ and \$\\angle BDC = 3 \\angle BAC\$. The perimeter of \$\\triangle ABC\$ can be written in the form \$a + \\sqrt{b}\$, where \$a\$ and \$b\$ are integers...	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 14, "tir": 11 }	25
omr-ws-3ece759e0b2fa85867d407cdc838f42e94438ea69eb34db37e71ba92a89501c6	Triangle $ABC$ is isosceles with $AB = AC$. A line $\ell$ is drawn from $A$ parallel to $BC$. Points $P$ and $Q$ are on the perpendicular bisectors of $AB$ and $AC$ respectively such that $PQ \perp BC$. Points $M$ and $N$ are on $\ell$ such that $\angle APM = \frac{\pi}{2}$ and \(\angle AQN ...	[ "\\dfrac{1}{5}" ]	[ "Triangle \$ABC\$ is isosceles with \$AB = AC\$. A line \$\\ell\$ is drawn from \$A\$ parallel to \$BC\$. Points \$P\$ and \$Q\$ are on the perpendicular bisectors of \$AB\$ and \$AC\$ respectively such that \$PQ \\perp BC\$. Points \$M\$ and \$N\$ are on \$\\ell\$ such that \\(\\angle APM...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-b7ec14335028def8b8c832d1ece1d8bf96429f79dde0883f6f9a81103bf1698c	Triangle $ABC$ is isosceles with $AB = AC$. $D$ is the midpoint of segment $BC$. $E$ is a point on the perpendicular from vertex $C$ to segment $AB$ such that $BE = BD$. $M$ is the midpoint of $BE$. The perpendicular from $M$ to $BE$ intersects the circumcircle of triangle $ABD$ at $F$. Calc...	[ "30" ]	[ "Triangle \$ABC\$ is isosceles with \$AB = AC\$. \$D\$ is the midpoint of segment \$BC\$. \$E\$ is a point on the perpendicular from vertex \$C\$ to segment \$AB\$ such that \$BE = BD\$. \$M\$ is the midpoint of \$BE\$. The perpendicular from \$M\$ to \$BE\$ intersects the circumcircle of tr...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 5 }	10
omr-ws-d4c8f2267cafb15d1d9c6e790418cd04f4c84cbc574f861e30ccb65885537506	Triangle $ABC$ is isosceles with $AC = BC$ and $\angle C = 120^\circ$. Points $D$ and $E$ are chosen on segment $AB$ such that $\|AD\| = \|DE\| = \|EB\|$. Find the sizes of the angles of triangle $CDE$.	[ "\$60^\\circ\$" ]	[ "Triangle \$ABC\$ is isosceles with \$AC = BC\$ and \$\\angle C = 120^\\circ\$. Points \$D\$ and \$E\$ are chosen on segment \$AB\$ such that \$\|AD\| = \|DE\| = \|EB\|\$. Find the sizes of the angles of triangle \$CDE\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 13 }	29
omr-ws-b8b63eddde7f20d7e3df68507f7d97deaa9faf8339f8dfe359d27c029464e8a7	Triangle $ABC$ is isosceles with $\angle B = \angle C = 40^\circ$ and $BC = 10$. Extend $AB$ to point $D$ so that $AD = 10$. Find $\angle ACD$.	[ "\$50^\\circ\$" ]	[ "Triangle \$ABC\$ is isosceles with \$\\angle B = \\angle C = 40^\\circ\$ and \$BC = 10\$. Extend \$AB\$ to point \$D\$ so that \$AD = 10\$. Find \$\\angle ACD\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 25 }	40
omr-ws-ae3fae46a757cf00dd60f7f0dcd2cc86f41d2c4b83e9291f76e2b568fd4a04dd	Triangle $ABC$ is placed on the Cartesian Plane with $AB$ horizontal and $C$ above $AB$. Angle bisectors $\ell_1$ and $\ell_2$ are drawn from $A$ and $B$ respectively, intersecting at point $D$. Points $E$ and $F$ are the intersections of $\ell_1$ with $BC$ and $\ell_2$ with $AC$, respecti...	[ "18" ]	[ "Triangle \$ABC\$ is placed on the Cartesian Plane with \$AB\$ horizontal and \$C\$ above \$AB\$. Angle bisectors \$\\ell_1\$ and \$\\ell_2\$ are drawn from \$A\$ and \$B\$ respectively, intersecting at point \$D\$. Points \$E\$ and \$F\$ are the intersections of \$\\ell_1\$ with \$BC\$ an...	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-85c9d8bbdedf25fc5ff0495cde862ee69d47f66a24c21877d35aa8f035ca9ef3	Triangle $ABC$ is right-angled at $A$. Among the points $P$ belonging to the perimeter of the triangle, find the one that minimizes the sum $AP + BP + CP$.	[ "\$A\$" ]	[ "Triangle \$ABC\$ is right-angled at \$A\$. Among the points \$P\$ belonging to the perimeter of the triangle, find the one that minimizes the sum \$AP + BP + CP\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-704722dd65f9f3204289356fd291dcfba204e9fcdf3731f10c44d0c5d5181456	Triangle $ABC$ is right-angled at $A$. Let $D$ be a point on $\overline{BC}$. Let $E$ and $F$ be the reflections of $D$ across $AC$ and $AB$, respectively. Calculate the ratio $\frac{[DEF]}{[ABC]}$ when $D$ is the midpoint of $\overline{BC}$.	[ "1" ]	[ "Triangle \$ABC\$ is right-angled at \$A\$. Let \$D\$ be a point on \$\\overline{BC}\$. Let \$E\$ and \$F\$ be the reflections of \$D\$ across \$AC\$ and \$AB\$, respectively. Calculate the ratio \$\\frac{[DEF]}{[ABC]}\$ when \$D\$ is the midpoint of \$\\overline{BC}\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-938ab0fbe4d46ceb9469325fad0e5795d1be5b5fb4b8352ddc7c259cf7e5ba2d	Triangle $ABC$ is right-angled such that $\angle ACB = 90^\circ$ and $\frac{AC}{BC} = 2$. Let the line parallel to side $AC$ intersect line segments $AB$ and $BC$ at $M$ and $N$ such that $\frac{CN}{BN} = 2$. Let $O$ be the intersection point of lines $CM$ and $AN$. On segment $ON$ lies point ...	[ "90^\\circ" ]	[ "Triangle \$ABC\$ is right-angled such that \$\\angle ACB = 90^\\circ\$ and \$\\frac{AC}{BC} = 2\$. Let the line parallel to side \$AC\$ intersect line segments \$AB\$ and \$BC\$ at \$M\$ and \$N\$ such that \$\\frac{CN}{BN} = 2\$. Let \$O\$ be the intersection point of lines \$CM\$ and \\(AN\...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 15, "genselect": 8 }	23
