id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-655e22117659bd225b640ce91b81b297d4143b00e30416f1256c505f6ce4c27f | Two circles $\omega_1$ and $\omega_2$ of different radii intersect at points $A$ and $B$. The straight line $CD$ touches the circles $\omega_1$ and $\omega_2$ at points $C$ and $D$, respectively, and the straight line $EF$ touches the circles $\omega_1$ and $\omega_2$ at points $E$ and $F$, respectively. Let $H_1, H_2,... | [
"0"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ of different radii intersect at points $A$ and $B$. The straight line $CD$ touches the circles $\\omega_1$ and $\\omega_2$ at points $C$ and $D$, respectively, and the straight line $EF$ touches the circles $\\omega_1$ and $\\omega_2$ at points $E$ and $F$, respectively. Let... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-cbb6b17ae53befd8c71c7661e872c6d1172fc439f2ed16cd3fcdc56667c56f66 | Two circles $\omega_1$ and $\omega_2$ touch externally at point $L$ and are inscribed into angle $BAC$. Circle $\omega_1$ touches ray $AB$ at point $E$, and circle $\omega_2$ touches ray $AC$ at point $M$. Line $EL$ meets $\omega_2$ for the second time at point $Q$. If the measure of angle $BAC$ is $60^\circ$ and the r... | [
"30^\\circ"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ touch externally at point $L$ and are inscribed into angle $BAC$. Circle $\\omega_1$ touches ray $AB$ at point $E$, and circle $\\omega_2$ touches ray $AC$ at point $M$. Line $EL$ meets $\\omega_2$ for the second time at point $Q$. If the measure of angle $BAC$ is $60^\\circ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-ae0344dc64a3d12ce5208efb141c6bde9502269936afc568b50c01298e799c01 | Two circles $\omega_1$ and $\omega_2$ touch externally at point $P$. Let $A$ be a point on $\omega_2$ not lying on the line through the centers of the two circles. Let $AB$ and $AC$ be the tangents to $\omega_1$. Lines $BP$ and $CP$ meet $\omega_2$ for the second time at points $E$ and $F$. Let $T$ be the intersection ... | [
"\\dfrac{5}{9}"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ touch externally at point $P$. Let $A$ be a point on $\\omega_2$ not lying on the line through the centers of the two circles. Let $AB$ and $AC$ be the tangents to $\\omega_1$. Lines $BP$ and $CP$ meet $\\omega_2$ for the second time at points $E$ and $F$. Let $T$ be the int... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-561b2710c772164c9bd44be4ffd649214d6b42a3e2524bed9bd140a5b60247e1 | Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ are externally tangent to each other at $C$. Another circle $\omega$ with center $O$ is externally tangent to $\omega_1$ and $\omega_2$. Let $l$ be the common tangent of $\omega_1$ and $\omega_2$ at point $C$ and $AB$ be a diameter of $\omega$ perpendic... | [
"6"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ with centers $O_1$ and $O_2$ are externally tangent to each other at $C$. Another circle $\\omega$ with center $O$ is externally tangent to $\\omega_1$ and $\\omega_2$. Let $l$ be the common tangent of $\\omega_1$ and $\\omega_2$ at point $C$ and $AB$ be a diameter of $\\ome... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-97c3e7ae0dd35c3e8f3de96c0552bc251d435489bd200eb92418e347db345126 | Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ intersect at points $A$ and $B$. An arbitrary line $\ell$ passing through $A$ intersects $\omega_1$ at $Y$ and $\omega_2$ at $Z$. The tangents to $\omega_1$ at $Y$ and to $\omega_2$ at $Z$ intersect at $X$. Let $\omega$ be the circumcircle of $\triangle... | [
"2R"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ with centers $O_1$ and $O_2$ intersect at points $A$ and $B$. An arbitrary line $\\ell$ passing through $A$ intersects $\\omega_1$ at $Y$ and $\\omega_2$ at $Z$. The tangents to $\\omega_1$ at $Y$ and to $\\omega_2$ at $Z$ intersect at $X$. Let $\\omega$ be the circumcircle ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-09c7550d56cfbc2479e6ccd6ad44ec9e50767f191ce2875f21233e25ac5a3df3 | Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ intersect at points $A$ and $B$. Points $C$ and $D$ lie on $\omega_1$ and $\omega_2$, respectively, on opposite sides of the line $AB$ and are equidistant from this line. If the distance from $C$ to $O_1$ is 10 units and the distance from $D$ to $O_2$ i... | [
"10"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ with centers $O_1$ and $O_2$ intersect at points $A$ and $B$. Points $C$ and $D$ lie on $\\omega_1$ and $\\omega_2$, respectively, on opposite sides of the line $AB$ and are equidistant from this line. If the distance from $C$ to $O_1$ is 10 units and the distance from $D$ t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-9f64ffae44ac9aec8ba79a06e21cb2352e85b6bbb90c198d514331f4f58166d1 | Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ intersect at points $A$ and $B$. The line $O_1A$ intersects $\omega_2$ at points $A$ and $D$, and the line $O_2A$ intersects $\omega_1$ at points $A$ and $C$. A line $k$ parallel to $CD$ passing through point $B$ intersects $\omega_1$ at point $E$ and $... | [
"12"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ with centers $O_1$ and $O_2$ intersect at points $A$ and $B$. The line $O_1A$ intersects $\\omega_2$ at points $A$ and $D$, and the line $O_2A$ intersects $\\omega_1$ at points $A$ and $C$. A line $k$ parallel to $CD$ passing through point $B$ intersects $\\omega_1$ at point... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-d612165f8b67bf6cfab32551fbc4a664597d7e2ec0791c23ee23541809a6dc55 | Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ respectively intersect each other at points $A$ and $B$, and point $O_1$ lies on $\omega_2$. Let $P$ be an arbitrary point lying on $\omega_1$. Lines $BP$, $AP$, and $O_1O_2$ cut $\omega_2$ for the second time at points $X$, $Y$, and $C$, respectively. ... | [
"10"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ with centers $O_1$ and $O_2$ respectively intersect each other at points $A$ and $B$, and point $O_1$ lies on $\\omega_2$. Let $P$ be an arbitrary point lying on $\\omega_1$. Lines $BP$, $AP$, and $O_1O_2$ cut $\\omega_2$ for the second time at points $X$, $Y$, and $C$, resp... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-d7bfbb2ea033bc3ea13c177473788fc8cbbe4494d764971fe35aa019bc437032 | Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ respectively intersect each other at points $A$ and $B$. The line $O_1B$ intersects $\omega_2$ again at $X$, and the line $O_2B$ intersects $\omega_1$ again at $Y$, such that $A$ and $B$ are on the same side of line $XY$. Let $P$ and $Q$ be the feet of ... | [
"15"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ with centers $O_1$ and $O_2$ respectively intersect each other at points $A$ and $B$. The line $O_1B$ intersects $\\omega_2$ again at $X$, and the line $O_2B$ intersects $\\omega_1$ again at $Y$, such that $A$ and $B$ are on the same side of line $XY$. Let $P$ and $Q$ be the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-3913db2779e996fc30d1fd85a19aeab16c500637b62a783e4ba645208490a523 | Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$, respectively, intersect at two points $A$ and $B$, and the angle $\angle O_1AO_2$ is obtuse. The line $O_2B$ intersects $\omega_1$ for the second time at $D$, and the straight line $O_1B$ intersects $\omega_2$ at the point $C$. If the radius of $\omega... | [
