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	blank_reference_answer_rows int64 0 25
omr-ws-70ee44acba482bcb9b4110f6451c7d143d9e280316d81f29ab17d4a0f06d07de	Two circles $(O_1)$ and $(O_2)$ intersect at points $A$ and $B$. $MN$ is a common tangent to $(O_1)$ and $(O_2)$ near $A$ (with $M \in (O_1)$ and $N \in (O_2)$). A line parallel to $MN$ intersects $(O_1)$ at $Q$ and $(O_2)$ at $P$. Let $S$ be the intersection of $MQ$ and $NP$. If the...	[ "(0,0)" ]	[ "Two circles \$(O_1)\$ and \$(O_2)\$ intersect at points \$A\$ and \$B\$. \$MN\$ is a common tangent to \$(O_1)\$ and \$(O_2)\$ near \$A\$ (with \$M \\in (O_1)\$ and \$N \\in (O_2)\$). A line parallel to \$MN\$ intersects \$(O_1)\$ at \$Q\$ and \$(O_2)\$ at \$P\$. Let \$S\$ be the in...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-6bd4b93a15b5065f6ccf7d4227981875a4c3d48d909ba92b5f23d0da0507a1a0	Two circles $(O_1)$ and $(O_2)$ intersect at points $A$ and $D$. The lines $O_1A$ and $O_2A$ intersect $(O_1)$ and $(O_2)$ again at points $B$ and $C$ respectively. Let $M$ be an arbitrary point on $AD$. The lines $BM$ and $CM$ intersect $AC$ and $AB$ at points $E$ and $F$ respective...	[ "1" ]	[ "Two circles \$(O_1)\$ and \$(O_2)\$ intersect at points \$A\$ and \$D\$. The lines \$O_1A\$ and \$O_2A\$ intersect \$(O_1)\$ and \$(O_2)\$ again at points \$B\$ and \$C\$ respectively. Let \$M\$ be an arbitrary point on \$AD\$. The lines \$BM\$ and \$CM\$ intersect \$AC\$ and \$AB\$...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 2 }	2	0
omr-ws-99c852a79052257bb8ccb30a8f77765e75e4e66d7fc878f50534148bd09d34ed	Two circles $(O_1, R)$ and $(O_2, r)$ intersect at points $P$ and $P'$. The line $O_1O_2$ intersects the circle $(O_1)$ at points $A$ and $B$ such that $A, O_1, O_2$ are in this order. The line $AP$ intersects the circle $(O_2)$ again at point $Q$. If $O_1O_2 = 10$, $AO_1 = 6$, and $R = 8$...	[ "15" ]	[ "Two circles \$(O_1, R)\$ and \$(O_2, r)\$ intersect at points \$P\$ and \$P'\$. The line \$O_1O_2\$ intersects the circle \$(O_1)\$ at points \$A\$ and \$B\$ such that \$A, O_1, O_2\$ are in this order. The line \$AP\$ intersects the circle \$(O_2)\$ again at point \$Q\$. If \\(O_1O_2 = 10\...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 4, "genselect": 5 }	9	0
omr-ws-34e9f93c6fea38e2822231771084c1a4dfb408a6231962e59e7a2cd5cc9aea95	Two circles $(w1)$ and $(w2)$ intersect at points $A$ and $B$. $MN$ is an external common tangent to the circles, with $M$ on $(w1)$ and $N$ on $(w2)$. Points $P$ on $(w1)$ and $Q$ on $(w2)$ are such that $PQ$ is parallel to $MN$. The lines $PM$ and $QN$ intersect at point $K$. If \(...	[ "\\theta" ]	[ "Two circles \$(w1)\$ and \$(w2)\$ intersect at points \$A\$ and \$B\$. \$MN\$ is an external common tangent to the circles, with \$M\$ on \$(w1)\$ and \$N\$ on \$(w2)\$. Points \$P\$ on \$(w1)\$ and \$Q\$ on \$(w2)\$ are such that \$PQ\$ is parallel to \$MN\$. The lines \$PM\$ and \...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-fa5578056ed0bd5a98853e40d899eb5f0e3ee1f7942d4bb56ec264dcfea538ef	Two circles $A$ and $B$ intersect at two distinct points $O$ and $P$. A line through $P$ intersects circle $A$ again at $C$ and circle $B$ again at $D$. Suppose that $CA = CD$. If the radius of circle $A$ is 10 units and the distance between the centers of circles $A$ and $B$ is 12 units, find...	[ "12" ]	[ "Two circles \$A\$ and \$B\$ intersect at two distinct points \$O\$ and \$P\$. A line through \$P\$ intersects circle \$A\$ again at \$C\$ and circle \$B\$ again at \$D\$. Suppose that \$CA = CD\$. If the radius of circle \$A\$ is 10 units and the distance between the centers of circles \\(A\\...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 2, "genselect": 7 }	9	0
omr-ws-5d2f0bada643dd238318a0c3c58e1ee5c6efebe6e8ae298f98ad8d37ac41604c	Two circles $A$ and $B$, both with radius 1, touch each other externally. Four circles $P, Q, R,$ and $S$, all with the same radius $r$, are arranged such that: - $P$ externally touches $A$, $B$, $Q$, and $S$, - $Q$ externally touches $P$, $B$, and $R$, - $R$ externally touches $A$, \(B\...	[ "2" ]	[ "Two circles \$A\$ and \$B\$, both with radius 1, touch each other externally. Four circles \$P, Q, R,\$ and \$S\$, all with the same radius \$r\$, are arranged such that:\n- \$P\$ externally touches \$A\$, \$B\$, \$Q\$, and \$S\$,\n- \$Q\$ externally touches \$P\$, \$B\$, and \$R\$,\n- ...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-b2f5da2381101a9da317a697a50d07f67f4a8692c3f6972b0d11b0c0edcb61ce	Two circles $C1$ and $C2$ with centers $O1$ and $O2$ respectively intersect at points $A$ and $B$. The line $O1B$ extended intersects $C2$ at $E$, and the line $O2B$ extended intersects $C1$ at $F$. A line $MN$ is drawn parallel to $EF$ through $B$, intersecting $C1$ at $N$ and $C2$ ...	[ "\\dfrac{305}{8}" ]	[ "Two circles \$C1\$ and \$C2\$ with centers \$O1\$ and \$O2\$ respectively intersect at points \$A\$ and \$B\$. The line \$O1B\$ extended intersects \$C2\$ at \$E\$, and the line \$O2B\$ extended intersects \$C1\$ at \$F\$. A line \$MN\$ is drawn parallel to \$EF\$ through \$B\$, inter...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7	0
omr-ws-cfebe56d0f91cf73ca25d5c1db0bffc6574b5e82acf62ec10fa453a2e5c5d0e2	Two circles $C_1$ and $C_2$ are internally tangent to a circle $C_3$ at points $A$ and $B$ respectively. A line $L$ is tangent to $C_1$ and $C_2$ at points $C$ and $D$ such that $C_1$ and $C_2$ are in different half-planes with respect to $CD$. Let $P$ and $Q$ be the intersections of \(C_3...	[ "12" ]	[ "Two circles \$C_1\$ and \$C_2\$ are internally tangent to a circle \$C_3\$ at points \$A\$ and \$B\$ respectively. A line \$L\$ is tangent to \$C_1\$ and \$C_2\$ at points \$C\$ and \$D\$ such that \$C_1\$ and \$C_2\$ are in different half-planes with respect to \$CD\$. Let \$P\$ and \\...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 3, "genselect": 4 }	7	0
omr-ws-e9ea266256985791ab56f115e561db06b6681eee4daa5d4e1380cb48422256e5	Two circles $C_1$ and $C_2$ intersect at points $A$ and $B$. Two lines are drawn from $A$ such that one line intersects $C_1$ at a point $P$ and the other intersects $C_2$ at a point $Q$, with $B$, $P$, and $Q$ being collinear. Three parallel lines are drawn within the triangle $APQ$, and a pe...	[ "AP = AQ" ]	[ "Two circles \$C_1\$ and \$C_2\$ intersect at points \$A\$ and \$B\$. Two lines are drawn from \$A\$ such that one line intersects \$C_1\$ at a point \$P\$ and the other intersects \$C_2\$ at a point \$Q\$, with \$B\$, \$P\$, and \$Q\$ being collinear. Three parallel lines are drawn within t...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 16 }	16	0
omr-ws-d871758d955b71b186cf63fdc90bf4b09cd9708a795c326795998eb8c910de7d	Two circles $C_1$ and $C_2$ intersect at points $P$ and $Q$. The common tangent to $C_1$ and $C_2$ nearer to $P$ than $Q$ touches $C_1$ at $A$ and $C_2$ at $B$. The tangent to $C_1$ at $P$ intersects $C_2$ at $E$ (distinct from $P$), and the tangent to $C_2$ at $P$ intersects \(C_1...	[ "12" ]	[ "Two circles \$C_1\$ and \$C_2\$ intersect at points \$P\$ and \$Q\$. The common tangent to \$C_1\$ and \$C_2\$ nearer to \$P\$ than \$Q\$ touches \$C_1\$ at \$A\$ and \$C_2\$ at \$B\$. The tangent to \$C_1\$ at \$P\$ intersects \$C_2\$ at \$E\$ (distinct from \$P\$), and the tange...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 2 }	7	0
omr-ws-c3891ddac1685cd8b9308ad63c429fddb5e7bfd1785613c9bf71ef98f8457395	Two circles $C_1$ and $C_2$ intersect at two distinct points in a plane. Let a line passing through $P$ meet the circles $C_1$ and $C_2$ at points $A$ and $B$ respectively. Let $Y$ be the midpoint of $AB$ and let $QY$ meet the circles $C_1$ and $C_2$ at points $X$ and $Z$ respectively. If \(...	[ "\\dfrac{25}{6}" ]	[ "Two circles \$C_1\$ and \$C_2\$ intersect at two distinct points in a plane. Let a line passing through \$P\$ meet the circles \$C_1\$ and \$C_2\$ at points \$A\$ and \$B\$ respectively. Let \$Y\$ be the midpoint of \$AB\$ and let \$QY\$ meet the circles \$C_1\$ and \$C_2\$ at points \\(X\\...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 3, "genselect": 4 }	7	0