omr-ws-9c52fe50968f0b09f9c1975b33f17c134b95faaa98edfb497e2bff71fecdf206	Triangle $ABC$ is scalene and acute. A circle passing through points $B$ and $C$ intersects $AC$ and $AB$ at points $M$ and $N$ respectively. Let $P$ be the intersection of $BM$ and $CN$, and $R$ be the intersection of $AP$ and $BC$. Let $Q$ be the isogonal conjugate of $P$ with respect to...	[ "\\dfrac{1}{2}" ]	[ "Triangle \$ABC\$ is scalene and acute. A circle passing through points \$B\$ and \$C\$ intersects \$AC\$ and \$AB\$ at points \$M\$ and \$N\$ respectively. Let \$P\$ be the intersection of \$BM\$ and \$CN\$, and \$R\$ be the intersection of \$AP\$ and \$BC\$. Let \$Q\$ be the isogonal c...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 6 }	11
omr-ws-980fc712e149ff013c8764cdf043fd252daeef0d39db46dade72ff68a2ed6922	Triangle $ABC$ satisfies $\angle C = 90^\circ$. A circle passing through $A$ and $B$ intersects segment $AC$ at $G$ ($\neq A, C$) and segment $BC$ at $D$ ($\neq B$). Segment $AD$ intersects segment $BG$ at $H$. Let $E$ be the circumcenter of $\triangle ABD$. The tangent to this circle at p...	[ "\\dfrac{48}{5}" ]	[ "Triangle \$ABC\$ satisfies \$\\angle C = 90^\\circ\$. A circle passing through \$A\$ and \$B\$ intersects segment \$AC\$ at \$G\$ (\$\\neq A, C\$) and segment \$BC\$ at \$D\$ (\$\\neq B\$). Segment \$AD\$ intersects segment \$BG\$ at \$H\$. Let \$E\$ be the circumcenter of \\(\\triangle...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 4, "genselect": 3 }	7
omr-ws-598406d6871d37afd664a6a6612454fa90f25a2bdb23e1b56178038286dcd915	Triangle $ABC$ satisfying $AB < AC$ has incenter $I$ and $A$-excenter $I_a$. Let the foot from $I_a$ to $BC$ be $D'$, and the circumcircles of $AI_aD'$ and $BIC$ meet again at $K$. $\ell$ is a line through $I$ parallel to $BC$. The tangent of $(BIC)$ through $K$ meets $\ell$ and $BC$...	[ "no_valid_answer_found" ]	[ "Triangle \$ABC\$ satisfying \$AB < AC\$ has incenter \$I\$ and \$A\$-excenter \$I_a\$. Let the foot from \$I_a\$ to \$BC\$ be \$D'\$, and the circumcircles of \$AI_aD'\$ and \$BIC\$ meet again at \$K\$. \$\\ell\$ is a line through \$I\$ parallel to \$BC\$. The tangent of \$(BIC)\$ thr...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-6afe192cd19c8bc3be6c087038a6ef945d29574bb5f406a4f2eba971fdccc8d2	Triangle $ABC$ with $AB = 50$ and $AC = 10$ has area 120. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$ respectively. What is the area of quadrilateral \...	[ "75" ]	[ "Triangle \$ABC\$ with \$AB = 50\$ and \$AC = 10\$ has area 120. Let \$D\$ be the midpoint of \$\\overline{AB}\$, and let \$E\$ be the midpoint of \$\\overline{AC}\$. The angle bisector of \$\\angle BAC\$ intersects \$\\overline{DE}\$ and \$\\overline{BC}\$ at \$F\$ and \$G\$ respectively. W...	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-4ca7d46009833ee2bdb803a275228f4c25b88519c5af0a0680f77be04a351366	Triangle $ABC$ with $AB=4$, $BC=5$, $CA=6$ has circumcircle $\Omega$ and incircle $\omega$. Let $\Gamma$ be the circle tangent to $\Omega$ and the sides $AB$, $AC$, and let $X = \Gamma \cap \Omega$. Let $Y$, $Z$ be distinct points on $\Omega$ such that $XY$, $YZ$ are tangent to \(\omega\...	[ "16" ]	[ "Triangle \$ABC\$ with \$AB=4\$, \$BC=5\$, \$CA=6\$ has circumcircle \$\\Omega\$ and incircle \$\\omega\$. Let \$\\Gamma\$ be the circle tangent to \$\\Omega\$ and the sides \$AB\$, \$AC\$, and let \$X = \\Gamma \\cap \\Omega\$. Let \$Y\$, \$Z\$ be distinct points on \$\\Omega\$ such tha...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 3, "genselect": 3 }	6
omr-ws-0177e2bdb0cf90d86c4851f43444ff5e3eb936bfe9d8c55a54f8327501fd7ca3	Triangle $ABC$ with $\angle ACB > 90^\circ$ is inscribed in circle $S$. Point $D$ is where $CS$ intersects $AB$. Given that $\|AC\| + \|BC\| = 2\|CS\|$, calculate the ratio of the radius of the incircle of triangle $ADC$ to the radius of the incircle of triangle $BDC$.	[ "1" ]	[ "Triangle \$ABC\$ with \$\\angle ACB > 90^\\circ\$ is inscribed in circle \$S\$. Point \$D\$ is where \$CS\$ intersects \$AB\$. Given that \$\|AC\| + \|BC\| = 2\|CS\|\$, calculate the ratio of the radius of the incircle of triangle \$ADC\$ to the radius of the incircle of triangle \$BDC\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 4 }	4
omr-ws-d53bc0618945a4e084ebaae1b466903a98ee48552f2051cd36785f1a237b375d	Triangle $ABE$ is a right triangle with hypotenuse $AB$, and point $A$ is to the right of point $B$. A line segment $BC$ is created such that point $C$ lies to the right of point $A$ and $D = AE \cap BC$. Point $C$ satisfies the conditions $AE = EC$, $AC = CD$, and $AB$ is tangent to the circumc...	[ "45 degrees" ]	[ "Triangle \$ABE\$ is a right triangle with hypotenuse \$AB\$, and point \$A\$ is to the right of point \$B\$. A line segment \$BC\$ is created such that point \$C\$ lies to the right of point \$A\$ and \$D = AE \\cap BC\$. Point \$C\$ satisfies the conditions \$AE = EC\$, \$AC = CD\$, and \\(A...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 14, "tir": 6 }	20