"5"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ with centers $O_1$ and $O_2$, respectively, intersect at two points $A$ and $B$, and the angle $\\angle O_1AO_2$ is obtuse. The line $O_2B$ intersects $\\omega_1$ for the second time at $D$, and the straight line $O_1B$ intersects $\\omega_2$ at the point $C$. If the radius ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4db42b62f10b0b374d6cf6f903da5c22690e1f65a52b6ffcbbc4fee6cf6da592 | Two circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. Point $X$ lies on the circle $\omega_2$. Point $Y$ is chosen on the circle $\omega_1$ such that $\angle XBY = 90^\circ$. The line $O_1X$ intersects the circle $\omega_2$ for the second time at the point $X'$. I... | [
"\\angle AO_2B"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. Point $X$ lies on the circle $\\omega_2$. Point $Y$ is chosen on the circle $\\omega_1$ such that $\\angle XBY = 90^\\circ$. The line $O_1X$ intersects the circle $\\omega_2$ for the second time at the p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-a11f38c1dc48cd849890301383617e938b7cadf906acdc7781a70ac89643fce2 | Two circles $\omega_1$ and $\omega_2$ with equal radii intersect each other exactly twice at points $E$ and $F$. Points $A$ and $D$ are selected on $\omega_1$ and $\omega_2$, respectively, such that line segment $\overline{AD}$ intersects both circles again at points $C$ and $B$. Given that $AB = 2$, $BC = 4$, $CD = 6$... | [
"25"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ with equal radii intersect each other exactly twice at points $E$ and $F$. Points $A$ and $D$ are selected on $\\omega_1$ and $\\omega_2$, respectively, such that line segment $\\overline{AD}$ intersects both circles again at points $C$ and $B$. Given that $AB = 2$, $BC = 4$... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-83f1cc9682a8ccd973169969403d35e587f16fd7b55dbd8932e39c93a1a4405a | 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 $A$, $P$, and $B$ are collinear. The tangents to $\omega_1$ at $A$ and to $\omega_2$ at $B$ meet at point $T$. Given that $\angle AQT =... | [
"107"
] | [
"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 $A$, $P$, and $B$ are collinear. The tangents to $\\omega_1$ at $A$ and to $\\omega_2$ at $B$ meet at point $T$. Given that $\... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef370eb44a431bb67a74af7c07c8d58642902a14ca2979cd1c1fe49c415caa84 | 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. The tangents to $\omega_1$ at $A$ and to $\omega_2$ at $B$ meet at point $T$. Given that $\angle AQT = \angle ... | [
"\\( 100\\sqrt{7} \\)"
] | [
"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. The tangents to $\\omega_1$ at $A$ and to $\\omega_2$ at $B$ meet at point $T$. Given that $\\angle A... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f4b9f114238db6d9ad92c0bd6531e078a4c66fe5645fdfd878c32ce370f1751 | 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. The tangents to $\omega_1$ at $A$ and to $\omega_2$ at $B$ meet at point $T$. Given that $\angle AQT = \angle ... | [
"107"
] | [
"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. The tangents to $\\omega_1$ at $A$ and to $\\omega_2$ at $B$ meet at point $T$. Given that $\\angle A... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-d639d7cfb2ed3c155a5c9ff6884a88d9eae36cceacba410ea33eef7071f8a1d2 | Two circles $\omega_1$ and $\omega_2$ with radii $r_1$ and $r_2$ ($r_2 > r_1$) are externally tangent. The line $t_1$ is tangent to the circles $\omega_1$ and $\omega_2$ at points $A$ and $D$ respectively. The parallel line $t_2$ to the line $t_1$ is tangent to the circle $\omega_1$ and intersects the circle $\omega_2$... | [
"15"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ with radii $r_1$ and $r_2$ ($r_2 > r_1$) are externally tangent. The line $t_1$ is tangent to the circles $\\omega_1$ and $\\omega_2$ at points $A$ and $D$ respectively. The parallel line $t_2$ to the line $t_1$ is tangent to the circle $\\omega_1$ and intersects the circle ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-831732a897ee3f9496486952fc515a7503225fa7ce8687e30ba76b9a142fcd3c | Two circles $\omega_1$ and $\omega_2$ with radii $r_1$ and $r_2$, respectively, are externally tangent at point $K$, with $r_1 > r_2$. Distinct points $A$ and $B$ lie on $\omega_1$ and $\omega_2$ respectively such that $AB$ is tangent to both circles. Ray $BK$ is extended past $K$ to intersect $\omega_1$ again at $P$. ... | [
"\\dfrac{1}{6}"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ with radii $r_1$ and $r_2$, respectively, are externally tangent at point $K$, with $r_1 > r_2$. Distinct points $A$ and $B$ lie on $\\omega_1$ and $\\omega_2$ respectively such that $AB$ is tangent to both circles. Ray $BK$ is extended past $K$ to intersect $\\omega_1$ agai... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-ab0c1da6902b28cbbbf2e52352bbb9d112083406a999d745b788f5f444ee754e | Two circles $\omega_1$ and $\omega_2$, both of radius $100$, are centered at $(0,31)$ and $(0,-31)$ respectively. Circle $\omega_1$ intersects the negative $y$-axis at point $P$. A variable line whose equation is $x=k$ intersects $\omega_1$ and $\omega_2$ in the first quadrant at points $A$ and $B$ respectively. Let th... | [
"\\dfrac{69\\pi}{2}"
] | [
"Two circles $\\omega_1$ and $\\omega_2$, both of radius $100$, are centered at $(0,31)$ and $(0,-31)$ respectively. Circle $\\omega_1$ intersects the negative $y$-axis at point $P$. A variable line whose equation is $x=k$ intersects $\\omega_1$ and $\\omega_2$ in the first quadrant at points $A$ and $B$ respective... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-914c4746df9efc10776a0a63fd3fa12425af62c503ecafca97dd4925f63397fb | Two circles $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ intersects $\omega_1$ again at $C$ and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ intersects the circle $\... | [
"2r"
] | [
"Two circles $\\omega_1$ and $\\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ intersects $\\omega_1$ again at $C$ and $\\omega_2$ again at $D$. The tangents to $\\omega_1$ and $\\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ intersects th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-c9bc64966821a6fb476511080c286071ae64a43a8cb4828baf558eee1f44503b | Two circles $\omega_1$ and $\omega_2$, each of radius $112$, are drawn such that their centers lie on each other's circumferences. Two smaller circles with equal radii are drawn such that they are both internally tangent to $\omega_1$ and $\omega_2$, and are externally tangent to each other. What is the radius of one o... | [
"42"
] | [
"Two circles $\\omega_1$ and $\\omega_2$, each of radius $112$, are drawn such that their centers lie on each other's circumferences. Two smaller circles with equal radii are drawn such that they are both internally tangent to $\\omega_1$ and $\\omega_2$, and are externally tangent to each other. What is the radius... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-9d58184e50c9ef7272513cabd4203c813344fdeb036ad013e85f0835cc6e5e98 | Two circles $\omega_1$ and $\omega_2$, of equal radius, intersect at different points $X_1$ and $X_2$. Consider a circle $\omega$ that is externally tangent to $\omega_1$ at $T_1$ and internally tangent to $\omega_2$ at point $T_2$. Let $P$ be the intersection point of lines $X_1T_1$ and $X_2T_2$. If the radius of $\om... | [