omr-ws-f6b3184d60285e4e5e15264e80d84bef1006f80af45a11bee15722a81f6aa46a	Two circles $C_1$ and $C_2$ with radii $r_1$ and $r_2$ respectively touch each other externally. Points $A$ and $B$ are on $C_1$ and points $C$ and $D$ are on $C_2$ such that $AD$ and $BC$ are common external tangents to the two circles. Let $r$ be the radius of the circle inscribed in the qua...	[ "\\dfrac{24}{5}" ]	[ "Two circles \$C_1\$ and \$C_2\$ with radii \$r_1\$ and \$r_2\$ respectively touch each other externally. Points \$A\$ and \$B\$ are on \$C_1\$ and points \$C\$ and \$D\$ are on \$C_2\$ such that \$AD\$ and \$BC\$ are common external tangents to the two circles. Let \$r\$ be the radius of ...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-514334ca5b698c9eac6d013ba66e9268d1bab047ef2a526c8da1dac2cd968545	Two circles $O$ and $O'$ intersect at points $A$ and $B$. A line through $A$ intersects circle $O$ at $C$ and circle $O'$ at $D$ (with $A$ between $C$ and $D$). Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (not containing $A$), respectively, and let $K$ be the midpoint ...	[ "\\dfrac{1}{4}" ]	[ "Two circles \$O\$ and \$O'\$ intersect at points \$A\$ and \$B\$. A line through \$A\$ intersects circle \$O\$ at \$C\$ and circle \$O'\$ at \$D\$ (with \$A\$ between \$C\$ and \$D\$). Let \$M\$ and \$N\$ be the midpoints of the arcs \$BC\$ and \$BD\$ (not containing \$A\$), respe...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 3 }	3	0
omr-ws-7ffad5a05e93824fa109ce78582ac00d9abba7bba548e46b65597129227b8ed1	Two circles $O_1$ and $O_2$ are externally tangent to each other. $AB$ is a common tangent with $A$ on $(O_2)$ and $B$ on $(O_1)$. $F$ is a point on $(O_1)$ and $l$ is a tangent from $F$ to $(O_1)$ parallel to $AB$. $M$ is the midpoint of $AB$. A line from $A$ intersects $l$ and \((O_2...	[ "1" ]	[ "Two circles \$O_1\$ and \$O_2\$ are externally tangent to each other. \$AB\$ is a common tangent with \$A\$ on \$(O_2)\$ and \$B\$ on \$(O_1)\$. \$F\$ is a point on \$(O_1)\$ and \$l\$ is a tangent from \$F\$ to \$(O_1)\$ parallel to \$AB\$. \$M\$ is the midpoint of \$AB\$. A line fro...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 3 }	3	0
omr-ws-566201fc54b357202405b78df51d283664c9bd3118e6a1f8c9a850a0b70dd394	Two circles $O_1$ and $O_2$ intersect at points $A$ and $B$. A straight line through $A$ intersects $O_1$ at $C$ and $O_2$ at $D$. If the centers of the circles are $R_1$ and $R_2$ respectively, find the value of $\angle CBD + \angle R_1BR_2$ in degrees.	[ "180" ]	[ "Two circles \$O_1\$ and \$O_2\$ intersect at points \$A\$ and \$B\$. A straight line through \$A\$ intersects \$O_1\$ at \$C\$ and \$O_2\$ at \$D\$. If the centers of the circles are \$R_1\$ and \$R_2\$ respectively, find the value of \$\\angle CBD + \\angle R_1BR_2\$ in degrees." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 6 }	6	0
omr-ws-531b45d3c8d6fb46e6214b8cda3fd935e02889f22853a82bb4fb069fbc909945	Two circles $O_1$ and $O_2$ intersect at points $A$ and $B$. Lines $\overline{AC}$ and $\overline{BD}$ are drawn such that $C$ is on $O_1$ and $D$ is on $O_2$, with $\overline{AC} \perp \overline{AB}$ and $\overline{BD} \perp \overline{AB}$. The minor arc $AB$ is $45^\circ$ relative to \(O_1...	[ "155" ]	[ "Two circles \$O_1\$ and \$O_2\$ intersect at points \$A\$ and \$B\$. Lines \$\\overline{AC}\$ and \$\\overline{BD}\$ are drawn such that \$C\$ is on \$O_1\$ and \$D\$ is on \$O_2\$, with \$\\overline{AC} \\perp \\overline{AB}\$ and \$\\overline{BD} \\perp \\overline{AB}\$. The minor arc \\(...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 6, "genselect": 3, "tir": 7 }	16	0
omr-ws-e58e490c997424a41ce2f6cf7b5fa47f6fc038ce5907fcc8fcf407b7a0598320	Two circles $O_1$ and $O_2$ intersect at points $A$ and $B$. The bisector of the outer angle $\angle O_1AO_2$ intersects circle $O_1$ at $C$ and circle $O_2$ at $D$. Let $P$ be a point on the circumcircle of $\triangle BCD$. If $CP$ intersects circle $O_1$ at $E$ and $DP$ intersects circle...	[ "1" ]	[ "Two circles \$O_1\$ and \$O_2\$ intersect at points \$A\$ and \$B\$. The bisector of the outer angle \$\\angle O_1AO_2\$ intersects circle \$O_1\$ at \$C\$ and circle \$O_2\$ at \$D\$. Let \$P\$ be a point on the circumcircle of \$\\triangle BCD\$. If \$CP\$ intersects circle \$O_1\$ at \...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7	0
omr-ws-5e47c4ac9e59bb74dab414ec1f3eb61e47f36229e00fe1378f35c47c1f59d929	Two circles $O_1$ and $O_2$ intersect at points $A$ and $B$. The diameter $AC$ of circle $O_1$ intersects circle $O_2$ at point $E$, and the diameter $AD$ of circle $O_2$ intersects circle $O_1$ at point $F$. The line $CF$ intersects circle $O_2$ at point $H$, and the line $DE$ intersect...	[ "20" ]	[ "Two circles \$O_1\$ and \$O_2\$ intersect at points \$A\$ and \$B\$. The diameter \$AC\$ of circle \$O_1\$ intersects circle \$O_2\$ at point \$E\$, and the diameter \$AD\$ of circle \$O_2\$ intersects circle \$O_1\$ at point \$F\$. The line \$CF\$ intersects circle \$O_2\$ at point \\(...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 1 }	6	0
omr-ws-70bbd2e2351760bcf8e22b10fee7c2878fbc76977e19531633846b364ec26f10	Two circles $O_1$ and $O_2$ intersect at points $A$ and $B$. The tangents to $O_1$ at $A$ and $B$ meet at point $K$. Point $M$ is on $O_1$, and line $KM$ intersects $O_1$ again at point $C$. Line $CA$ intersects $O_2$ again at point $Q$, and line $AM$ intersects $O_2$ again at point ...	[ "20" ]	[ "Two circles \$O_1\$ and \$O_2\$ intersect at points \$A\$ and \$B\$. The tangents to \$O_1\$ at \$A\$ and \$B\$ meet at point \$K\$. Point \$M\$ is on \$O_1\$, and line \$KM\$ intersects \$O_1\$ again at point \$C\$. Line \$CA\$ intersects \$O_2\$ again at point \$Q\$, and line \\(A...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 7, "genselect": 7 }	14	0
omr-ws-b2cb971875687094e1c83f982dc5283256b86dadda1a1699dd55d780448c90f5	Two circles $O_1$ and $O_2$ intersect at points $A$ and $B$. Two lines $CD$ and $EF$ pass through $B$, with $C$ and $E$ on circle $O_1$ and $D$ and $F$ on circle $O_2$. Line $CF$ intersects circle $O_1$ at $P$ and circle $O_2$ at $Q$. $M$ is the midpoint of arc $PB$ and $N$ is ...	[ "90" ]	[ "Two circles \$O_1\$ and \$O_2\$ intersect at points \$A\$ and \$B\$. Two lines \$CD\$ and \$EF\$ pass through \$B\$, with \$C\$ and \$E\$ on circle \$O_1\$ and \$D\$ and \$F\$ on circle \$O_2\$. Line \$CF\$ intersects circle \$O_1\$ at \$P\$ and circle \$O_2\$ at \$Q\$. \$M\$ ...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 2, "genselect": 2 }	4	0
omr-ws-89351a2e5af960aa99444f069f5e2d7f68bca18d514117d46e51ddab38579abd	Two circles $O_1$ and $O_2$ intersect at points $M$ and $N$. Let $AB$ be the line tangent to these circles at $A$ and $B$, respectively, such that $M$ is closer to $AB$ than $N$. Let $CD$ be the line parallel to $AB$ and passing through $M$, with $C$ on $O_1$ and $D$ on $O_2$. Let \(T\...	[ "30" ]	[ "Two circles \$O_1\$ and \$O_2\$ intersect at points \$M\$ and \$N\$. Let \$AB\$ be the line tangent to these circles at \$A\$ and \$B\$, respectively, such that \$M\$ is closer to \$AB\$ than \$N\$. Let \$CD\$ be the line parallel to \$AB\$ and passing through \$M\$, with \$C\$ on \\(O_...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 4 }	4	0
omr-ws-9e262d813756551df28d5c96a66ecb7c52abc9d4b54a1298f18d7d7675213fb3	Two circles $O_1$ and $O_2$ intersect at points $M$ and $N$. The common tangent line nearer to $M$ touches $O_1$ at $A$ and $O_2$ at $B$. Let $C$ and $D$ be the symmetric points of $A$ and $B$ with respect to $M$, respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ ...	[ "35" ]	[ "Two circles \$O_1\$ and \$O_2\$ intersect at points \$M\$ and \$N\$. The common tangent line nearer to \$M\$ touches \$O_1\$ at \$A\$ and \$O_2\$ at \$B\$. Let \$C\$ and \$D\$ be the symmetric points of \$A\$ and \$B\$ with respect to \$M\$, respectively. The circumcircle of triangle \\...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 4, "genselect": 5 }	9	0