omr-ws-d371b4213491d43d3002c9596984c08a9d0ba9111250c3721f393b74bebcebc3	Triangle $ADF$ is a right triangle with $\angle D = 30^\circ$ and $\angle F = 60^\circ$. Line $CF$ is the angle bisector of $\angle F$ such that $C$ is on $AD$. Line $EC$ is the angle bisector of $\angle C$ such that $E$ is on $AF$. Lastly, line $BE$ is the angle bisector of $\angle E$ such th...	[ "10" ]	[ "Triangle \$ADF\$ is a right triangle with \$\\angle D = 30^\\circ\$ and \$\\angle F = 60^\\circ\$. Line \$CF\$ is the angle bisector of \$\\angle F\$ such that \$C\$ is on \$AD\$. Line \$EC\$ is the angle bisector of \$\\angle C\$ such that \$E\$ is on \$AF\$. Lastly, line \$BE\$ is the ang...	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 6 }	22
omr-ws-cc64f656a3fcb4a6c0ace61f51456ef9e85d56ec05aaf854453d620f7ddcc6f0	Triangle $AMC$ is isosceles with $AM = AC$. Medians $\overline{MV}$ and $\overline{CU}$ are perpendicular to each other, and $MV = CU = 12$. What is the area of $\triangle AMC$?	[ "96" ]	[ "Triangle \$AMC\$ is isosceles with \$AM = AC\$. Medians \$\\overline{MV}\$ and \$\\overline{CU}\$ are perpendicular to each other, and \$MV = CU = 12\$. What is the area of \$\\triangle AMC\$?" ]	[ "aops_c5_contests_amp_programs" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 1 }	17
omr-ws-211a8ddc755548d524cb1eed5d5de0e3eec1fa0ba34482cec534ec6064792beb	Triangle $A_1A_2A_3$ is inscribed in a circle. The medians through $A_1$ and $A_2$ meet the circle again at $M_1$ and $M_2$ respectively. The angle bisectors through $A_1$ and $A_2$ meet the circle again at $T_1$ and $T_2$ respectively. If $A_1A_2 = 12$, $A_1A_3 = 16$, and $A_2A_3 = 20$, calcula...	[ "4" ]	[ "Triangle \$A_1A_2A_3\$ is inscribed in a circle. The medians through \$A_1\$ and \$A_2\$ meet the circle again at \$M_1\$ and \$M_2\$ respectively. The angle bisectors through \$A_1\$ and \$A_2\$ meet the circle again at \$T_1\$ and \$T_2\$ respectively. If \$A_1A_2 = 12\$, \$A_1A_3 = 16\$, a...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems", "genselect" ]	{ "additional_problems": 1, "genselect": 3 }	4
omr-ws-141d3b318a0aef3cf1fc72d25a8cf38a4d7c573b56039f80e72a4ba59bbdc374	Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA = FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA = DC$ and $AC$ is the bisector of $\angle DAB$. Point $E$ is chosen so that $EA = ED$ and $AD$ is the bisector of $\angle EAC$....	[ "no_valid_answer_found" ]	[ "Triangle \$BCF\$ has a right angle at \$B\$. Let \$A\$ be the point on line \$CF\$ such that \$FA = FB\$ and \$F\$ lies between \$A\$ and \$C\$. Point \$D\$ is chosen so that \$DA = DC\$ and \$AC\$ is the bisector of \$\\angle DAB\$. Point \$E\$ is chosen so that \$EA = ED\$ and \\(AD\\...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-b57de7e73e823384add1948a1593618b67ac8bce0d92e43d2b20af788d3cee13	Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA = FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA = DC$ and $AC$ is the bisector of $\angle DAB$. Point $E$ is chosen so that $EA = ED$ and $AD$ is the bisector of $\angle EAC$....	[ "\\dfrac{11}{4}" ]	[ "Triangle \$BCF\$ has a right angle at \$B\$. Let \$A\$ be the point on line \$CF\$ such that \$FA = FB\$ and \$F\$ lies between \$A\$ and \$C\$. Point \$D\$ is chosen so that \$DA = DC\$ and \$AC\$ is the bisector of \$\\angle DAB\$. Point \$E\$ is chosen so that \$EA = ED\$ and \\(AD\\...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 2 }	2
omr-ws-075a8cb6366ef5bf36aff5f83a10e216ae9b3dd46619ee8cdd7a1238c8bd256f	Triangle $OAB$ has $O=(0,0)$, $B=(5,0)$, and $A$ in the first quadrant. In addition, $\angle ABO=90^\circ$ and $\angle AOB=30^\circ$. Suppose that $OA$ is rotated $90^\circ$ counterclockwise about $O$. What are the coordinates of the image of $A$?	[ "\$\\left( -\\frac{5 \\sqrt{3}}{3}, 5 \\right)\$" ]	[ "Triangle \$OAB\$ has \$O=(0,0)\$, \$B=(5,0)\$, and \$A\$ in the first quadrant. In addition, \$\\angle ABO=90^\\circ\$ and \$\\angle AOB=30^\\circ\$. Suppose that \$OA\$ is rotated \$90^\\circ\$ counterclockwise about \$O\$. What are the coordinates of the image of \$A\$?" ]	[ "aops_c5_contests_amp_programs" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 21 }	37
omr-ws-84879eb6b1e670939d07890ec4bf5dc8d9d9fb216567d10d98f632e3ca8f0473	Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside $\triangle ABC$, $\angle ABC = 40^\circ$, and $\angle ADC = 140^\circ$. What is the degree measure of $\angle BAD$?	[ "50^{\\circ}" ]	[ "Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside $\\triangle ABC$, $\\angle ABC = 40^\\circ$, and $\\angle ADC = 140^\\circ$. What is the degree measure of $\\angle\nBAD$?" ]	[ "MATH_training_set" ]	[ "cot", "genselect" ]	{ "cot": 15, "genselect": 8 }	23