"\\dfrac{4\\sqrt{10}}{5}"
] | [
"Two circles $\\omega_1$ and $\\omega_2$, of equal radius, intersect at different points $X_1$ and $X_2$. Consider a circle $\\omega$ that is externally tangent to $\\omega_1$ at $T_1$ and internally tangent to $\\omega_2$ at point $T_2$. Let $P$ be the intersection point of lines $X_1T_1$ and $X_2T_2$. If the radi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-7b815355832b441bf2211ef64ac0c9286f418918e20c9f61141ffc0600d46ba6 | Two circles $\omega_1$ and $\omega_2$, which do not have common points, are inscribed in the angle $\angle BAC$, with $B \in \omega_1$ and $C \in \omega_2$, and the radius of $\omega_1$ is less than the radius of $\omega_2$. The line $BC$ intersects the circles $\omega_1$ and $\omega_2$ for the second time at points $K... | [
"0"
] | [
"Two circles $\\omega_1$ and $\\omega_2$, which do not have common points, are inscribed in the angle $\\angle BAC$, with $B \\in \\omega_1$ and $C \\in \\omega_2$, and the radius of $\\omega_1$ is less than the radius of $\\omega_2$. The line $BC$ intersects the circles $\\omega_1$ and $\\omega_2$ for the second t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cffb82e814bd9ed5eb443260eb181cbc46ff82cd5b3b88ec8df471158bf2099c | Two circles $\omega_1$ and $\omega_2$, which do not have common points, are internally tangent to the circle $\omega$. The two inner common tangents of circles $\omega_1$ and $\omega_2$ intersect the circle $\omega$ at four points. Let $M$ and $N$ be any two of these points lying on the same arc of the circle $\omega$ ... | [
"2\\sqrt{r(r - r_1 - r_2)}"
] | [
"Two circles $\\omega_1$ and $\\omega_2$, which do not have common points, are internally tangent to the circle $\\omega$. The two inner common tangents of circles $\\omega_1$ and $\\omega_2$ intersect the circle $\\omega$ at four points. Let $M$ and $N$ be any two of these points lying on the same arc of the circl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-ed3b61c1992de80e3a83dabb93172b2f3d06774092325eb0893603f945365d33 | Two circles $\omega_1$ and $\omega_2$, with centers $O_1$ and $O_2$, respectively, intersect at points $A$ and $B$. A line $\ell$ passing through $B$ intersects $\omega_1$ again at $C$ and $\omega_2$ again at $D$ such that points $C, B, D$ appear in that order. The tangents to $\omega_1$ at $C$ and to $\omega_2$ at $D$... | [
"no_valid_answer_found"
] | [
"Two circles $\\omega_1$ and $\\omega_2$, with centers $O_1$ and $O_2$, respectively, intersect at points $A$ and $B$. A line $\\ell$ passing through $B$ intersects $\\omega_1$ again at $C$ and $\\omega_2$ again at $D$ such that points $C, B, D$ appear in that order. The tangents to $\\omega_1$ at $C$ and to $\\ome... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c387d750e6d6083811277ad4952290279d022a0faebd78268e2eca1650643c41 | Two circles $\omega_1$ and $\omega_2$, with centers $O_1$ and $O_2$, respectively, intersect at points $A$ and $B$. A line $\ell$ passing through $B$ intersects $\omega_1$ again at $C$ and $\omega_2$ again at $D$, such that points $C, B, D$ appear in that order. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$,... | [
"\\dfrac{r_1 r_2}{d}"
] | [
"Two circles $\\omega_1$ and $\\omega_2$, with centers $O_1$ and $O_2$, respectively, intersect at points $A$ and $B$. A line $\\ell$ passing through $B$ intersects $\\omega_1$ again at $C$ and $\\omega_2$ again at $D$, such that points $C, B, D$ appear in that order. The tangents to $\\omega_1$ and $\\omega_2$ at ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-09b1769dedf2b81768de95eaf868238b3a2b3da743d7532093704e177eef451a | Two circles $\omega_1(O)$ and $\omega_2$ intersect at points $A$ and $B$, with $O$ lying on $\omega_2$. Let $S$ be a point on $AB$ such that $OS \perp AB$. Line $OS$ intersects $\omega_2$ at $P$ (other than $O$). The bisector of $\angle ASP$ intersects $\omega_1$ at $L$ (with $A$ and $L$ on the same side of line $OP$).... | [
"\\sqrt{15 - 3\\sqrt{21}}"
] | [
"Two circles $\\omega_1(O)$ and $\\omega_2$ intersect at points $A$ and $B$, with $O$ lying on $\\omega_2$. Let $S$ be a point on $AB$ such that $OS \\perp AB$. Line $OS$ intersects $\\omega_2$ at $P$ (other than $O$). The bisector of $\\angle ASP$ intersects $\\omega_1$ at $L$ (with $A$ and $L$ on the same side of... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-ed6674aea49b25a7de4a76b9e56f7f6c6d94ffef05a4929a33b7a072bc254f16 | Two circles $\omega_1, \omega_2$ have centers $O_1, O_2$ and radii $25, 39$ respectively. The smallest distance between a point on $\omega_1$ and a point on $\omega_2$ is $1$. Tangents from $O_2$ to $\omega_1$ meet $\omega_1$ at $S_1, T_1$, and tangents from $O_1$ to $\omega_2$ meet $\omega_2$ at $S_2, T_2$, such that ... | [
"787"
] | [
"Two circles $\\omega_1, \\omega_2$ have centers $O_1, O_2$ and radii $25, 39$ respectively. The smallest distance between a point on $\\omega_1$ and a point on $\\omega_2$ is $1$. Tangents from $O_2$ to $\\omega_1$ meet $\\omega_1$ at $S_1, T_1$, and tangents from $O_1$ to $\\omega_2$ meet $\\omega_2$ at $S_2, T_2... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9967d1257d223da6e9fcc3e0ce0d8423de4f16904aa6cd41f61447ccc054ee7e | Two circles $\omega_1, \omega_2$ with radii $28, 112$ respectively intersect at $P, Q$. $A$ is on $\omega_1$ and $B$ on $\omega_2$ such that $A, P, B$ are collinear. Tangents to $\omega_1, \omega_2$ at $A, B$ respectively meet at $T$. Suppose $\angle AQT=\angle BQT=60^\circ$. The length of $TQ$ can be expressed in the ... | [
"171"
] | [
"Two circles $\\omega_1, \\omega_2$ with radii $28, 112$ respectively intersect at $P, Q$. $A$ is on $\\omega_1$ and $B$ on $\\omega_2$ such that $A, P, B$ are collinear. Tangents to $\\omega_1, \\omega_2$ at $A, B$ respectively meet at $T$. Suppose $\\angle AQT=\\angle BQT=60^\\circ$. The length of $TQ$ can be exp... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-2d940ad0762eee8a03ce901aefea9d82cdb7238bad340c0f5b03f2962bd6e950 | Two circles $\omega_{1}$ and $\omega_{2}$ are orthogonal. Two lines $l$ and $m$ meet at point $P$ on $\omega_{1}$ and intersect $\omega_{2}$ at points $A$, $B$ on line $l$ and $C$, $D$ on line $m$. If the coordinates of $P$ are $(3, 4)$ and the equations of $l$ and $m$ are $y = x + 1$ and $y = -x + 7$ respectively, fin... | [
"(5, 5)"
] | [
"Two circles $\\omega_{1}$ and $\\omega_{2}$ are orthogonal. Two lines $l$ and $m$ meet at point $P$ on $\\omega_{1}$ and intersect $\\omega_{2}$ at points $A$, $B$ on line $l$ and $C$, $D$ on line $m$. If the coordinates of $P$ are $(3, 4)$ and the equations of $l$ and $m$ are $y = x + 1$ and $y = -x + 7$ respecti... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-038dc559c3eea5240232f831637658b836695859816910ae9e5d8bb5648820e9 | Two circles $\omega_{1}$ and $\omega_{2}$ intersect at two points $A$ and $B$. Construct a line $EF$ through the point $A$ such that $E$ lies on $\omega_{1}$ and $F$ lies on $\omega_{2}$, and such that the value $EA \cdot AF$ is maximal. | [
"EF \\text{ is the angle bisector of } \\angle O_1AO_2"
] | [