omr-ws-43681b4f49306407e21b9f5644f99dbe0fb0c003a40888464313ab0bc5e78063	Two circles $O_1$ and $O_2$ intersect at points $M$ and $N$. The common tangent to the circles nearer to $M$ touches $O_1$ at $A$ and $O_2$ at $B$. Let $C$ and $D$ be the reflections of $A$ and $B$ respectively with respect to $M$. The circumcircle of triangle $DCM$ intersects circles \(O_...	[ "1" ]	[ "Two circles \$O_1\$ and \$O_2\$ intersect at points \$M\$ and \$N\$. The common tangent to the circles nearer to \$M\$ touches \$O_1\$ at \$A\$ and \$O_2\$ at \$B\$. Let \$C\$ and \$D\$ be the reflections of \$A\$ and \$B\$ respectively with respect to \$M\$. The circumcircle of triangl...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-06767f06fba6d6b07caf882ac864c30125a7a699521be09bd7aef6e83cc58480	Two circles $\Gamma$ and $\Sigma$ intersect at two distinct points $A$ and $B$. A line through $B$ intersects $\Gamma$ and $\Sigma$ again at $C$ and $D$, respectively. Suppose that $CA = CD$. If the radius of $\Gamma$ is 10 and the distance between the centers of $\Gamma$ and $\Sigma$ is 12, f...	[ "14" ]	[ "Two circles \$\\Gamma\$ and \$\\Sigma\$ intersect at two distinct points \$A\$ and \$B\$. A line through \$B\$ intersects \$\\Gamma\$ and \$\\Sigma\$ again at \$C\$ and \$D\$, respectively. Suppose that \$CA = CD\$. If the radius of \$\\Gamma\$ is 10 and the distance between the centers of \\...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 4, "genselect": 2 }	6	0
omr-ws-6057bfbb810e5f25d49354f9f5a59aa37812eb1532c5f77b674ad2364402a303	Two circles $\Gamma_1$ and $\Gamma_2$ centered at the points $O_1$ and $O_2$ respectively intersect at the points $A$ and $B$. The line $O_1A$ intersects $\Gamma_2$ for the second time at the point $C$, and line $O_2A$ intersects $\Gamma_1$ for the second time at point $D$. Line $\ell$ paralle...	[ "90" ]	[ "Two circles \$\\Gamma_1\$ and \$\\Gamma_2\$ centered at the points \$O_1\$ and \$O_2\$ respectively intersect at the points \$A\$ and \$B\$. The line \$O_1A\$ intersects \$\\Gamma_2\$ for the second time at the point \$C\$, and line \$O_2A\$ intersects \$\\Gamma_1\$ for the second time at poi...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 2 }	2	0
omr-ws-02bef33b2bd3508ae5b374ab51631ee13c5ba53714629702e738c5f10dec8030	Two circles $\alpha$ and $\beta$ with centers $A$ and $B$ respectively intersect at points $C$ and $D$. The segment $AB$ intersects $\alpha$ and $\beta$ at points $K$ and $L$ respectively. The ray $DK$ intersects the circle $\beta$ for the second time at the point $N$, and the ray $DL$ int...	[ "(6, 2\\sqrt{3})" ]	[ "Two circles \$\\alpha\$ and \$\\beta\$ with centers \$A\$ and \$B\$ respectively intersect at points \$C\$ and \$D\$. The segment \$AB\$ intersects \$\\alpha\$ and \$\\beta\$ at points \$K\$ and \$L\$ respectively. The ray \$DK\$ intersects the circle \$\\beta\$ for the second time at the...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-e124df818707dfd2b2b213113c9aee1d89f3a3240b9dadf302ec8f7ff6f4a9bb	Two circles $\gamma_1$ and $\gamma_2$ intersect at points $A$ and $B$. A line $r$ through $B$ intersects $\gamma_1$ at $C$ and $\gamma_2$ at $D$ such that $B$ is between $C$ and $D$. A line $s$ parallel to $AD$ is tangent to $\gamma_1$ at $E$ and is the line with the minimal distance f...	[ "\\dfrac{2}{7}" ]	[ "Two circles \$\\gamma_1\$ and \$\\gamma_2\$ intersect at points \$A\$ and \$B\$. A line \$r\$ through \$B\$ intersects \$\\gamma_1\$ at \$C\$ and \$\\gamma_2\$ at \$D\$ such that \$B\$ is between \$C\$ and \$D\$. A line \$s\$ parallel to \$AD\$ is tangent to \$\\gamma_1\$ at \$E\$...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-9acda62e834c0c47e2c51e3c85ccf6c529831aedddadefe4618475c54ec166ac	Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. The tangent to circle $\omega_1$ drawn at point $A$ and the tangent to circle $\omega_2$ drawn at point $B$ intersect at point $C$. The line $AC$ intersects $\omega_2$ for the second time at point $T$. Point $X$ is selected...	[ "m" ]	[ "Two circles \$\\omega_1\$ and \$\\omega_2\$ intersect at points \$A\$ and \$B\$. The tangent to circle \$\\omega_1\$ drawn at point \$A\$ and the tangent to circle \$\\omega_2\$ drawn at point \$B\$ intersect at point \$C\$. The line \$AC\$ intersects \$\\omega_2\$ for the second time at poin...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 4 }	4	0
omr-ws-cd6a6ccb64a3f2455277d6ac88ee03064412625ce147af9006099cb76903135b	Two circles $\omega_1$ and $\omega_2$ intersect at points $B$ and $C$. Circle $\omega_3$ is tangent to $BC$ at $A$ and $\omega_1$ at $N$, and intersects $\omega_2$ at $S$ and $T$. $NA$ intersects $ST$ and $\omega_1$ at $M$ and $Q$, respectively. $P$ is the point on $\omega_1$ diame...	[ "1" ]	[ "Two circles \$\\omega_1\$ and \$\\omega_2\$ intersect at points \$B\$ and \$C\$. Circle \$\\omega_3\$ is tangent to \$BC\$ at \$A\$ and \$\\omega_1\$ at \$N\$, and intersects \$\\omega_2\$ at \$S\$ and \$T\$. \$NA\$ intersects \$ST\$ and \$\\omega_1\$ at \$M\$ and \$Q\$, respectiv...	[ "aops_c5_contests_amp_programs" ]	[ "cot", "genselect" ]	{ "cot": 7, "genselect": 5 }	12	0
omr-ws-1839c52d0a207ce3bd9f51f555ca5f7444c150d0e7adfd5cda9667e2cdde1398	Two circles $\omega_1$ and $\omega_2$ with equal radius intersect at two points $E$ and $X$. Arbitrary points $C$ and $D$ lie on $\omega_1$ and $\omega_2$, respectively. Parallel lines to $XC$ and $XD$ from $E$ intersect $\omega_2$ and $\omega_1$ at $A$ and $B$, respectively. Suppose that ...	[ "60^\\circ" ]	[ "Two circles \$\\omega_1\$ and \$\\omega_2\$ with equal radius intersect at two points \$E\$ and \$X\$. Arbitrary points \$C\$ and \$D\$ lie on \$\\omega_1\$ and \$\\omega_2\$, respectively. Parallel lines to \$XC\$ and \$XD\$ from \$E\$ intersect \$\\omega_2\$ and \$\\omega_1\$ at \$A\$...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-b02b03b0f70a4d6c4963bb3c80fdd076d11b9f3f42a9adf995c5a97870eeb5cf	Two circles $\omega_1$ and $\omega_2$ with radii 28 and 112 respectively intersect at points $P$ and $Q$. Points $A$ and $B$ are on $\omega_1$ and $\omega_2$ respectively such that $A, P, B$ are collinear. Tangents to $\omega_1$ at $A$ and to $\omega_2$ at $B$ meet at $T$. Given that \(\angl...	[ "59" ]	[ "Two circles \$\\omega_1\$ and \$\\omega_2\$ with radii 28 and 112 respectively intersect at points \$P\$ and \$Q\$. Points \$A\$ and \$B\$ are on \$\\omega_1\$ and \$\\omega_2\$ respectively such that \$A, P, B\$ are collinear. Tangents to \$\\omega_1\$ at \$A\$ and to \$\\omega_2\$ at \\(B...	[ "aops_c5_contests_amp_programs" ]	[ "additional_problems", "genselect" ]	{ "additional_problems": 1, "genselect": 2 }	3	0
omr-ws-31945426a1f9493390cd8d0284a53684806993dbfc98c2b9806dc3cece2b8dfd	Two circles $\omega_1$ and $\omega_2$ with radii $r_1$ and $r_2$ are externally tangent at point $A$. A line $\ell$ intersects $\omega_1$ at points $P$ and $Q$ and $\omega_2$ at points $R$ and $S$ such that $P, Q, R,$ and $S$ are in that order on the line, and \(\frac{PQ}{RS} = \frac{r_1}{r_...	[ "90" ]	[ "Two circles \$\\omega_1\$ and \$\\omega_2\$ with radii \$r_1\$ and \$r_2\$ are externally tangent at point \$A\$. A line \$\\ell\$ intersects \$\\omega_1\$ at points \$P\$ and \$Q\$ and \$\\omega_2\$ at points \$R\$ and \$S\$ such that \$P, Q, R,\$ and \$S\$ are in that order on the lin...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7	0
omr-ws-91cc22620db198b1ae78de6d37ed27cb683221477f9b3a8a1d8a96b7e1efcf16	Two circles $\omega_{1}$ and $\omega_{2}$ intersect in points $A$ and $B$. Let $PQ$ and $RS$ be segments of common tangents to these circles (points $P$ and $R$ lie on $\omega_{1}$, points $Q$ and $S$ lie on $\omega_{2}$). It appears that $RB \parallel PQ$. Ray $RB$ intersects $\omega_{2}$...	[ "1" ]	[ "Two circles \$\\omega_{1}\$ and \$\\omega_{2}\$ intersect in points \$A\$ and \$B\$. Let \$PQ\$ and \$RS\$ be segments of common tangents to these circles (points \$P\$ and \$R\$ lie on \$\\omega_{1}\$, points \$Q\$ and \$S\$ lie on \$\\omega_{2}\$). It appears that \$RB \\parallel PQ\$. ...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 12, "genselect": 3 }	15	0