omr-ws-0feb185e9413c21dc56860945676c84abb314ec7b4dc49d00fb0ea6a869fa18b	Triangles $ABC$ and $ADE$ have areas $2007$ and $7002,$ respectively, with $B=(0,0), C=(223,0), D=(680,380),$ and $E=(689,389).$ What is the sum of all possible $x$-coordinates of $A$?	[ "1200" ]	[ "Triangles $ABC$ and $ADE$ have areas $2007$ and $7002,$ respectively, with $B=(0,0), C=(223,0), D=(680,380),$ and $E=(689,389).$ What is the sum of all possible $x$-coordinates of $A$?" ]	[ "MATH_training_set" ]	[ "cot", "genselect" ]	{ "cot": 15, "genselect": 8 }	23
omr-ws-35890137aa21c6377bb9dcc3a61ee0c423e05c6d29c0ef495e64d26f69e880d9	Triangles $ABC$ and $AEF$ are such that $B$ is the midpoint of $\overline{EF}.$ Also, $AB = EF = 1,$ $BC = 6,$ $CA = \sqrt{33},$ and \[\overrightarrow{AB} \cdot \overrightarrow{AE} + \overrightarrow{AC} \cdot \overrightarrow{AF} = 2.\]Find the cosine of the angle between vectors $\overrightarrow{EF}$ and $\overrightarr...	[ "\\frac{2}{3}" ]	[ "Triangles $ABC$ and $AEF$ are such that $B$ is the midpoint of $\\overline{EF}.$ Also, $AB = EF = 1,$ $BC = 6,$ $CA = \\sqrt{33},$ and\n\\[\\overrightarrow{AB} \\cdot \\overrightarrow{AE} + \\overrightarrow{AC} \\cdot \\overrightarrow{AF} = 2.\\]Find the cosine of the angle between vectors $\\overrightarrow{EF}$ ...	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-9f07a9aeefb0ae31419a9386fc0fe8df2e087883553431742dd7614ead9d09e9	Triangles $ABC$ and $A_1 B_1 C_1$ are such that $\sin A_1 = \cos A$, $\sin B_1 = \cos B$, and $\sin C_1 = \cos C$. Find the measure of the largest of the angles $A, B, C, A_1, B_1, C_1$.	[ "135^\\circ" ]	[ "Triangles $ABC$ and $A_1 B_1 C_1$ are such that $\\sin A_1 = \\cos A$, $\\sin B_1 = \\cos B$, and $\\sin C_1 = \\cos C$. Find the measure of the largest of the angles $A, B, C, A_1, B_1, C_1$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-cce9f926db03a9f29ca8afda455f685a85c8087222d908a7dd121148a3b8456a	Triangles $ABC$ and $A_1B_1C_1$ are similar and differently oriented. On the segment $AA_1$, a point $A'$ is taken such that $AA' / A_1A' = BC / B_1C_1$. Similarly, points $B'$ and $C'$ are constructed on segments $BB_1$ and $CC_1$, respectively. If $A'$, $B'$, and $C'$ lie on one straight line, and the ratio of simila...	[ "\\dfrac{3}{2}" ]	[ "Triangles $ABC$ and $A_1B_1C_1$ are similar and differently oriented. On the segment $AA_1$, a point $A'$ is taken such that $AA' / A_1A' = BC / B_1C_1$. Similarly, points $B'$ and $C'$ are constructed on segments $BB_1$ and $CC_1$, respectively. If $A'$, $B'$, and $C'$ lie on one straight line, and the ratio of s...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 6 }	6
omr-ws-cc18ffa7bc7b9f861d9b8daa3586f2bcd43947e661f8709eba3f743d2f442c45	Triangles $ABC$ and $A_1B_1C_1$ have corresponding, parallel sides: $AB \parallel A_1B_1$, $CB \parallel C_1B_1$, and $AC \parallel A_1C_1$. If the angles of triangle $ABC$ are $\alpha$, $\beta$, and $\gamma$, and the angles of triangle $A_1B_1C_1$ are $\alpha_1$, $\beta_1$, and $\gamma_1$, calculate the value of $\alp...	[ "0" ]	[ "Triangles $ABC$ and $A_1B_1C_1$ have corresponding, parallel sides: $AB \\parallel A_1B_1$, $CB \\parallel C_1B_1$, and $AC \\parallel A_1C_1$. If the angles of triangle $ABC$ are $\\alpha$, $\\beta$, and $\\gamma$, and the angles of triangle $A_1B_1C_1$ are $\\alpha_1$, $\\beta_1$, and $\\gamma_1$, calculate the ...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-68220fec7bc7b2e7eab6585a9e713ea09dc88a76b694778bb83f93bc31b64898	Triangles $ABC$ and $DEF$ are such that $\angle A = \angle D$, $AB = DE = 17$, $BC = EF = 10$, and $AC - DF = 12$. What is $AC + DF$?	[ "30" ]	[ "Triangles $ABC$ and $DEF$ are such that $\\angle A = \\angle D$, $AB = DE = 17$, $BC = EF = 10$, and $AC - DF = 12$. What is $AC + DF$?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 10 }	25
omr-ws-269c6d444d0d3a77779664885d25f4de991ca1cb3d93850c0cb1e8f10fe0ec32	Triangles $ABC$ and $DEF$ have pairwise parallel sides: $EF \parallel BC$, $FD \parallel CA$, and $DE \parallel AB$. The line $m_A$ is the reflection of $EF$ through $BC$, similarly $m_B$ is the reflection of $FD$ through $CA$, and $m_C$ is the reflection of $DE$ through $AB$. Assume that the lines $m_A$, $m_B$, and $m...	[ "\\dfrac{1}{4}" ]	[ "Triangles $ABC$ and $DEF$ have pairwise parallel sides: $EF \\parallel BC$, $FD \\parallel CA$, and $DE \\parallel AB$. The line $m_A$ is the reflection of $EF$ through $BC$, similarly $m_B$ is the reflection of $FD$ through $CA$, and $m_C$ is the reflection of $DE$ through $AB$. Assume that the lines $m_A$, $m_B$...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 14, "genselect": 6 }	20