"Two circles $\\omega_{1}$ and $\\omega_{2}$ intersect at two points $A$ and $B$. Construct a line $EF$ through the point $A$ such that $E$ lies on $\\omega_{1}$ and $F$ lies on $\\omega_{2}$, and such that the value $EA \\cdot AF$ is maximal."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-74b7e29bde9e87f3c29eaaa0c4ecb36c99c92283f40f05f66a053a93a4b16532 | Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively touch externally at $A$. Let $M$ be a point inside $k_2$ and outside the line $O_1O_2$. Find a line $d$ through $M$ which intersects $k_1$ and $k_2$ again at $B$ and $C$ respectively so that the circumcircle of $\Delta ABC$ is tangent to $O_1O_2$. | [
"AM"
] | [
"Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively touch externally at $A$. Let $M$ be a point inside $k_2$ and outside the line $O_1O_2$. Find a line $d$ through $M$ which intersects $k_1$ and $k_2$ again at $B$ and $C$ respectively so that the circumcircle of $\\Delta ABC$ is tangent to $O_1O_... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-96c83ce4f5227d29a4dd8dc4fe20cbc9803180a6c82159d355e78c2ba57db24a | Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ touch each other outside at point $Q$. The other endpoints of the diameters through $Q$ are $P$ on $k_1$ and $R$ on $k_2$. We choose two points $A$ and $B$, one on each of the arcs $PQ$ of $k_1$ ($PBQA$ is a convex quadrangle). Further, let $C$ be the second point ... | [
"\\left( \\dfrac{r_1 + r_2}{2}, -\\dfrac{r_1}{2} \\right)"
] | [
"Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ touch each other outside at point $Q$. The other endpoints of the diameters through $Q$ are $P$ on $k_1$ and $R$ on $k_2$. We choose two points $A$ and $B$, one on each of the arcs $PQ$ of $k_1$ ($PBQA$ is a convex quadrangle). Further, let $C$ be the second p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-823cf7fdae5c2df5244efe8650d2d7f6c28a7b3c86250e29d252bf7cbb9e79f7 | Two circles $w_1$ and $w_2$ with centers at the points $O_1$ and $O_2$ respectively intersect at two different points $A$ and $B$, $P_1P_2$ is the common tangent of these two circles, $P_i \in w_i$, $i = 1, 2$. The projections of the points $P_1$ and $P_2$ on the line $O_1O_2$ are denoted by $M_1$ and $M_2$, respective... | [
"14\\sqrt{2}"
] | [
"Two circles $w_1$ and $w_2$ with centers at the points $O_1$ and $O_2$ respectively intersect at two different points $A$ and $B$, $P_1P_2$ is the common tangent of these two circles, $P_i \\in w_i$, $i = 1, 2$. The projections of the points $P_1$ and $P_2$ on the line $O_1O_2$ are denoted by $M_1$ and $M_2$, resp... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-4d52ae08c7c212b8ceb33b1f03dd873387f63f71c1ba33caca17e7b10d1e82e4 | Two circles ${{\gamma} _ {1}}$ and ${{\gamma} _ {2}}$ of the same radius intersect at points $A$ and $B$. The circle $\gamma$, centered at point $A$, intersects circle ${{\gamma} _ {1}}$ at points $C$ and $D$. Let $E$ and $F$ be the points of intersection of circles $\gamma$ and ${{\gamma} _ {2}}$. If the radius of $\g... | [
"\\sqrt{4r^2 - d^2}"
] | [
"Two circles ${{\\gamma} _ {1}}$ and ${{\\gamma} _ {2}}$ of the same radius intersect at points $A$ and $B$. The circle $\\gamma$, centered at point $A$, intersects circle ${{\\gamma} _ {1}}$ at points $C$ and $D$. Let $E$ and $F$ be the points of intersection of circles $\\gamma$ and ${{\\gamma} _ {2}}$. If the ra... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4098818cbf4f1b37fb00796d49bc8088c141778f37e8f8a0fca25bb539adf5a3 | Two circles (C) and (E) intersect at points A and B. A line (d) passes through point A and intersects circle (C) at point P and circle (E) at point F. Let M be the midpoint of segment PF. If the coordinates of A are (0,0), B are (2,0), P are (4,3), and F are (-2,3), find the coordinates of the point through which the m... | [
"(1,0)"
] | [
"Two circles (C) and (E) intersect at points A and B. A line (d) passes through point A and intersects circle (C) at point P and circle (E) at point F. Let M be the midpoint of segment PF. If the coordinates of A are (0,0), B are (2,0), P are (4,3), and F are (-2,3), find the coordinates of the point through which ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-6cfca613ad0b8124f0bf199fd868826bab577620b5e798dfcc06c404a23ffd6e | Two circles (K) and (L) are externally tangent at point \( T \). A chord \( AB \) of circle (K), when extended, touches circle (L) at point \( C \). Line \( CT \) intersects circle (K) again at point \( D \). If the length of \( AB \) is 10 units and the radius of circle (K) is 5 units, find the length of \( DA \). | [
"5\\sqrt{2}"
] | [
"Two circles (K) and (L) are externally tangent at point \\( T \\). A chord \\( AB \\) of circle (K), when extended, touches circle (L) at point \\( C \\). Line \\( CT \\) intersects circle (K) again at point \\( D \\). If the length of \\( AB \\) is 10 units and the radius of circle (K) is 5 units, find the length... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23cdd02c14a4c9c0d0cb63890bf405edfe19119b8dba4cd7b7ee7c2af7c7791b | Two circles (not equally large) intersect on the outside. Points $A$ and $A_1$ are on the small circle, while $B$ and $B_1$ are on the large circle. The lines $PAB$ and $PA_1B_1$ are common tangent lines of the two circles. If $PA = AB = 4$, then find the area of the small circle. | [
"2\\pi"
] | [
"Two circles (not equally large) intersect on the outside. Points $A$ and $A_1$ are on the small circle, while $B$ and $B_1$ are on the large circle. The lines $PAB$ and $PA_1B_1$ are common tangent lines of the two circles. If $PA = AB = 4$, then find the area of the small circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-99fe562e495a30d2bd569bc5f781c076058b376ee66389fc7c9a5549accc53dc | Two circles 1 and 2 are externally tangent to each other and are both internally tangent to another circle 3 at points P and Q, respectively. A chord AB of circle 3 touches circles 1 and 2 at points M and N, respectively. Let ℓ be the common tangent of circles 1 and 2. If the radius of circle 3 is 10 units and the radi... | [
"8\\sqrt{6}"
] | [
"Two circles 1 and 2 are externally tangent to each other and are both internally tangent to another circle 3 at points P and Q, respectively. A chord AB of circle 3 touches circles 1 and 2 at points M and N, respectively. Let ℓ be the common tangent of circles 1 and 2. If the radius of circle 3 is 10 units and the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-06df7bc0bd7aebf571c253cce1379a6810efcfa0a9ee5e22e41a2305b3b1dd7d | Two circles A and B touch each other internally at point P. O is the center of circle B, and X is a point on the circumference of B such that XO is tangent to circle A. Extend XO to meet the circumference of B at point Y, with PY > PX. Let PY intersect circle A at point Z. Given that YZ = 2PZ, find the magnitude of ang... | [
"\\(15^\\circ\\)"
] | [
"Two circles A and B touch each other internally at point P. O is the center of circle B, and X is a point on the circumference of B such that XO is tangent to circle A. Extend XO to meet the circumference of B at point Y, with PY > PX. Let PY intersect circle A at point Z. Given that YZ = 2PZ, find the magnitude o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 8,
"tir": 16
} | 62 | 0 | 0 |