omr-ws-068a11ba2e676333dde3d89a0b9683ec08b52f283b19c68600b0d23a26c2b6bd	Two circles and a straight line are given. Construct a square so that two of its opposite vertices lie on the circles, and the other two lie on the straight line.	[ "\\text{Construct the square by reflecting one circle, intersecting with the other, and using perpendiculars from the intersections to the line.}" ]	[ "Two circles and a straight line are given. Construct a square so that two of its opposite vertices lie on the circles, and the other two lie on the straight line." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-21d622c542fe2b785bfeb3b77cb62ad01557d780301d0d7a87b0e9e20c476565	Two circles are concentric. A chord AC of the outer circle touches the inner circle at Q. P is the midpoint of AQ. A line through A intersects the inner circle at R and S. The perpendicular bisectors of PR and CS meet at T on the line AC. What is the ratio AT/TC?	[ "\$\\frac{5}{3}\$" ]	[ "Two circles are concentric. A chord AC of the outer circle touches the inner circle at Q. P is the midpoint of AQ. A line through A intersects the inner circle at R and S. The perpendicular bisectors of PR and CS meet at T on the line AC. What is the ratio AT/TC?" ]	[ "aops_c5_contests_amp_programs" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-f4393e250f219220f9d8d92f57553b5990df9a24cf8701ed99b326a0c1380e27	Two circles are concentric. A chord $ AC $ of the outer circle touches the inner circle at $ Q $. $ P $ is the midpoint of $ AQ $. A line through $ A $ intersects the inner circle at $ R $ and $ S $. The perpendicular bisectors of $ PR $ and $ CS $ meet at $ T $ on the line $ AC $. What is the rat...	[ "\\frac{1}{3}" ]	[ "Two circles are concentric. A chord \$ AC \$ of the outer circle touches the inner circle at \$ Q \$. \$ P \$ is the midpoint of \$ AQ \$. A line through \$ A \$ intersects the inner circle at \$ R \$ and \$ S \$. The perpendicular bisectors of \$ PR \$ and \$ CS \$ meet at \$ T \$ on the line ...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-b3e18f87b4045e051cac388441f4696c15c435ffee00fdc0a732bf7707284a23	Two circles are drawn in a 12-inch by 14-inch rectangle. Each circle has a diameter of 6 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles?	[ "10\\text{ inches}" ]	[ "Two circles are drawn in a 12-inch by 14-inch rectangle. Each circle has a diameter of 6 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles?" ]	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 16 }	16	0
omr-ws-d843c8e9655db060495293df0cc0c93da24ad85f59de27e84c69f4435a0691c0	Two circles are drawn through the points $(a, 5a)$ and $(4a, a)$ and touch the $y$-axis. Let $\theta$ be the angle at which these circles intersect. Calculate $\tan(\theta)$.	[ "\\dfrac{40}{9}" ]	[ "Two circles are drawn through the points \$(a, 5a)\$ and \$(4a, a)\$ and touch the \$y\$-axis. Let \$\\theta\$ be the angle at which these circles intersect. Calculate \$\\tan(\\theta)\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-a1ea93560143f701ce0950781509b05b59f6b3072caf5ec99f00138145ca5cad	Two circles are drawn through the points $(a, 5a)$ and $(4a, a)$ and touch the y-axis. Find the angle at which these circles intersect.	[ "\$\\tan^{-1}\\left(\\frac{40}{9}\\right)\$" ]	[ "Two circles are drawn through the points \$(a, 5a)\$ and \$(4a, a)\$ and touch the y-axis. Find the angle at which these circles intersect." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 2, "tir": 1 }	3	0
omr-ws-2a56deb7c599f21a94d114be1018dcdbbcc63eabdd1f934f783fcf6529a926c8	Two circles are drawn through the points $(a, 5a)$ and $(4a, a)$ and touch the y-axis. Let $\theta$ be the angle at which these circles intersect. Calculate $\tan(\theta)$.	[ "\\dfrac{40}{9}" ]	[ "Two circles are drawn through the points \$(a, 5a)\$ and \$(4a, a)\$ and touch the y-axis. Let \$\\theta\$ be the angle at which these circles intersect. Calculate \$\\tan(\\theta)\$." ]	[ "aops_c7_college_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-7f1b9039e32260e6082f959237842f77f8fa217c76da50b97a12385b990b59bd	Two circles are externally tangent at point $A$. Let $BC$ be a common tangent. A line through $A$ and parallel to $BC$ meets the circles at $D$ and $E$. If $M$ is the midpoint of $DE$, find the value of $\frac{MB^2 + MC^2}{BC^2}$.	[ "1" ]	[ "Two circles are externally tangent at point $A$. Let $BC$ be a common tangent. A line through $A$ and parallel to $BC$ meets the circles at $D$ and $E$. If $M$ is the midpoint of $DE$, find the value of $\\frac{MB^2 + MC^2}{BC^2}$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-7a0e353ecc1d9aa85fdde1e0acc321566e04b72c58bfc9bb4c0b34a9812b3fc4	Two circles are externally tangent at point $ P $. One of their other common tangents touches the circles at points $ A $ and $ B $. A line is drawn through $ P $ parallel to $ AB $, meeting the circles again at points $ C $ and $ D $. If the radii of the circles are 2 and 18, find the distance between th...	[ "10" ]	[ "Two circles are externally tangent at point \$ P \$. One of their other common tangents touches the circles at points \$ A \$ and \$ B \$. A line is drawn through \$ P \$ parallel to \$ AB \$, meeting the circles again at points \$ C \$ and \$ D \$. If the radii of the circles are 2 and 18, find the ...	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-177114ff072be37f8e29c9600cf3025aa6d2744d98c1edd326113c1cc03bdb2f	Two circles are externally tangent at point $ P $. Segment $ \overline{CPD} $ is parallel to the common external tangent $ \overline{AB} $. Let $ M $ and $ N $ be the midpoints of $ \overline{AB} $ and $ \overline{CD} $ respectively. If the length of $ \overline{AB} $ is 12, calculate the length of \( \...	[ "6" ]	[ "Two circles are externally tangent at point \$ P \$. Segment \$ \\overline{CPD} \$ is parallel to the common external tangent \$ \\overline{AB} \$. Let \$ M \$ and \$ N \$ be the midpoints of \$ \\overline{AB} \$ and \$ \\overline{CD} \$ respectively. If the length of \$ \\overline{AB} \$ is 12, ca...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-701e1f4b49ee04847627561d4ce599942a5cefe12c56820dfa07c286632387e7	Two circles are externally tangent at the point $A$. A common tangent of the circles meets one circle at the point $B$ and another at the point $C$ ($B \ne C$). Line segments $BD$ and $CE$ are diameters of the circles. If the radius of the first circle is $r_1$ and the radius of the second circle is $r_2$, find the rat...	[ "" ]	[ "Two circles are externally tangent at the point $A$. A common tangent of the circles meets one circle at the point $B$ and another at the point $C$ ($B \\ne C$). Line segments $BD$ and $CE$ are diameters of the circles. If the radius of the first circle is $r_1$ and the radius of the second circle is $r_2$, find t...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	1
omr-ws-fe59975c0fc263b1eaef33a4d6b4d4ec43698feae42e31ff3b783eb6d00ffa98	Two circles are externally tangent to each other at a point $A$ and internally tangent to a third circle $\Gamma$ at points $B$ and $C$. Let $D$ be the midpoint of the secant of $\Gamma$ which is tangent to the smaller circles at $A$. If the centers of the circles are not collinear, calculate the value of $\frac{BD + C...	[ "2" ]	[ "Two circles are externally tangent to each other at a point $A$ and internally tangent to a third circle $\\Gamma$ at points $B$ and $C$. Let $D$ be the midpoint of the secant of $\\Gamma$ which is tangent to the smaller circles at $A$. If the centers of the circles are not collinear, calculate the value of $\\fra...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 3 }	3	0
omr-ws-89339c765a9d7d0e93618985e4921df1c1eb14f10c791f3a2615f5f96a214058	Two circles are externally tangent with radii of $8$ and $11$. What is the length of the common external tangent segment?	[ "\$4\\sqrt{22}\$" ]	[ "Two circles are externally tangent with radii of $8$ and $11$. What is the length of the common external tangent segment?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 22 }	38	0
omr-ws-0963ab292d73b0b98d4c2463ee60f7c75d54523a23e75bd249915b78802dbdcb	Two circles are externally tangent, and have a common external tangent line, with points of tangency P and Q. The radii of the circles are $ r_1 $ and $ r_2 $. The center of the smaller circle (the one which contains point Q) is S, and a line through S is drawn tangent to the larger circle. The point of tangency of...	[ "\\dfrac{d^2 - r_1^2}{2r_1}" ]	[ "Two circles are externally tangent, and have a common external tangent line, with points of tangency P and Q. The radii of the circles are \$ r_1 \$ and \$ r_2 \$. The center of the smaller circle (the one which contains point Q) is S, and a line through S is drawn tangent to the larger circle. The point of ta...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16	0