omr-ws-0ab91152ae8780a5b1f5e8425e76e407fbcc3d1deb217dc5cf932f66a593a436	Triangles $ABC$ and $DEF$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A,F,B,D,C,$ and $E$ occur in this order along $\Omega$. Let $\Delta_A$ be the triangle formed by lines $AB, AC,$ and $EF,$ and define triangles $\Delta_B, \Delta_C, \ldots, \Delta_F$ similarly. Furthermore, let $\Omega_A$ and $\...	[ "2" ]	[ "Triangles $ABC$ and $DEF$ share circumcircle $\\Omega$ and incircle $\\omega$ so that points $A,F,B,D,C,$ and $E$ occur in this order along $\\Omega$. Let $\\Delta_A$ be the triangle formed by lines $AB, AC,$ and $EF,$ and define triangles $\\Delta_B, \\Delta_C, \\ldots, \\Delta_F$ similarly. Furthermore, let $\\O...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 5 }	5
omr-ws-d525ebb66fc2828cb53c9c103bc4a97e67cdd0b33f7e949dd530424b48688da7	Triangles $ABC$ and $XYZ$ have a common inscribed circle, lines $BC$ and $YZ$ coincide, and line $AX$ is parallel to $BC$. Let $P$ be the point of contact of the common inscribed circle with the line $BC$. If $O_1$ and $O_2$ are the centers of the circles circumscribed around triangles $ABC$ and $XYZ$ respectively, and...	[ "no_valid_answer_found" ]	[ "Triangles $ABC$ and $XYZ$ have a common inscribed circle, lines $BC$ and $YZ$ coincide, and line $AX$ is parallel to $BC$. Let $P$ be the point of contact of the common inscribed circle with the line $BC$. If $O_1$ and $O_2$ are the centers of the circles circumscribed around triangles $ABC$ and $XYZ$ respectively...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-7da0b62d1738e4f10c4168478ed4a4e61337afc18f75f5261d81b0c764dc99e2	Triangles $AMB$ and $AMC$ lie on different sides of the segment $AM$. It is known that $\angle BAM=30^\circ$, $\angle BMA=110^\circ$, $\angle CAM=20^\circ$, $\angle CMA=150^\circ$. Find the angles of triangle $BMC$.	[ "\$\\angle BMC = 100^\\circ\$, \$\\angle MBC = 40^\\circ\$, \$\\angle MCB = 10^\\circ\$" ]	[ "Triangles $AMB$ and $AMC$ lie on different sides of the segment $AM$. It is known that $\\angle BAM=30^\\circ$, $\\angle BMA=110^\\circ$, $\\angle CAM=20^\\circ$, $\\angle CMA=150^\\circ$. Find the angles of triangle $BMC$." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-a7eb12c9c9ea82ad8ade2d3ebc722414204605e748be3f607df9e4134cc0de71	Triangles $A_1B_1C_1$ and $A_2B_2C_2$ are inscribed into triangle $ABC$ such that $C_1A_1 \perp BC$, $A_1B_1 \perp CA$, $B_1C_1 \perp AB$, $B_2A_2 \perp BC$, $C_2B_2 \perp CA$, $A_2C_2 \perp AB$. If $A_1B_1C_1$ and $A_2B_2C_2$ have equal areas, find the ratio of the perimeter of triangle $A_1B_1C_1$ to the perimeter of...	[ "1" ]	[ "Triangles $A_1B_1C_1$ and $A_2B_2C_2$ are inscribed into triangle $ABC$ such that $C_1A_1 \\perp BC$, $A_1B_1 \\perp CA$, $B_1C_1 \\perp AB$, $B_2A_2 \\perp BC$, $C_2B_2 \\perp CA$, $A_2C_2 \\perp AB$. If $A_1B_1C_1$ and $A_2B_2C_2$ have equal areas, find the ratio of the perimeter of triangle $A_1B_1C_1$ to the p...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-b90c5911dfacc85c362c1823a8e869b681a8b004bce93e8a16ffba683e7141b9	Triangles $A_1B_1C_1$ and $A_2B_2C_2$ have parallel sides: $A_1B_1 \parallel A_2B_2$, $B_1C_1 \parallel B_2C_2$, and $C_1A_1 \parallel C_2A_2$. Triangle $A_2B_2C_2$ is contained inside triangle $A_1B_1C_1$. A third triangle $ABC$ is inscribed in $A_1B_1C_1$ and circumscribed around $A_2B_2C_2$, such that points $A, B, ...	[ "21" ]	[ "Triangles $A_1B_1C_1$ and $A_2B_2C_2$ have parallel sides: $A_1B_1 \\parallel A_2B_2$, $B_1C_1 \\parallel B_2C_2$, and $C_1A_1 \\parallel C_2A_2$. Triangle $A_2B_2C_2$ is contained inside triangle $A_1B_1C_1$. A third triangle $ABC$ is inscribed in $A_1B_1C_1$ and circumscribed around $A_2B_2C_2$, such that points...	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 2 }	18
omr-ws-6cb6d37c5f29cf81a575292494fd04b4ea794fd81c7f4fcb77b4c963bca971ad	Triangles $BCD$ and $ACE$ are erected externally to a triangle $ABC$ such that $AE = BD$ and ∠$BDC$ + ∠$AEC$ = $180^{\circ}$. Let $F$ be a point on segment $AB$ such that $\frac{AF}{FB} = \frac{CD}{CE}$. If $CD = 12$, $CE = 18$, and $BC = 24$, calculate the value of $\frac{DE}{CD+CE} + \frac{EF}{BC} + \frac{FD}{AC}$.	[ "2" ]	[ "Triangles $BCD$ and $ACE$ are erected externally to a triangle $ABC$ such that $AE = BD$ and ∠$BDC$ + ∠$AEC$ = $180^{\\circ}$. Let $F$ be a point on segment $AB$ such that $\\frac{AF}{FB} = \\frac{CD}{CE}$. If $CD = 12$, $CE = 18$, and $BC = 24$, calculate the value of $\\frac{DE}{CD+CE} + \\frac{EF}{BC} + \\frac{...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 3 }	3