omr-ws-7547ca969a44b38b12bbf1742477099d6ad9806095be9472962d29344e084be4 | Two circles P and Q intersect at points X and Y. Point A is located on XY such that AP = 10 and AQ = 15. If the radius of circle Q is 7, find the radius of circle P. | [
"2\\sqrt{19}"
] | [
"Two circles P and Q intersect at points X and Y. Point A is located on XY such that AP = 10 and AQ = 15. If the radius of circle Q is 7, find the radius of circle P."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-9b272db4ed6aa95179d2503925a7ca96c968f4ebd1e237d749de2e99e996c46d | Two circles P and Q intersect at points X and Y. Point A is located on line segment XY such that AP = 10 and AQ = 12. If the radius of circle Q is 7, find the radius of circle P. | [
"$\\sqrt{5}$"
] | [
"Two circles P and Q intersect at points X and Y. Point A is located on line segment XY such that AP = 10 and AQ = 12. If the radius of circle Q is 7, find the radius of circle P."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-ed933ef75468c02a8fdd495ce96cf6de18c0cecf0fef15583c0d1e8cdca889fc | Two circles P and Q intersect at points X and Y. Point A is located on line segment XY such that AP = 10 and AQ = 15. If the radius of circle Q is 7, find the radius of circle P. | [
"2\\sqrt{19}"
] | [
"Two circles P and Q intersect at points X and Y. Point A is located on line segment XY such that AP = 10 and AQ = 15. If the radius of circle Q is 7, find the radius of circle P."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-903ec13a41e9d3b3dd6ca542235c0dd6de6eae8c0d5b9f762815a95b507498c4 | Two circles S and T touch at point X. They have a common tangent which meets circle S at point A and circle T at point B, where A and B are different points. Let AP be a diameter of circle S. If the radius of circle S is 5 units and the radius of circle T is 3 units, find the length of segment BX. | [
"\\dfrac{3\\sqrt{10}}{2}"
] | [
"Two circles S and T touch at point X. They have a common tangent which meets circle S at point A and circle T at point B, where A and B are different points. Let AP be a diameter of circle S. If the radius of circle S is 5 units and the radius of circle T is 3 units, find the length of segment BX."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-588f831c3d19fb2d8359e65792278be58c1d4a79ec0d1635f4aa54d95b460aea | Two circles X and Y intersect at point P. Let the tangent line to circle X at P meet circle Y at point B, and let the tangent line to circle Y at P meet circle X at point A. Let circle Z pass through points P, B, and A. Let the tangent line to circle Z at P meet circle X at point C and circle Y at point D. If the radiu... | [
"\\dfrac{60\\sqrt{13}}{13}"
] | [
"Two circles X and Y intersect at point P. Let the tangent line to circle X at P meet circle Y at point B, and let the tangent line to circle Y at P meet circle X at point A. Let circle Z pass through points P, B, and A. Let the tangent line to circle Z at P meet circle X at point C and circle Y at point D. If the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d96418a0b8e4d3b99b0a85d4d397e1478457ac4b1d13d90a9ea79d6e34121507 | Two circles X and Y with the same radius intersect at points A and B. Points P and Q are on circles X and Y, respectively. Point R is chosen such that PAQR forms a parallelogram. Given that BRPQ is a cyclic quadrilateral, find the length of PQ if the distance between the centers of circles X and Y is 10 units. | [
"10"
] | [
"Two circles X and Y with the same radius intersect at points A and B. Points P and Q are on circles X and Y, respectively. Point R is chosen such that PAQR forms a parallelogram. Given that BRPQ is a cyclic quadrilateral, find the length of PQ if the distance between the centers of circles X and Y is 10 units."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-dbfe9f851952f3a5ebd1371063a2d1e6ef202eb3d2779401d49a34b831904c8f | Two circles X and Y with the same radius intersect at points A and B. Points P and Q are on circles X and Y, respectively. Point R is chosen such that PAQR forms a parallelogram. Given that points B, R, P, and Q are concyclic, find the length of PQ if the distance between the centers of circles X and Y is 10 units. | [
"10"
] | [
"Two circles X and Y with the same radius intersect at points A and B. Points P and Q are on circles X and Y, respectively. Point R is chosen such that PAQR forms a parallelogram. Given that points B, R, P, and Q are concyclic, find the length of PQ if the distance between the centers of circles X and Y is 10 units... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-66df004462b92515cc1cba8495210fd52f9434b1df6226e5c9c482ade6640bbf | Two circles \( C \) and \( C' \) have centers \( O \) and \( O' \) and intersect at points \( M \) and \( N \). The common tangent closer to \( M \) touches \( C \) at \( A \) and \( C' \) at \( B \). The line through \( B \) perpendicular to \( AM \) meets the line \( OO' \) at \( D \). \( BO'B' \) is a diameter of \(... | [
"90^\\circ"
] | [
"Two circles \\( C \\) and \\( C' \\) have centers \\( O \\) and \\( O' \\) and intersect at points \\( M \\) and \\( N \\). The common tangent closer to \\( M \\) touches \\( C \\) at \\( A \\) and \\( C' \\) at \\( B \\). The line through \\( B \\) perpendicular to \\( AM \\) meets the line \\( OO' \\) at \\( D \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-e646f48f195ad6bba6f947c04581938f61aadff4103f39a1600c3696f0dbed6f | Two circles \( C \) and \( C' \) have centers \( O \) and \( O' \) and intersect at points \( M \) and \( N \). The common tangent closer to \( M \) touches \( C \) at \( A \) and \( C' \) at \( B \). The line through \( B \) perpendicular to \( AM \) meets the line \( OO' \) at \( D \). \( BO'B' \) is a diameter of \(... | [
"\\dfrac{\\sqrt{11}}{7}"
] | [
"Two circles \\( C \\) and \\( C' \\) have centers \\( O \\) and \\( O' \\) and intersect at points \\( M \\) and \\( N \\). The common tangent closer to \\( M \\) touches \\( C \\) at \\( A \\) and \\( C' \\) at \\( B \\). The line through \\( B \\) perpendicular to \\( AM \\) meets the line \\( OO' \\) at \\( D \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-c6a019f63722b63b482e2d365c05816dc2d0a7a249549a18a3ae434c2786f39c | Two circles \( C \) and \( C' \) intersect at points \( A \) and \( B \). A straight line passing through \( B \) intersects the circles \( C \) and \( C' \) at points \( M \) and \( N \), respectively, with \( B \) between \( M \) and \( N \). Let \( I \) be the midpoint of the arc \( AM \) of circle \( C \) not conta... | [
"90"
] | [
"Two circles \\( C \\) and \\( C' \\) intersect at points \\( A \\) and \\( B \\). A straight line passing through \\( B \\) intersects the circles \\( C \\) and \\( C' \\) at points \\( M \\) and \\( N \\), respectively, with \\( B \\) between \\( M \\) and \\( N \\). Let \\( I \\) be the midpoint of the arc \\( A... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-90666a65bf15de42339088f4dc8399e73cc5e8a9b144877319eb140355141341 | Two circles \( C \) and \( C_1 \) intersect at points \( A \) and \( B \). A chord through \( A \) intersects \( C \) at \( M \) and \( C_1 \) at \( P \). A chord through \( B \) intersects \( C \) at \( N \) and \( C_1 \) at \( Q \). If the radius of circle \( C \) is 10 and the radius of circle \( C_1 \) is 15, and t... | [
"8"
] | [