omr-ws-40535aeef38d718599b9d47c7ca51db438a1b9a58a0e80705496dea2e03a0919	Two circles are externally tangent. Lines $\overline{PAB}$ and $\overline{PA'B'}$ are common tangents with $A$ and $A'$ on the smaller circle and $B$ and $B'$ on the larger circle. If $PA = AB = 4$, then what is the area of the smaller circle?	[ "\$ 2\\pi \$" ]	[ "Two circles are externally tangent. Lines $\\overline{PAB}$ and $\\overline{PA'B'}$ are common tangents with $A$ and $A'$ on the smaller circle and $B$ and $B'$ on the larger circle. If $PA = AB = 4$, then what is the area of the smaller circle?" ]	[ "aops_c5_contests_amp_programs" ]	[ "cot", "tir" ]	{ "cot": 41, "tir": 10 }	51	0
omr-ws-78cb047c1b4ae42f99d04a1ca5f0325d20bca3831354bf1329bb5092d5860b25	Two circles are given in the plane, $\Omega$ and inside it $\omega$. The center of $\omega$ is $I$. Let $P$ be a point on $\Omega$. The second intersections of the tangents from $P$ to $\omega$ with circle $\Omega$ are $Q$ and $R$. The second intersections of circle $IQR$ with lines $PI$, $PQ$, and $PR$ are $J$, $S$, a...	[ "\\dfrac{1}{3}" ]	[ "Two circles are given in the plane, $\\Omega$ and inside it $\\omega$. The center of $\\omega$ is $I$. Let $P$ be a point on $\\Omega$. The second intersections of the tangents from $P$ to $\\omega$ with circle $\\Omega$ are $Q$ and $R$. The second intersections of circle $IQR$ with lines $PI$, $PQ$, and $PR$ are ...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 1 }	1	0
omr-ws-e322ef6126072002b18338ef228cffdb1b23c13bf2a7387f565b6029faad125a	Two circles are given such that circle A is internally touching circle B at point P. Let XY be the diameter of circle B which is tangent to circle A at point Z. If PY > PX and PY = 2PZ, find the measure of angle PYX. Given that the center of circle B lies outside circle A.	[ "15 degrees" ]	[ "Two circles are given such that circle A is internally touching circle B at point P. Let XY be the diameter of circle B which is tangent to circle A at point Z. If PY > PX and PY = 2PZ, find the measure of angle PYX. Given that the center of circle B lies outside circle A." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-cca24b6b416962ebbafeda47123bec52a002d0c7d858e1c19a11b6715836e858	Two circles are given tangent internally at the point $N$. The chords $BA$ and $BC$ of the outer circle touch the interior circle at the points $K$ and $M$, respectively. Let $Q$ and $P$ be, respectively, the midpoints of the arcs $AB$ and $BC$ that do not contain the point $N$. The circumcircles of the triangles $BQK$...	[ "120" ]	[ "Two circles are given tangent internally at the point $N$. The chords $BA$ and $BC$ of the outer circle touch the interior circle at the points $K$ and $M$, respectively. Let $Q$ and $P$ be, respectively, the midpoints of the arcs $AB$ and $BC$ that do not contain the point $N$. The circumcircles of the triangles ...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 1 }	6	0
omr-ws-ef79617a4777cb599a8bc5d29e9a190f1a560afa665046553e76ac0ab6bb8cde	Two circles are in contact internally at a point $T$. Let the chord $AB$ of the larger circle be tangent to the smaller circle at a point $P$. If the measure of $\angle ATB$ is $120^\circ$, find the measure of $\angle ATP$.	[ "30^\\circ" ]	[ "Two circles are in contact internally at a point $T$. Let the chord $AB$ of the larger circle be tangent to the smaller circle at a point $P$. If the measure of $\\angle ATB$ is $120^\\circ$, find the measure of $\\angle ATP$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 2 }	2	0
omr-ws-279933984745a48ff3ee1142197f0426ecfe1ee041cbacedff481a61cd573067	Two circles are inscribed in a $60^\circ$ angle and are tangent to one another. The radius of the larger circle is $120$. What is the radius of the smaller circle?	[ "40" ]	[ "Two circles are inscribed in a $60^\\circ$ angle and are tangent to one another. The radius of the larger circle is $120$. What is the radius of the smaller circle?" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16	0
omr-ws-19c1c7315abd11102cbea38cbd6637346f62e5d149aea241f5b21ea3a85c3c4a	Two circles are inscribed in an acute angle $\alpha$, with the first circle having radius $R_1$ and the second circle having radius $R_2$. The second circle touches one of the sides forming the angle $\alpha$ at the same point as the first circle. Let $A$ and $B$ be the points where the second circle intersects the oth...	[ "8\\sqrt{3}" ]	[ "Two circles are inscribed in an acute angle $\\alpha$, with the first circle having radius $R_1$ and the second circle having radius $R_2$. The second circle touches one of the sides forming the angle $\\alpha$ at the same point as the first circle. Let $A$ and $B$ be the points where the second circle intersects ...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 7, "genselect": 8 }	15	0
omr-ws-1c983cb550eb824609a7e439c55adbda4ad88c30a3e4873eb1023b24e196c641	Two circles are internally tangent at $N$. The chords $BA$ and $BC$ of the larger circle are tangent to the smaller circle at $K$ and $M$ respectively. $Q$ and $P$ are the midpoints of arcs $AB$ and $BC$ respectively. The circumcircles of triangles $BQK$ and $BPM$ intersect at $L$. If the length of $BP$ is $10$ and the...	[ "24" ]	[ "Two circles are internally tangent at $N$. The chords $BA$ and $BC$ of the larger circle are tangent to the smaller circle at $K$ and $M$ respectively. $Q$ and $P$ are the midpoints of arcs $AB$ and $BC$ respectively. The circumcircles of triangles $BQK$ and $BPM$ intersect at $L$. If the length of $BP$ is $10$ an...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7	0
omr-ws-a164f5f1d363ee1634f71090fca4c6cf6bc3a2e121de6f14363a84e76545b526	Two circles are internally tangent at a point $A$. Let $C$ be a point on the smaller circle other than $A$. The tangent line to the smaller circle at $C$ meets the bigger circle at $D$ and $E$; and the line $AC$ meets the bigger circle at $A$ and $P$. If the radius of the smaller circle is $r$ and the radius of the big...	[ "4" ]	[ "Two circles are internally tangent at a point $A$. Let $C$ be a point on the smaller circle other than $A$. The tangent line to the smaller circle at $C$ meets the bigger circle at $D$ and $E$; and the line $AC$ meets the bigger circle at $A$ and $P$. If the radius of the smaller circle is $r$ and the radius of th...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 4 }	4	0
omr-ws-7a765291f46c174b69f519e370019280e3d9337a4fe1e263386b0f6775f1b49e	Two circles are internally tangent at point $K$. Chord $\overline{AB}$ of the larger circle is tangent to the smaller circle at point $L$. Given $AL = 10$ and $AK:BK = 2:5$, find $BL$.	[ "25" ]	[ "Two circles are internally tangent at point $K$. Chord $\\overline{AB}$ of the larger circle is tangent to the smaller circle at point $L$. Given $AL = 10$ and $AK:BK = 2:5$, find $BL$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 9 }	9	0
omr-ws-7635af75fc447265a3660f9a1986d92370f89b44ea57d0a3d680d29751e99935	Two circles are internally tangent at point $T$. Construct segment $AB$ on the larger circle tangent to the smaller circle at $P$. If the radius of the larger circle is $10$ and the radius of the smaller circle is $4$, find the length of segment $TP$.	[ "8" ]	[ "Two circles are internally tangent at point $T$. Construct segment $AB$ on the larger circle tangent to the smaller circle at $P$. If the radius of the larger circle is $10$ and the radius of the smaller circle is $4$, find the length of segment $TP$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 2, "genselect": 8 }	10	0
omr-ws-7836061f3966bb0e56ef134e0aa002461dc4859fc233e45b5d210094e56dd45d	Two circles are internally tangent at point $ M $. The tangent to the smaller circle at point $ P $ intersects the larger circle at points $ Q $ and $ R $. If the radius of the smaller circle is $ r $ and the radius of the larger circle is $ R $, find the value of $ \frac{R}{r} $ given that \( \angle QMP ...	[ "1 + \\sqrt{2}" ]	[ "Two circles are internally tangent at point \$ M \$. The tangent to the smaller circle at point \$ P \$ intersects the larger circle at points \$ Q \$ and \$ R \$. If the radius of the smaller circle is \$ r \$ and the radius of the larger circle is \$ R \$, find the value of \$ \\frac{R}{r} \$ given...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 3 }	3	0