omr-ws-020a0072da185c72a196233010f6b6698365a9d201aff52373d6ae3006c08b41	Triangles $BCD$ and $ACE$ are erected externally to a triangle $ABC$ with $\angle AEC + \angle BDC = 180^\circ$ and $AE = BD$. Point $F$ lies on segment $AB$ such that $\frac{AF}{FB} = \frac{CD}{CE}$. If $DE = 12$, $CD = 5$, and $CE = 7$, find the value of $\frac{EF}{CB} + \frac{DF}{CA}$.	[ "2" ]	[ "Triangles $BCD$ and $ACE$ are erected externally to a triangle $ABC$ with $\\angle AEC + \\angle BDC = 180^\\circ$ and $AE = BD$. Point $F$ lies on segment $AB$ such that $\\frac{AF}{FB} = \\frac{CD}{CE}$. If $DE = 12$, $CD = 5$, and $CE = 7$, find the value of $\\frac{EF}{CB} + \\frac{DF}{CA}$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-3961e3bc46b5fd88dceef08b64ee1eb338dd3e2b56e44a85045f9c1ec16d4d4c	Triangles $BCD$, $CEA$, and $AFB$ are constructed outside triangle $ABC$ such that $\angle ABF = \angle CBD$, $\angle BCD = \angle ACE$, and $\angle CAE = \angle BAF$. Let $P$ be the intersection point of lines $AD$, $BE$, and $CF$. If $AB = 13$, $BC = 14$, and $CA = 15$, calculate the ratio of the area of triangle $PB...	[ "\\dfrac{4}{21}" ]	[ "Triangles $BCD$, $CEA$, and $AFB$ are constructed outside triangle $ABC$ such that $\\angle ABF = \\angle CBD$, $\\angle BCD = \\angle ACE$, and $\\angle CAE = \\angle BAF$. Let $P$ be the intersection point of lines $AD$, $BE$, and $CF$. If $AB = 13$, $BC = 14$, and $CA = 15$, calculate the ratio of the area of t...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 3, "genselect": 1 }	4
omr-ws-2c708484e22edb155d207aa1e2e589a52cf3cc0f4a7a302679355cd497da78a5	Triangles $OAB$, $OBC$, and $OCD$ are isosceles triangles with $\angle OAB = \angle OBC = \angle OCD = 90^\circ$. If the area of triangle $OCD$ is 12, find the area of triangle $OAB$.	[ "3" ]	[ "Triangles $OAB$, $OBC$, and $OCD$ are isosceles triangles with $\\angle OAB = \\angle OBC = \\angle OCD = 90^\\circ$. If the area of triangle $OCD$ is 12, find the area of triangle $OAB$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 15, "genselect": 4 }	19
omr-ws-7ed92ca9cd6b4e5ea574c2708a0a27997c0938916a340aeb200c4e85b0eb7cb4	Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$ where $0<m<180$, will transform $\triangle ABC$ to $\triangle A'B'C'$. Find $m+x+y$.	[ "108" ]	[ "Triangles $\\triangle ABC$ and $\\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$ where $0<m<180$, will transform $\\triangle ABC$ to $\\triangle A'B'C'$. Find $m+x+y$.", ...	[ "MATH_training_set" ]	[ "cot", "tir" ]	{ "cot": 32, "tir": 2 }	34
omr-ws-2e8434cc14f352430d49359d3ecde593271e9e08f295b8f1d065032b358cb9cf	Triangles ABC and CED are equilateral. Lines AD and BE are drawn. Given that angle ACE is 70 degrees, calculate angle AFB.	[ "60 degrees" ]	[ "Triangles ABC and CED are equilateral. Lines AD and BE are drawn. Given that angle ACE is 70 degrees, calculate angle AFB." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 4, "genselect": 1, "tir": 6 }	11
omr-ws-339430661d3f2f6f4d6d9ee5501afb90999fce0df9a071477beafce6501fa2f0	Triangles ABD and ACE are equilateral. Angle CAB is 90 degrees. If AB = AC, what is the measure of angle CED?	[ "\$ 30^\\circ \$" ]	[ "Triangles ABD and ACE are equilateral. Angle CAB is 90 degrees. If AB = AC, what is the measure of angle CED?" ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-35eef8f75d74fbf43035a112789d0bf4592d0cb82148bda34068be0984ca0354	Triangles ACD and BCD are inscribed in a semicircle with diameter CD. Given AD=14, BD=40, and CD=50, calculate the area of the union of the interiors of triangles ACD and BCD.	[ "\\dfrac{1347}{2}" ]	[ "Triangles ACD and BCD are inscribed in a semicircle with diameter CD. Given AD=14, BD=40, and CD=50, calculate the area of the union of the interiors of triangles ACD and BCD." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-82e0a13fb04dc112ed75b73bb64151e96f386d94f51d180862e5560513333301	Triangles $ T_1 $ and $ T_2 $ are coplanar, and each point of $ T_2 $ is interior to and 2 units from $ T_1 $. If the side lengths of $ T_1 $ are 13, 14, and 15, find the area of the region that is the intersection of the interior of $ T_1 $ and the exterior of $ T_2 $.	[ "63." ]	[ "Triangles \$ T_1 \$ and \$ T_2 \$ are coplanar, and each point of \$ T_2 \$ is interior to and 2 units from \$ T_1 \$. If the side lengths of \$ T_1 \$ are 13, 14, and 15, find the area of the region that is the intersection of the interior of \$ T_1 \$ and the exterior of \$ T_2 \$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 14, "genselect": 7, "tir": 8 }	29
omr-ws-ab3865f6403939b42f42bf6694b959331d0c052a62ec5f39d3aedbf307248d38	Triangles $ABC$ and $A'B'C'$ lie on the same circle and have the same orthocenter $H$. The lines $AA'$, $BB'$, and $CC'$ intersect at points $X$, $Y$, and $Z$, respectively. If the circumradius of triangle $XYZ$ is $R$, find the value of $R$ given that the circumradius of triangle $ABC$ is \(1...	[ "5" ]	[ "Triangles \$ABC\$ and \$A'B'C'\$ lie on the same circle and have the same orthocenter \$H\$. The lines \$AA'\$, \$BB'\$, and \$CC'\$ intersect at points \$X\$, \$Y\$, and \$Z\$, respectively. If the circumradius of triangle \$XYZ\$ is \$R\$, find the value of \$R\$ given that the circumradi...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 4, "genselect": 8 }	12