"Two circles \\( C \\) and \\( C_1 \\) intersect at points \\( A \\) and \\( B \\). A chord through \\( A \\) intersects \\( C \\) at \\( M \\) and \\( C_1 \\) at \\( P \\). A chord through \\( B \\) intersects \\( C \\) at \\( N \\) and \\( C_1 \\) at \\( Q \\). If the radius of circle \\( C \\) is 10 and the radi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0585da5fc658d40d6db180db21efb0f699785e3dca402b3d6ac8768db8ad3abe | Two circles \( C \) and \( D \) are externally tangent at point \( P \). Point \( A \) is inside circle \( C \). Tangents \( AM \) and \( AN \) are drawn from \( A \) to circle \( D \), with \( M \) and \( N \) being the points of tangency. The second intersection points of \( AM \) and \( AN \) with circle \( C \) are... | [
"1"
] | [
"Two circles \\( C \\) and \\( D \\) are externally tangent at point \\( P \\). Point \\( A \\) is inside circle \\( C \\). Tangents \\( AM \\) and \\( AN \\) are drawn from \\( A \\) to circle \\( D \\), with \\( M \\) and \\( N \\) being the points of tangency. The second intersection points of \\( AM \\) and \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-7c52973e00aae499e8d4344fd01c378e38b027d94def7dfaf1c0c524c7a1e558 | Two circles \( C1 \) and \( C2 \) are externally tangent at point \( C \) and have centers \( O1 \) and \( O2 \) respectively. A third circle is tangent to \( C1 \) and \( C2 \) at points \( M \) and \( N \) respectively. The diameter \( AB \) of this third circle is perpendicular to the common tangent \( t \) of \( C1... | [
"15"
] | [
"Two circles \\( C1 \\) and \\( C2 \\) are externally tangent at point \\( C \\) and have centers \\( O1 \\) and \\( O2 \\) respectively. A third circle is tangent to \\( C1 \\) and \\( C2 \\) at points \\( M \\) and \\( N \\) respectively. The diameter \\( AB \\) of this third circle is perpendicular to the common... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-f1ebe187e965babc916b4383eec25c4e9107f4f32095a423a15fd14b8309fc0d | Two circles \( C1 \) and \( C2 \) intersect at points \( A \) and \( B \). A point \( C \) lies on \( C1 \). The line \( CA \) intersects \( C2 \) again at \( X \), and the line \( CB \) intersects \( C2 \) again at \( Y \). The line \( BX \) intersects \( C1 \) again at \( E \), and the line \( AY \) intersects \( C1 ... | [
"12"
] | [
"Two circles \\( C1 \\) and \\( C2 \\) intersect at points \\( A \\) and \\( B \\). A point \\( C \\) lies on \\( C1 \\). The line \\( CA \\) intersects \\( C2 \\) again at \\( X \\), and the line \\( CB \\) intersects \\( C2 \\) again at \\( Y \\). The line \\( BX \\) intersects \\( C1 \\) again at \\( E \\), and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a71c930547d447c1c1a7e857e4a68fb8e4012f1da3114e1a25ef492aad8e85e9 | Two circles \( C1 \) and \( C2 \) intersect at points \( A \) and \( B \). Point \( C \) is on circle \( C1 \). Line \( BC \) intersects circle \( C2 \) at point \( D \). The tangents to \( C1 \) at \( C \) and to \( C2 \) at \( D \) meet at point \( M \). Line \( AM \) intersects line \( CD \) at point \( P \). Point ... | [
"\\dfrac{28\\sqrt{66}}{13}"
] | [
"Two circles \\( C1 \\) and \\( C2 \\) intersect at points \\( A \\) and \\( B \\). Point \\( C \\) is on circle \\( C1 \\). Line \\( BC \\) intersects circle \\( C2 \\) at point \\( D \\). The tangents to \\( C1 \\) at \\( C \\) and to \\( C2 \\) at \\( D \\) meet at point \\( M \\). Line \\( AM \\) intersects lin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-03a70f309958d0b3db5904b4ae1499c0f2d9980888615a0f664d33d8b78bbf69 | Two circles \( C1 \) and \( C2 \) intersect at points \( P \) and \( Q \). A line through \( P \) intersects \( C1 \) and \( C2 \) again at points \( A \) and \( B \), respectively, and \( X \) is the midpoint of \( AB \). The line through \( Q \) and \( X \) intersects \( C1 \) and \( C2 \) again at \( Y \) and \( Z \... | [
"10"
] | [
"Two circles \\( C1 \\) and \\( C2 \\) intersect at points \\( P \\) and \\( Q \\). A line through \\( P \\) intersects \\( C1 \\) and \\( C2 \\) again at points \\( A \\) and \\( B \\), respectively, and \\( X \\) is the midpoint of \\( AB \\). The line through \\( Q \\) and \\( X \\) intersects \\( C1 \\) and \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-14cd67dc96c1ecbb99d1b32f96ce8d0e93971ad213a920acef7d3e9ae7ae9f33 | Two circles \( C1 \) and \( C2 \) meet at points \( A \) and \( B \). \( P \) is a point on the circumference of \( C1 \) such that \( PA \) is tangent to \( C2 \) at \( A \). \( Q \) is a point on the circumference of \( C2 \) outside \( C1 \). \( QA \) is produced to meet \( C1 \) again at \( T \) so that \( \angle P... | [
"\\( 75^\\circ \\)"
] | [
"Two circles \\( C1 \\) and \\( C2 \\) meet at points \\( A \\) and \\( B \\). \\( P \\) is a point on the circumference of \\( C1 \\) such that \\( PA \\) is tangent to \\( C2 \\) at \\( A \\). \\( Q \\) is a point on the circumference of \\( C2 \\) outside \\( C1 \\). \\( QA \\) is produced to meet \\( C1 \\) aga... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-0f60a7785ee6be0c6c02ab66e2e75f789581fda5c177abbb830dfc595272842e | Two circles \( C_1 \) and \( C_2 \) are externally tangent at point \( P \). Point \( A \) is on \( C_1 \). Tangents \( AM \) and \( AM' \) are drawn from \( A \) to \( C_2 \). The second points of intersection of \( AM \) and \( AM' \) with \( C_1 \) are \( N \) and \( N' \), respectively. If the radii of \( C_1 \) an... | [
"1"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) are externally tangent at point \\( P \\). Point \\( A \\) is on \\( C_1 \\). Tangents \\( AM \\) and \\( AM' \\) are drawn from \\( A \\) to \\( C_2 \\). The second points of intersection of \\( AM \\) and \\( AM' \\) with \\( C_1 \\) are \\( N \\) and \\( N' \\), respectiv... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-526d605d349b88eea8406f356cc3732adb1e691eebdaa5289932bd591b8803af | Two circles \( C_1 \) and \( C_2 \) are tangent at \( O \). Another two circles \( C_3 \) and \( C_4 \) are also tangent at \( O \). Let \( A = C_1 \cap C_3 \), \( B = C_2 \cap C_3 \), \( C = C_1 \cap C_4 \), and \( D = C_2 \cap C_4 \) such that each of \( A, B, C, D \) is distinct from \( O \). If \( OA = 3 \), \( OB ... | [
"\\dfrac{2}{5}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) are tangent at \\( O \\). Another two circles \\( C_3 \\) and \\( C_4 \\) are also tangent at \\( O \\). Let \\( A = C_1 \\cap C_3 \\), \\( B = C_2 \\cap C_3 \\), \\( C = C_1 \\cap C_4 \\), and \\( D = C_2 \\cap C_4 \\) such that each of \\( A, B, C, D \\) is distinct from \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-b02c9ef258abe336731b3e8fe7a22b073304b7023d4d5ef14c39d687727b046b | Two circles \( C_1 \) and \( C_2 \) are tangent at a point \( P \). A straight line \( D \) is tangent to \( C_1 \) at point \( A \) and intersects \( C_2 \) at points \( B \) and \( C \). If the measure of \( \angle BPC \) is \( 120^\circ \), calculate the measure of \( \angle BPA \). | [
"60^\\circ"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) are tangent at a point \\( P \\). A straight line \\( D \\) is tangent to \\( C_1 \\) at point \\( A \\) and intersects \\( C_2 \\) at points \\( B \\) and \\( C \\). If the measure of \\( \\angle BPC \\) is \\( 120^\\circ \\), calculate the measure of \\( \\angle BPA \\)."
... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-51493700634a3f2aaa356f7906f88c7b77ca2a3466466d3b1fd74bce5fa5ab84 | Two circles \( C_1 \) and \( C_2 \) are tangent at a point \( P \). A straight line is tangent to circle \( C_1 \) at point \( A \) and intersects circle \( C_2 \) at points \( B \) and \( C \). If the lengths of segments \( PA \), \( PB \), and \( PC \) are 6, 8, and 10 respectively, find the length of the segment \( ... | [
"4\\sqrt{5}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) are tangent at a point \\( P \\). A straight line is tangent to circle \\( C_1 \\) at point \\( A \\) and intersects circle \\( C_2 \\) at points \\( B \\) and \\( C \\). If the lengths of segments \\( PA \\), \\( PB \\), and \\( PC \\) are 6, 8, and 10 respectively, find th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48c79ce4aab556b331ea9b00026044fa7c5241a9bbe864762f5bb7598ac1bfc9 | Two circles \( C_1 \) and \( C_2 \) are tangent externally at a point \( X \). A common tangent to the two circles meets \( C_1 \) at \( Y \) and \( C_2 \) at \( Z \) (with \( Y \ne Z \)). Let \( T \) be such that \( [YT] \) is a diameter of \( C_1 \). If the radius of \( C_1 \) is 5 and the radius of \( C_2 \) is 3, f... | [
"4\\sqrt{10}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) are tangent externally at a point \\( X \\). A common tangent to the two circles meets \\( C_1 \\) at \\( Y \\) and \\( C_2 \\) at \\( Z \\) (with \\( Y \\ne Z \\)). Let \\( T \\) be such that \\( [YT] \\) is a diameter of \\( C_1 \\). If the radius of \\( C_1 \\) is 5 and t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7ad387e066e99531ce5a9e108f8641fc5866bbfb2be1cc6971fbef797d17b43c | Two circles \( C_1 \) and \( C_2 \) have a common center with \( C_1 \) larger than \( C_2 \). Point \( A \) lies on \( C_1 \) and \( AB \) is tangent to \( C_2 \) at point \( B \). Line \( AB \) intersects \( C_1 \) again at point \( C \), and \( D \) is the midpoint of \( AB \). A line from \( A \) intersects \( C_2 ... | [
"1"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) have a common center with \\( C_1 \\) larger than \\( C_2 \\). Point \\( A \\) lies on \\( C_1 \\) and \\( AB \\) is tangent to \\( C_2 \\) at point \\( B \\). Line \\( AB \\) intersects \\( C_1 \\) again at point \\( C \\), and \\( D \\) is the midpoint of \\( AB \\). A lin... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-c18a69d94ab252b5e6811b08d428379875e82a151fa3f8d42bc26c304cba3d04 | Two circles \( C_1 \) and \( C_2 \) intersect at \( S \). The tangent at \( S \) to \( C_1 \) intersects \( C_2 \) at \( A \) (different from \( S \)). The tangent at \( S \) to \( C_2 \) intersects \( C_1 \) at \( B \) (different from \( S \)). Another circle \( C_3 \) passes through \( A, B, \) and \( S \). The tange... | [
""
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at \\( S \\). The tangent at \\( S \\) to \\( C_1 \\) intersects \\( C_2 \\) at \\( A \\) (different from \\( S \\)). The tangent at \\( S \\) to \\( C_2 \\) intersects \\( C_1 \\) at \\( B \\) (different from \\( S \\)). Another circle \\( C_3 \\) passes through \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-de21abaf300ff5b0f13fb517aa53e619c035c81316a2efa760a49549c1b9c473 | Two circles \( C_1 \) and \( C_2 \) intersect at points \( A \) and \( B \). A circle \( C \) centered at \( A \) intersects \( C_1 \) at points \( M \) and \( P \), and \( C_2 \) at points \( N \) and \( Q \), such that \( N \) and \( Q \) are on different sides with respect to \( MP \) and \( AB > AM \). Let \( \thet... | [
"1"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at points \\( A \\) and \\( B \\). A circle \\( C \\) centered at \\( A \\) intersects \\( C_1 \\) at points \\( M \\) and \\( P \\), and \\( C_2 \\) at points \\( N \\) and \\( Q \\), such that \\( N \\) and \\( Q \\) are on different sides with respect to \\( MP ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bbcd53a41168c7a66de08bc40c850bc9492e34bcf6701c9055addac31b876649 | Two circles \( C_1 \) and \( C_2 \) intersect at points \( A \) and \( B \). A line through \( B \) intersects \( C_1 \) at \( C \) and \( C_2 \) at \( D \). Another line through \( B \) intersects \( C_1 \) at \( E \) and \( C_2 \) at \( F \). The line \( CF \) intersects \( C_1 \) at \( P \) and \( C_2 \) at \( Q \).... | [
"90"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at points \\( A \\) and \\( B \\). A line through \\( B \\) intersects \\( C_1 \\) at \\( C \\) and \\( C_2 \\) at \\( D \\). Another line through \\( B \\) intersects \\( C_1 \\) at \\( E \\) and \\( C_2 \\) at \\( F \\). The line \\( CF \\) intersects \\( C_1 \\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-bae93bf54c3e7a76bd232ce06d1bedffa349956dc20d3989689d4429c5b2e7fb | Two circles \( C_1 \) and \( C_2 \) intersect at points \( A \) and \( B \). A smaller circle \( C \) is internally tangent to both \( C_1 \) and \( C_2 \) at points \( D \) and \( E \) respectively. Let \( K \) be one of the points of intersection of circle \( C \) with line segment \( AB \). Define points \( F = EK \... | [
"1"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at points \\( A \\) and \\( B \\). A smaller circle \\( C \\) is internally tangent to both \\( C_1 \\) and \\( C_2 \\) at points \\( D \\) and \\( E \\) respectively. Let \\( K \\) be one of the points of intersection of circle \\( C \\) with line segment \\( AB \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-2fe3e4a0005d9fb674ff382565091da3bdbb8cd72377b1c4336b6f38f1ce08c9 | Two circles \( C_1 \) and \( C_2 \) intersect at points \( A \) and \( B \). Circle \( C_3 \) is drawn to touch \( C_1 \) and \( C_2 \) internally, with \( C_1 \) and \( C_2 \) lying inside \( C_3 \). The tangent points of \( C_1 \) and \( C_3 \) are \( P \), and the tangent points of \( C_2 \) and \( C_3 \) are \( Q \... | [
"4\\sqrt{6}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at points \\( A \\) and \\( B \\). Circle \\( C_3 \\) is drawn to touch \\( C_1 \\) and \\( C_2 \\) internally, with \\( C_1 \\) and \\( C_2 \\) lying inside \\( C_3 \\). The tangent points of \\( C_1 \\) and \\( C_3 \\) are \\( P \\), and the tangent points of \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b12b6fa1b3171d8bed77e16668446bb32649caa6180d99c4aafe88ded6404e89 | Two circles \( C_1 \) and \( C_2 \) intersect at points \( A \) and \( B \). Points \( C \) and \( D \) are on \( C_1 \), and points \( E \) and \( F \) are defined as \( E = AC \cap C_2 \) and \( F = AD \cap C_2 \). Let \( P = CD \cap EF \). Let \( M \) and \( N \) be the midpoints of \( CD \) and \( EF \), respective... | [
"90"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at points \\( A \\) and \\( B \\). Points \\( C \\) and \\( D \\) are on \\( C_1 \\), and points \\( E \\) and \\( F \\) are defined as \\( E = AC \\cap C_2 \\) and \\( F = AD \\cap C_2 \\). Let \\( P = CD \\cap EF \\). Let \\( M \\) and \\( N \\) be the midpoints ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef006bfd228cc9bf4c30f5118861b8402415d612f41fbd21e5da0e5e78b834a2 | Two circles \( C_1 \) and \( C_2 \) intersect at points \( A \) and \( B \). Points \( P \) and \( Q \) are on circles \( C_1 \) and \( C_2 \) respectively, such that \( |AP| = |AQ| \). The segment \( PQ \) intersects circles \( C_1 \) and \( C_2 \) at points \( M \) and \( N \) respectively. Let \( C \) be the center ... | [