omr-ws-028ca2fcacda29655f05a2af0c54ba5f2538406572b5a6241a3c170b37cc9946	Two circles are internally tangent at point $ T $. A chord $ AB $ of the larger circle is tangent to the smaller circle at point $ P $. If $ AT = 12 $ and $ BT = 16 $, find the length of $ PT $.	[ "no_valid_answer_found" ]	[ "Two circles are internally tangent at point \$ T \$. A chord \$ AB \$ of the larger circle is tangent to the smaller circle at point \$ P \$. If \$ AT = 12 \$ and \$ BT = 16 \$, find the length of \$ PT \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-0ed980ebc90e8bd17b2cf8740b09424b9340a1a21186f894cfcf0b8ad4f8fe83	Two circles are internally tangent at point $ T $. The smaller circle has a radius of 1, and the larger circle has a radius of 3. What is the maximum area possible for a triangle with one vertex at $ T $, another vertex on the smaller circle, and the third vertex on the larger circle?	[ "\$\\frac{9\\sqrt{3}}{4}\$" ]	[ "Two circles are internally tangent at point \$ T \$. The smaller circle has a radius of 1, and the larger circle has a radius of 3. What is the maximum area possible for a triangle with one vertex at \$ T \$, another vertex on the smaller circle, and the third vertex on the larger circle?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 26 }	41	0
omr-ws-e561f8c6596573be3ab0c5a04d5b363de7ea67a01346efba03cd642712017b96	Two circles are internally tangent with radii 1 and 3. A triangle is formed with one vertex on each circle (not at the tangent point) and the third at the tangent point. Find the maximum area of such a triangle.	[ "\$\\frac{9\\sqrt{3}}{4}\$" ]	[ "Two circles are internally tangent with radii 1 and 3. A triangle is formed with one vertex on each circle (not at the tangent point) and the third at the tangent point. Find the maximum area of such a triangle." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 23 }	38	0
omr-ws-0e796ddd6bb3759b4c843db6c9bac9c2c145ac627ee30705ccaab2d743eba2e5	Two circles are internally tangent. A line passing through the center of the larger circle intersects it at points $A$ and $D$. The same line intersects the smaller circle at points $B$ and $C$. Given that $\|AB\| : \|BC\| : \|CD\| = 3 : 7 : 2$, find the ratio of the radii of the circles.	[ "\$\\frac{2}{3}\$" ]	[ "Two circles are internally tangent. A line passing through the center of the larger circle intersects it at points \$A\$ and \$D\$. The same line intersects the smaller circle at points \$B\$ and \$C\$. Given that \$\|AB\| : \|BC\| : \|CD\| = 3 : 7 : 2\$, find the ratio of the radii of the circles." ]	[ "aops_c4_high_school_math" ]	[ "tir" ]	{ "tir": 4 }	4	0
omr-ws-80a549da07c3dc3deccd3ac5f498419fa29c762eb93966baec3a381770a5b8a1	Two circles are located on the coordinate plane such that the distance between their centers is $\sqrt{2}$ units. If both circles have a radius of $1$ unit, what is the area of the region enclosed by their intersection?	[ "\$\\frac{\\pi}{2} - 1\$" ]	[ "Two circles are located on the coordinate plane such that the distance between their centers is $\\sqrt{2}$ units. If both circles have a radius of $1$ unit, what is the area of the region enclosed by their intersection?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 28 }	44	0
omr-ws-58f70eddf3128d6f1db87bae53149d0ee6e839418a42df92459b335c9666cbe1	Two circles are tangent at point K, with the smaller circle inside the larger one. Two parallel lines $\overline{AB}$ and $\overline{CD}$ are tangent to the smaller circle at points X and Y, respectively, and intersect the larger circle at points A, B, C, and D. If the radius of the smaller circle is $r$ and the radius...	[ "2r" ]	[ "Two circles are tangent at point K, with the smaller circle inside the larger one. Two parallel lines $\\overline{AB}$ and $\\overline{CD}$ are tangent to the smaller circle at points X and Y, respectively, and intersect the larger circle at points A, B, C, and D. If the radius of the smaller circle is $r$ and the...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 6 }	6	0
omr-ws-51d93b4ef0a2f3f04bfe65759f52b448037755a8ec7910da357fef526a7c2fa1	Two circles are tangent externally at a point $B$. A line tangent to one of the circles at a point $A$ intersects the other circle at points $C$ and $D$. If the lengths of $BC$ and $BD$ are $12$ and $16$ respectively, find the length of $AB$.	[ "\\dfrac{48\\sqrt{2}}{5}" ]	[ "Two circles are tangent externally at a point $B$. A line tangent to one of the circles at a point $A$ intersects the other circle at points $C$ and $D$. If the lengths of $BC$ and $BD$ are $12$ and $16$ respectively, find the length of $AB$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-d8394f8920b48611726c6bf208a077d8fc69ca029109c6c6bc8b902b06eb743b	Two circles are tangent externally at a point $ B $. A line tangent to one of the circles at a point $ A $ intersects the other circle at points $ C $ and $ D $. If the length of $ AB $ is 10 units, find the value of $ AC \cdot AD $.	[ "100" ]	[ "Two circles are tangent externally at a point \$ B \$. A line tangent to one of the circles at a point \$ A \$ intersects the other circle at points \$ C \$ and \$ D \$. If the length of \$ AB \$ is 10 units, find the value of \$ AC \\cdot AD \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 4 }	4	0
omr-ws-a2b591c5b475984d13af80611c9d5772c0d461f093d45339475f1c960e1d4e29	Two circles are tangent internally at point $ A $. A secant intersects the circles at points $ M, N, P, Q $ in that order. If the measure of $ \angle MAP $ is $ x $ degrees, find the measure of $ \angle NAQ $ in terms of $ x $.	[ "x" ]	[ "Two circles are tangent internally at point \$ A \$. A secant intersects the circles at points \$ M, N, P, Q \$ in that order. If the measure of \$ \\angle MAP \$ is \$ x \$ degrees, find the measure of \$ \\angle NAQ \$ in terms of \$ x \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 6 }	6	0
omr-ws-c6d6f1a57af4085009345e3a8bfc0c001cb3440368c3912b704279fc8888c880	Two circles are tangent internally at point $ P $. A chord $ AB $ of the larger circle is tangent to the smaller circle at point $ C $. The lines $ PB $ and $ PA $ cut the smaller circle at points $ E $ and $ D $ respectively. Given that $ AB = 15 $, $ PE = 2 $, and $ PD = 3 $, determine the length ...	[ "9" ]	[ "Two circles are tangent internally at point \$ P \$. A chord \$ AB \$ of the larger circle is tangent to the smaller circle at point \$ C \$. The lines \$ PB \$ and \$ PA \$ cut the smaller circle at points \$ E \$ and \$ D \$ respectively. Given that \$ AB = 15 \$, \$ PE = 2 \$, and \\( PD = 3 \...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 17, "tir": 2 }	19	0
omr-ws-8d69779a01f030ffe56d1f69d7702e533dddfe1743a796cdf5e266fe66a227de	Two circles are tangent to each other externally at point $ A $. Points $ X $ and $ Y $ are on the respective circles such that $ \angle XAY = 90^\circ $. Let $ O_1 $ and $ O_2 $ be the centers of the two circles with radii $ r_1 $ and $ r_2 $ respectively. If the distance between $ O_1 $ and \( O_2 \...	[ "\\sqrt{2(r_1^2 + r_2^2)}" ]	[ "Two circles are tangent to each other externally at point \$ A \$. Points \$ X \$ and \$ Y \$ are on the respective circles such that \$ \\angle XAY = 90^\\circ \$. Let \$ O_1 \$ and \$ O_2 \$ be the centers of the two circles with radii \$ r_1 \$ and \$ r_2 \$ respectively. If the distance between...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 4 }	4	0
omr-ws-4237d42e534fc5c706a1da95bb0cace84ff1ba653351a15424ac4acc1cf3f14e	Two circles are tangent to each other externally, and both are internally tangent to the third. The centers of all three circles lie on the same straight line $k$. Another fourth circle is drawn, touching all three given circles. Its center is at a distance $a$ from the straight line $k$. Find the radius of this circle...	[ "\$ a \$" ]	[ "Two circles are tangent to each other externally, and both are internally tangent to the third. The centers of all three circles lie on the same straight line $k$. Another fourth circle is drawn, touching all three given circles. Its center is at a distance $a$ from the straight line $k$. Find the radius of this c...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 1 }	1	0
omr-ws-c4bf766951ff222a4dfa26bd93a7eb0d83b5a1deb67fcd1e6af19e034059287c	Two circles are tangent to each other in the xy-plane. One of the circles has center (1,1) and radius 1. The other circle has center (0,-1). What are the possible radii of the other circle?	[ "\$ \\sqrt{5} - 1 \$ or \$ \\sqrt{5} + 1 \$" ]	[ "Two circles are tangent to each other in the xy-plane. One of the circles has center (1,1) and radius 1. The other circle has center (0,-1). What are the possible radii of the other circle?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 2, "tir": 30 }	32	0