omr-ws-863903bc743734241249225ac000a8f0743905bfda713f86a2c97759c0f9dfb7	Triangles $ABC$ and $A'B'C'$ lie on the same circle with circumcenter $O$ and have the same orthocenter $H$. The lines $AA'$, $BB'$, and $CC'$ intersect at points $X$, $Y$, and $Z$ respectively. Calculate the ratio of the distance from the circumcenter of triangle $XYZ$ to $O$ to the distance fr...	[ "\\dfrac{1}{2}" ]	[ "Triangles \$ABC\$ and \$A'B'C'\$ lie on the same circle with circumcenter \$O\$ and have the same orthocenter \$H\$. The lines \$AA'\$, \$BB'\$, and \$CC'\$ intersect at points \$X\$, \$Y\$, and \$Z\$ respectively. Calculate the ratio of the distance from the circumcenter of triangle \$XYZ\$ ...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-dc070505e4c7c31796f864fc637a0bff4bda77d7bc64b55879e9d992bba66df4	Triangles $ABC$ and $ADE$ are equilateral, and $AD$ is an altitude of $ABC$. The area of the intersection of these triangles is $3$. Find the area of the larger triangle $ABC$.	[ "8" ]	[ "Triangles \$ABC\$ and \$ADE\$ are equilateral, and \$AD\$ is an altitude of \$ABC\$. The area of the intersection of these triangles is \$3\$. Find the area of the larger triangle \$ABC\$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-13bc711f381602e06bff6911fa63c301d1f002feb2e2597a3db636849f3212be	Triangles $ABC$ and $ADE$ have areas 2007 and 7002, respectively, with $B = (0,0)$, $C = (223,0)$, $D = (680,380)$, and $E = (689,389)$. What is the sum of all possible x-coordinates of $A$?	[ "1200" ]	[ "Triangles \$ABC\$ and \$ADE\$ have areas 2007 and 7002, respectively, with \$B = (0,0)\$, \$C = (223,0)\$, \$D = (680,380)\$, and \$E = (689,389)\$. What is the sum of all possible x-coordinates of \$A\$?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 26 }	42
omr-ws-36f452b1834fbdee9cd2e4b4f84cf6b643bce3960227fa31f2a5c8834f820d06	Triangles $ABC$ and $A_1B_1C_1$ are non-congruent with $AC = A_1C_1 = b$, $BC = B_1C_1 = a$, and $BH = B_1H_1$, where $BH$ and $B_1H_1$ are the altitudes. Given $a = 5$ and $b = 12$, find the maximum possible value of $a \cdot AB + b \cdot A_1B_1$.	[ "169\\sqrt{2}" ]	[ "Triangles \$ABC\$ and \$A_1B_1C_1\$ are non-congruent with \$AC = A_1C_1 = b\$, \$BC = B_1C_1 = a\$, and \$BH = B_1H_1\$, where \$BH\$ and \$B_1H_1\$ are the altitudes. Given \$a = 5\$ and \$b = 12\$, find the maximum possible value of \$a \\cdot AB + b \\cdot A_1B_1\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 7, "genselect": 7 }	14
omr-ws-7c16e94b353f45f4e1609c16b1c550644f7fbe4d162501409537edf6daa79d75	Triangles $ABC$ and $DEF$ are similar but not congruent, with $[AB] \parallel [DE]$, $[AC] \parallel [DF]$, and $[BC] \parallel [EF]$. Let $P$ be the point of intersection of lines $AD$, $BE$, and $CF$. If the coordinates of $A$, $B$, $C$, $D$, $E$, and $F$ are $(1,1)$, $(4,1)$, \((1,4...	[ "(0,0)" ]	[ "Triangles \$ABC\$ and \$DEF\$ are similar but not congruent, with \$[AB] \\parallel [DE]\$, \$[AC] \\parallel [DF]\$, and \$[BC] \\parallel [EF]\$. Let \$P\$ be the point of intersection of lines \$AD\$, \$BE\$, and \$CF\$. If the coordinates of \$A\$, \$B\$, \$C\$, \$D\$, \$E\$, and \\...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-01e6159f5c9d4c45f4f396c178aa6c4b8e32f7ec0c31306e95d31d37df8a27f7	Triangles $ABC$ and $DEF$ have a common incircle of radius $R$ and intersect at points $X_1, X_2, \ldots, X_6$, forming six triangles. Let $r_1, r_2, \ldots, r_6$ be the radii of the inscribed circles of these six triangles, and let $R_1, R_2, \ldots, R_6$ be the radii of the inscribed circles of the triang...	[ "6" ]	[ "Triangles \$ABC\$ and \$DEF\$ have a common incircle of radius \$R\$ and intersect at points \$X_1, X_2, \\ldots, X_6\$, forming six triangles. Let \$r_1, r_2, \\ldots, r_6\$ be the radii of the inscribed circles of these six triangles, and let \$R_1, R_2, \\ldots, R_6\$ be the radii of the inscribed c...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-b52e16480359a725fb16f85273504f2fc762cce1c6a525561eebb1fbe04be7c4	Triangles $ABC$ and $DEF$ have pairwise parallel sides: $EF \parallel BC$, $FD \parallel CA$, and $DE \parallel AB$. The line $m_A$ is the reflection of $EF$ through $BC$, $m_B$ is the reflection of $FD$ through $CA$, and $m_C$ is the reflection of $DE$ through $AB$. Assume that the lines \(...	[ "4" ]	[ "Triangles \$ABC\$ and \$DEF\$ have pairwise parallel sides: \$EF \\parallel BC\$, \$FD \\parallel CA\$, and \$DE \\parallel AB\$. The line \$m_A\$ is the reflection of \$EF\$ through \$BC\$, \$m_B\$ is the reflection of \$FD\$ through \$CA\$, and \$m_C\$ is the reflection of \$DE\$ throug...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 8, "genselect": 7, "tir": 4 }	19
omr-ws-8f0745d486ebbaf2fbd5735649dedea2c40cabaf1b4e8355a6e524b8c67288ea	Triangles $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB = 4$, what is the perimeter of figure $ABCDEFG$?	[ "15" ]	[ "Triangles \$ABC\$, \$ADE\$, and \$EFG\$ are all equilateral. Points \$D\$ and \$G\$ are midpoints of \$\\overline{AC}\$ and \$\\overline{AE}\$, respectively. If \$AB = 4\$, what is the perimeter of figure \$ABCDEFG\$?" ]	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-fa7a5f2be69748d02b702844ea14f1be043c972ca0ce965eecfb23ff9a941ad5	Trickster Rabbit agrees to double Foolish Fox's money every time Fox crosses the bridge by Rabbit's house, but Fox must pay 40 coins in toll to Rabbit after each crossing. After crossing the bridge three times, Fox discovers that all his money is gone. How many coins did Fox have at the beginning?	[ "35" ]	[ "Trickster Rabbit agrees to double Foolish Fox's money every time Fox crosses the bridge by Rabbit's house, but Fox must pay 40 coins in toll to Rabbit after each crossing. After crossing the bridge three times, Fox discovers that all his money is gone. How many coins did Fox have at the beginning?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 5 }	21