omr-ws-9c81094e978de9ab16964ca2b42a577d95c31fece74f98b80b4fb00aaeb7f796	Two circles are tangent to each other internally at a point T. Let the chord AB of the larger circle be tangent to the smaller circle at a point P. If the measure of ∠ATB is 120°, find the measure of ∠ATP.	[ "30^\\circ" ]	[ "Two circles are tangent to each other internally at a point T. Let the chord AB of the larger circle be tangent to the smaller circle at a point P. If the measure of ∠ATB is 120°, find the measure of ∠ATP." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-4624bd28dd13633ce06ebb2167c4aeac6e43c87f2a01e6d00e13c2abb13980ef	Two circles are tangent to each other internally at a point $ T $. Let the chord $ AB $ of the larger circle be tangent to the smaller circle at a point $ P $. If $ \angle ATB = 120^\circ $, calculate the measure of $ \angle ATP $.	[ "30^\\circ" ]	[ "Two circles are tangent to each other internally at a point \$ T \$. Let the chord \$ AB \$ of the larger circle be tangent to the smaller circle at a point \$ P \$. If \$ \\angle ATB = 120^\\circ \$, calculate the measure of \$ \\angle ATP \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 5 }	5	0
omr-ws-d2c7149a8b92ab71c59f6f6ea8fe98fd83a3e2550f80fc80702f741f92d53f41	Two circles are tangent to each other internally at a point $ T $. Let the chord $ AB $ of the larger circle be tangent to the smaller circle at a point $ P $. If the measure of $ \angle ATB $ is $ 120^\circ $, find the measure of $ \angle ATP $.	[ "60^\\circ" ]	[ "Two circles are tangent to each other internally at a point \$ T \$. Let the chord \$ AB \$ of the larger circle be tangent to the smaller circle at a point \$ P \$. If the measure of \$ \\angle ATB \$ is \$ 120^\\circ \$, find the measure of \$ \\angle ATP \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 4, "genselect": 4 }	8	0
omr-ws-7490fab883119d86f03ee93aff5c3ce78f2e250facc1a91629bec9e6f024afb2	Two circles are tangent to each other. Let their external common tangents touch them at points $A$, $B$, $C$, and $D$. Find the area of quadrilateral $ABCD$ in terms of the radii $R_1$ and $R_2$.	[ "\$(R_1 + R_2)(2\\sqrt{R_1 \\cdot R_2})\$" ]	[ "Two circles are tangent to each other. Let their external common tangents touch them at points \$A\$, \$B\$, \$C\$, and \$D\$. Find the area of quadrilateral \$ABCD\$ in terms of the radii \$R_1\$ and \$R_2\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 4, "genselect": 2, "tir": 2 }	8	0
omr-ws-89f5e9dff757e3e9e616af114258f09547ee21dac6bb88bafa932b90e2ad6eb4	Two circles centered at $ O_1 $ and $ O_2 $ are tangent to each other externally at point $ C $. These circles are internally tangent to a third circle at points $ A $ and $ B $, respectively. Given $ AB = 2 $, $\angle AO_1C = 140^\circ$, and $\angle CO_2B = 130^\circ$, what is the radius of the third c...	[ "\$\\sqrt{2}\$" ]	[ "Two circles centered at \$ O_1 \$ and \$ O_2 \$ are tangent to each other externally at point \$ C \$. These circles are internally tangent to a third circle at points \$ A \$ and \$ B \$, respectively. Given \$ AB = 2 \$, \$\\angle AO_1C = 140^\\circ\$, and \$\\angle CO_2B = 130^\\circ\$, what is ...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 8, "genselect": 8, "tir": 4 }	20	0
omr-ws-1b25e24204b2f9ae96a9e3c21487421b5d75f6755477856d61f9b9449e9b9bf6	Two circles centered at $O_1$ and $O_2$ intersect at points $M$ and $N$. The line $O_1M$ intersects the circle centered at $O_1$ at $A_1$ and the circle centered at $O_2$ at $A_2$. The line $O_2M$ intersects the circle centered at $O_1$ at $B_1$ and the circle centered at $O_2$ at $B_2$. Let...	[ "2" ]	[ "Two circles centered at \$O_1\$ and \$O_2\$ intersect at points \$M\$ and \$N\$. The line \$O_1M\$ intersects the circle centered at \$O_1\$ at \$A_1\$ and the circle centered at \$O_2\$ at \$A_2\$. The line \$O_2M\$ intersects the circle centered at \$O_1\$ at \$B_1\$ and the circle center...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 4 }	9	0
omr-ws-745c42e9ea23f5569105b9fed4b0a55ee6ed90f757e43dd54e4addb77e12dc21	Two circles each with radius 2 are inscribed into a larger circle of radius 4 so that each is mutually tangent to the other. Find the radius of a circle that is tangent to the three other circles.	[ "\$\\frac{4}{3}\$" ]	[ "Two circles each with radius 2 are inscribed into a larger circle of radius 4 so that each is mutually tangent to the other. Find the radius of a circle that is tangent to the three other circles." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 19 }	35	0
omr-ws-0e2e0692a2561409dd9d29f8bace9e02d47bb6dcd64dbaa93260f3089a710d68	Two circles have a common chord $AB$. Through point $B$, a straight line intersects the circles at points $C$ and $D$, with $B$ between $C$ and $D$. The tangents to the circles at points $C$ and $D$ intersect at a point $E$. Compare $AD \cdot AC$ with $AB \cdot AE$.	[ "\$ AD \\cdot AC = AB \\cdot AE \$" ]	[ "Two circles have a common chord $AB$. Through point $B$, a straight line intersects the circles at points $C$ and $D$, with $B$ between $C$ and $D$. The tangents to the circles at points $C$ and $D$ intersect at a point $E$. Compare $AD \\cdot AC$ with $AB \\cdot AE$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 14, "tir": 15 }	29	0
omr-ws-f218ecf6436b4a00355b2bd330eb3d3830c53ed5ae47f79896b9a4634d900a60	Two circles have a common external tangent of length 11 and a common internal tangent of length 7. Find the product of the radii of the two circles.	[ "18" ]	[ "Two circles have a common external tangent of length 11 and a common internal tangent of length 7. Find the product of the radii of the two circles." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 16 }	16	0
omr-ws-ad5788d327c98976ee8b07d3001a2fd09a660ebbaef3ab5cf34573fe7d8e63e6	Two circles have a common external tangent of length 8 and a common internal tangent of length 6. Find the product of the circles' areas.	[ "\$ 49\\pi^2 \$" ]	[ "Two circles have a common external tangent of length 8 and a common internal tangent of length 6. Find the product of the circles' areas." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 11 }	27	0
omr-ws-2fc8a6129784d62ac8ed7c1a4a513ff1c8eeff311d9bed0372621a6a83f23aa3	Two circles have a radius of 10 cm. They overlap such that each circle contains exactly 25% of the other's circumference. Find the area of the overlapped region.	[ "\$ 50\\pi - 100 \$" ]	[ "Two circles have a radius of 10 cm. They overlap such that each circle contains exactly 25% of the other's circumference. Find the area of the overlapped region." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 28 }	44	0
omr-ws-da7780811b43e0b1dee503e5faebfbe6395de06b257e79072f6f7e4697d45861	Two circles have an external tangent with a length of 36 cm. The shortest distance between the circles is 14 cm. Find the radius, in cm, of the larger circle if its radius is four times that of the smaller circle.	[ "20" ]	[ "Two circles have an external tangent with a length of 36 cm. The shortest distance between the circles is 14 cm. Find the radius, in cm, of the larger circle if its radius is four times that of the smaller circle." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 31, "tir": 9 }	40	0
omr-ws-bbbd7ac1d38fff788de43c416c4a870c3a08e2c0384eca8a3a7d44fc20f8fedc	Two circles have centers $P$ and $Q$. A common external tangent touches the first circle at $A$ and the second circle at $B$. If $AP = 3$ and $BQ = 5$, calculate the length of $AB$.	[ "2\\sqrt{15}" ]	[ "Two circles have centers $P$ and $Q$. A common external tangent touches the first circle at $A$ and the second circle at $B$. If $AP = 3$ and $BQ = 5$, calculate the length of $AB$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 8 }	13	0
omr-ws-79b91824362323a4a71fa7fa64196e317532499f54494350d83369968bd5886d	Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degrees, how far apart are the centers of the circles?	[ "160" ]	[ "Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degrees, how far apart are the centers of the circles?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 14, "genselect": 3, "tir": 7 }	24	0
omr-ws-66e5c1b5587594e16a42e5e0b9c34a72dd21783a02f1f841f74e2668e8e4361f	Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at 60 degrees, how far apart are the centers of the circles?	[ "160" ]	[ "Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at 60 degrees, how far apart are the centers of the circles?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 14, "genselect": 4, "tir": 3 }	21	0
omr-ws-8059611c08af15e770b6403d14f7b99c8584a40caa1d1244a86689baeb85ab20	Two circles have radii $a$ and $b$. The distance between their centers is $d$. They intersect at points $P$ and $Q$. Find the distance between $P$ and $Q$.	[ " \\dfrac{ \\sqrt{ (a + b + d)(-a + b + d)(a - b + d)(a + b - d) } }{d} " ]	[ "Two circles have radii $a$ and $b$. The distance between their centers is $d$. They intersect at points $P$ and $Q$. Find the distance between $P$ and $Q$." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems", "genselect" ]	{ "additional_problems": 1, "genselect": 4 }	5	0
omr-ws-076e424d4e2d22b090519346cbced8e9940e73dd8a99d0489e4e7db6b5b56b2d	Two circles have radii 13 and $\sqrt{65}$, respectively. The line passing through their points of intersection is given by the equation $x + y = c$. What is the value of $c$?	[ "3" ]	[ "Two circles have radii 13 and $\\sqrt{65}$, respectively. The line passing through their points of intersection is given by the equation $x + y = c$. What is the value of $c$?" ]	[ "aops_c5_contests_amp_programs" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-3a7427273d7cbf4446e6cabdf20fdf209efa5cc43e9c948ff65991edf639c104	Two circles have radii $ x $ and $ y $, and their centers are $ q $ units apart. A rope is looped around the circles such that it contains the internal tangents and parts of the circumference (that is, it crosses itself once). What is the length of the rope?	[ "2\\sqrt{q^2 - (x + y)^2} + 2(x + y) \\arccos\\left(\\frac{x + y}{q}\\right)" ]	[ "Two circles have radii \$ x \$ and \$ y \$, and their centers are \$ q \$ units apart. A rope is looped around the circles such that it contains the internal tangents and parts of the circumference (that is, it crosses itself once). What is the length of the rope?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 7, "genselect": 7 }	14	0
omr-ws-b8f9b786345cd16db9045ae4dc285b86972a3fbfd81eaf2224099dfe75abea3d	Two circles have radii $ x $ and $ y $, and their centers are $ q $ units apart. A rope is looped around the outside of the circles (that is, it doesn't cross itself). What is the length of the rope?	[ "2\\sqrt{q^2 - (x - y)^2} + \\pi(x + y)" ]	[ "Two circles have radii \$ x \$ and \$ y \$, and their centers are \$ q \$ units apart. A rope is looped around the outside of the circles (that is, it doesn't cross itself). What is the length of the rope?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 4, "genselect": 2 }	6	0
omr-ws-1b9b6fca3674ce557a96523769f4bb9f17044b582c8c2bf4b7ae53ab7ccc23b4	Two circles have radii $R_1$ and $R_2$ and intersect at points $A$ and $B$. Let $CD$ be their common tangent. Let $R$ be the radius of the circumcircles of triangles $CAD$ and $CBD$. Calculate the value of $R^2$ in terms of $R_1$ and $R_2$.	[ "R_1 R_2" ]	[ "Two circles have radii \$R_1\$ and \$R_2\$ and intersect at points \$A\$ and \$B\$. Let \$CD\$ be their common tangent. Let \$R\$ be the radius of the circumcircles of triangles \$CAD\$ and \$CBD\$.\n\nCalculate the value of \$R^2\$ in terms of \$R_1\$ and \$R_2\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 4 }	4	0
omr-ws-02017e8af7d353b74f25d9ce87d9b481a1598fae5118497db19f596c38630b07	Two circles have radii $a$ and $b$. The distance between their centers is $d$. They intersect at points $P$ and $Q$. What is the distance from $P$ to $Q$? Express your answer using $a$, $b$, and/or $d$.	[ "\$\\frac{\\sqrt{d^2 - (a-b)^2} \\sqrt{(a+b)^2 - d^2}}{d}\$" ]	[ "Two circles have radii \$a\$ and \$b\$. The distance between their centers is \$d\$. They intersect at points \$P\$ and \$Q\$. What is the distance from \$P\$ to \$Q\$? Express your answer using \$a\$, \$b\$, and/or \$d\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 5, "genselect": 5, "tir": 5 }	15	0
omr-ws-1ee0512280c49c91a48561a947dae5c9a6fe1d7bb2789ec23277354d83239e1d	Two circles have radii of $a$ and $b$, respectively, and are externally tangent. Their common internal tangent intersects their common external tangents at points $E$ and $F$. Find the length of $EF$ in terms of $a$ and $b$.	[ "\$2\\sqrt{ab}\$" ]	[ "Two circles have radii of $a$ and $b$, respectively, and are externally tangent. Their common internal tangent intersects their common external tangents at points $E$ and $F$. Find the length of $EF$ in terms of $a$ and $b$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 9, "genselect": 7, "tir": 7 }	23	0
omr-ws-fde3a25e568d7608cdc977885f2d21bb244c270f54e87d6b2beb2a254a09b7e9	Two circles have radii of 6 and 10. The distance between their centers is 24. What is the difference between the lengths of their common internal and external tangents?	[ "4\\sqrt{35} - 8\\sqrt{5}" ]	[ "Two circles have radii of 6 and 10. The distance between their centers is 24. What is the difference between the lengths of their common internal and external tangents?" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16	0
omr-ws-5d45a147bce3393441f036a7e29c65885a372e9e938b5ad08bb7f340e0870b26	Two circles have radii of 7 and 15, and their centers are 24 units apart. What is the length of a common internal tangent?	[ "\$2\\sqrt{23}\$" ]	[ "Two circles have radii of 7 and 15, and their centers are 24 units apart. What is the length of a common internal tangent?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 14, "genselect": 7, "tir": 24 }	45	0
omr-ws-ae74876cfacd5fb101de2911c81cfe468ab05ca61df731cd839adb4c80909576	Two circles have radius $ r $. Their centers are $ r $ apart. What is the area of the intersection of the two circles?	[ "\$ r \\left( \\frac{\\pi}{3} - \\frac{\\sqrt{3}}{4} \\right) \$" ]	[ "Two circles have radius \$ r \$. Their centers are \$ r \$ apart. What is the area of the intersection of the two circles?" ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-4c428b8cd7024f23d3c3e4767f212d624087474d39e76372a1508e9910037bb1	Two circles in a plane intersect at point $A$. Starting simultaneously from $A$, two points $M$ and $N$ move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Let $P$ be a fixed point in the plane such that $M$ and $N$ are alwa...	[ "\\dfrac{\|r_1^2 - r_2^2\|}{d}" ]	[ "Two circles in a plane intersect at point $A$. Starting simultaneously from $A$, two points $M$ and $N$ move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Let $P$ be a fixed point in the plane such that $M$ and $N$ are...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 3 }	3	0
omr-ws-fddc6fa284032fa9263ab54ead1a856f63ea460f0d48fc5aff237394a577c06f	Two circles in a plane intersect at points $A$ and $B$. Starting simultaneously from $A$, two points $P$ and $Q$ move with constant speeds around different circles, each point traveling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. If the distance between $P$ ...	[ "AB" ]	[ "Two circles in a plane intersect at points $A$ and $B$. Starting simultaneously from $A$, two points $P$ and $Q$ move with constant speeds around different circles, each point traveling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. If the distance between...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1	0
omr-ws-bdb3f553c4725b31f6353b972564a0b8738295567f18b617fc92b23aedf2729d	Two circles in a plane intersect at points $A$ and $B$. Starting simultaneously from $A$, two points $P$ and $Q$ move with constant speeds around different circles, each point traveling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Let $S$ be a fixed point in ...	[ "\\dfrac{\|r_1^2 - r_2^2\|}{d}" ]	[ "Two circles in a plane intersect at points $A$ and $B$. Starting simultaneously from $A$, two points $P$ and $Q$ move with constant speeds around different circles, each point traveling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Let $S$ be a fixed poin...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 3 }	3	0

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