omr-ws-f5bddc39b07fd7999eb7b622fe5db8bcaeed119effb8e4906e76d6b6a257ad4d	Trickster Rabbit agrees to double Foolish Fox's money every time Fox crosses the bridge, but Fox must pay 40 coins in toll to Rabbit after each doubling. After crossing the bridge three times, Fox discovers that all his money is gone. How many coins did Fox have at the beginning?	[ "35" ]	[ "Trickster Rabbit agrees to double Foolish Fox's money every time Fox crosses the bridge, but Fox must pay 40 coins in toll to Rabbit after each doubling. After crossing the bridge three times, Fox discovers that all his money is gone. How many coins did Fox have at the beginning?" ]	[ "aops_c5_contests_amp_programs" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 4 }	20
omr-ws-b6388585455315f940a5f2086f736337cf6fe890f4c5ca06a7186ce13aabab10	Trinni rearranged the numbers 1 through 12 around a clock face so that each adjacent pair adds up to a triangle number. The number 12 is in its usual place. What number is where the 6 would usually be?	[ "11" ]	[ "Trinni rearranged the numbers 1 through 12 around a clock face so that each adjacent pair adds up to a triangle number. The number 12 is in its usual place. What number is where the 6 would usually be?" ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-24d6e522b4088ac51f87c9afdd9999bebe6351e3b12470a831ab948a219bfb35	Trisect each side of an equilateral triangle with side length 2 and construct an equilateral triangle on the center third of each side. Omit the side of each new triangle that was once part of the original triangle. Repeat the trisection, construction, and omission steps on each segment formed, creating a figure. What ...	[ "\$\\frac{9\\sqrt{3}}{5}\$" ]	[ "Trisect each side of an equilateral triangle with side length 2 and construct an equilateral triangle on the center third of each side. Omit the side of each new triangle that was once part of the original triangle. Repeat the trisection, construction, and omission steps on each segment formed, creating a figure. ...	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-66f0cf5d07da28688c15a71e04e43d66f522346cb791646ed41c0af7ec99ffde	Trisha's store sells pineapples in packs of 7 and 9. What is the largest number of pineapples that cannot be bought at Trisha's store with integral combinations of these pack sizes? In other words, what is the largest number of pineapples that cannot be written in the form $7a + 9b$, where $a$ and $b$ are positiv...	[ "47" ]	[ "Trisha's store sells pineapples in packs of 7 and 9. What is the largest number of pineapples that cannot be bought at Trisha's store with integral combinations of these pack sizes? In other words, what is the largest number of pineapples that cannot be written in the form \$7a + 9b\$, where \$a\$ and \$b\$ ...	[ "aops_c4_high_school_math" ]	[ "tir" ]	{ "tir": 8 }	8
omr-ws-ba90f73bcd132735b15cd360d9e475d5a1f53168257d6d8d4789aed4f462eacc	Trophies at Trophies trophies shop cost 3 trophies each and trollphies cost 2 trophies each. If 5 trophies is equal to 6 trollphies, which is all equal to 7 dollars, how many dollars will it cost to purchase 5 trophies and 8 trollphies?	[ "\\dfrac{49}{3}" ]	[ "Trophies at Trophies trophies shop cost 3 trophies each and trollphies cost 2 trophies each. If 5 trophies is equal to 6 trollphies, which is all equal to 7 dollars, how many dollars will it cost to purchase 5 trophies and 8 trollphies?" ]	[ "aops_c5_contests_amp_programs" ]	[ "cot", "genselect" ]	{ "cot": 11, "genselect": 2 }	13
omr-ws-72012e127733ad6362e6db6ccd6939acd65cfe55b82bb243897e1d9d3e346db6	Truman and Colin regularly walk on a 2-mile trail that has one single entry point. To complete the trail, they must walk a round trip. If both randomly choose a time on the hour between 8 AM and 12 Noon and they both walk at 2 mph, what is the probability that their paths will intersect? Assume Truman and Colin can sta...	[ "\\dfrac{7}{32}" ]	[ "Truman and Colin regularly walk on a 2-mile trail that has one single entry point. To complete the trail, they must walk a round trip. If both randomly choose a time on the hour between 8 AM and 12 Noon and they both walk at 2 mph, what is the probability that their paths will intersect? Assume Truman and Colin ca...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 4, "genselect": 4 }	8

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