"2"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at points \\( A \\) and \\( B \\). Points \\( P \\) and \\( Q \\) are on circles \\( C_1 \\) and \\( C_2 \\) respectively, such that \\( |AP| = |AQ| \\). The segment \\( PQ \\) intersects circles \\( C_1 \\) and \\( C_2 \\) at points \\( M \\) and \\( N \\) respect... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-dc1e33ddfb9534bbbf1a63a8d2ff7fff3b27e16b19743be501cf9075d4b71476 | Two circles \( C_1 \) and \( C_2 \) intersect at points \( A \) and \( B \). The common tangent of \( C_1 \) and \( C_2 \) that is closer to \( B \) than \( A \) meets \( C_1 \) at \( P \) and \( C_2 \) at \( Q \). The line \( QB \) extended meets \( C_1 \) at \( S \), and the line \( PB \) extended meets \( C_2 \) at ... | [
"1"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at points \\( A \\) and \\( B \\). The common tangent of \\( C_1 \\) and \\( C_2 \\) that is closer to \\( B \\) than \\( A \\) meets \\( C_1 \\) at \\( P \\) and \\( C_2 \\) at \\( Q \\). The line \\( QB \\) extended meets \\( C_1 \\) at \\( S \\), and the line \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-559e0566a55ee8feebdb1aaf368eeabff3a4377b1f10eaf17dca836d998e9c95 | Two circles \( C_1 \) and \( C_2 \) intersect at points \( A \) and \( B \). The tangent to \( C_1 \) at \( A \) cuts \( C_2 \) again at the point \( P \) and the tangent to \( C_2 \) at \( A \) cuts \( C_1 \) again at the point \( Q \). Knowing that \( PB = 640 \) and \( QB = 1000 \), calculate \( AB \). | [
"800"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at points \\( A \\) and \\( B \\). The tangent to \\( C_1 \\) at \\( A \\) cuts \\( C_2 \\) again at the point \\( P \\) and the tangent to \\( C_2 \\) at \\( A \\) cuts \\( C_1 \\) again at the point \\( Q \\). Knowing that \\( PB = 640 \\) and \\( QB = 1000 \\), ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-97a5fd20dd510389cd519502c6b5161216874f67dcdee4ed11d047fd88f54d01 | Two circles \( C_1 \) and \( C_2 \) intersect at points \( A \) and \( B \). The tangent to circle \( C_2 \) at \( A \) intersects circle \( C_1 \) at point \( C \), and the tangent to circle \( C_1 \) at \( A \) intersects circle \( C_2 \) at point \( D \). The half-line \( (d) \) passing through \( A \) (internal to ... | [
"12"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at points \\( A \\) and \\( B \\). The tangent to circle \\( C_2 \\) at \\( A \\) intersects circle \\( C_1 \\) at point \\( C \\), and the tangent to circle \\( C_1 \\) at \\( A \\) intersects circle \\( C_2 \\) at point \\( D \\). The half-line \\( (d) \\) passin... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-230347bd8521259998e22d6243fdfdfa86008161c533a6c3a9ecef0229fb5aeb | Two circles \( C_1 \) and \( C_2 \) intersect at points \( A \) and \( B \). Their external common tangent (closer to \( B \)) touches \( C_1 \) at \( P \) and \( C_2 \) at \( Q \). Let \( C \) be the reflection of \( B \) in line \( PQ \). If \( \angle CAP = \angle BAQ = \theta \), calculate \( \cos(2\theta) \) given ... | [
"\\dfrac{1}{2}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at points \\( A \\) and \\( B \\). Their external common tangent (closer to \\( B \\)) touches \\( C_1 \\) at \\( P \\) and \\( C_2 \\) at \\( Q \\). Let \\( C \\) be the reflection of \\( B \\) in line \\( PQ \\). If \\( \\angle CAP = \\angle BAQ = \\theta \\), ca... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-3ebb9437143a6585d796b7d870ef835bead280f63edd70c9314e27cef4a1fc32 | Two circles \( C_1 \) and \( C_2 \) intersect at points \( P \) and \( Q \). Their common tangent, closer to \( P \) than to \( Q \), touches \( C_1 \) at \( A \) and \( C_2 \) at \( B \). The tangents to \( C_1 \) and \( C_2 \) at \( P \) meet the other circle at points \( E \neq P \) and \( F \neq P \), respectively.... | [
"\\dfrac{120}{13}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at points \\( P \\) and \\( Q \\). Their common tangent, closer to \\( P \\) than to \\( Q \\), touches \\( C_1 \\) at \\( A \\) and \\( C_2 \\) at \\( B \\). The tangents to \\( C_1 \\) and \\( C_2 \\) at \\( P \\) meet the other circle at points \\( E \\neq P \\)... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e1521c9cc3024fc1baea09a3d495e0aeaae5269eb443a26f1a9a7234b957004 | Two circles \( C_1 \) and \( C_2 \) intersect at two distinct points \( P \) and \( Q \). A line passing through \( P \) intersects \( C_1 \) at \( A \) and \( C_2 \) at \( B \). Let \( Y \) be the midpoint of \( AB \). The line \( QY \) intersects \( C_1 \) at \( X \) and \( C_2 \) at \( Z \). If the length of \( AB \... | [
"10"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at two distinct points \\( P \\) and \\( Q \\). A line passing through \\( P \\) intersects \\( C_1 \\) at \\( A \\) and \\( C_2 \\) at \\( B \\). Let \\( Y \\) be the midpoint of \\( AB \\). The line \\( QY \\) intersects \\( C_1 \\) at \\( X \\) and \\( C_2 \\) a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-a4a8c16478da177f5ed37286bf68317881d530ed486598e866981c394d79707b | Two circles \( C_1 \) and \( C_2 \) intersect at two distinct points \( P \) and \( Q \). Points \( R \) and \( S \) lie on \( C_1 \) and \( C_2 \) respectively such that \( R, Q, \) and \( S \) are collinear. The lines \( RP \) and \( SP \) intersect \( C_2 \) and \( C_1 \) at \( N \) and \( M \) respectively. Let \( ... | [
"90"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) intersect at two distinct points \\( P \\) and \\( Q \\). Points \\( R \\) and \\( S \\) lie on \\( C_1 \\) and \\( C_2 \\) respectively such that \\( R, Q, \\) and \\( S \\) are collinear. The lines \\( RP \\) and \\( SP \\) intersect \\( C_2 \\) and \\( C_1 \\) at \\( N \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-8a68072cba44b1342762b32e60a81436738abb7e92831779263cd5f8bc88dd59 | Two circles \( C_1 \) and \( C_2 \) touch each other externally at point \( P \). A tangent line at point \( Q \) on \( C_1 \) intersects \( C_2 \) at points \( A \) and \( B \). The line \( QP \) intersects \( C_2 \) again at point \( C \). If the radius of \( C_1 \) is 5 units and the radius of \( C_2 \) is 12 units,... | [
"2\\sqrt{102}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) touch each other externally at point \\( P \\). A tangent line at point \\( Q \\) on \\( C_1 \\) intersects \\( C_2 \\) at points \\( A \\) and \\( B \\). The line \\( QP \\) intersects \\( C_2 \\) again at point \\( C \\). If the radius of \\( C_1 \\) is 5 units and the rad... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f17589f7b177d950bb84a95c5916ed6c475181769c229c743792c65d227fa220 | Two circles \( C_1 \) and \( C_2 \) with different radii touch each other externally at point \( T \). For any points \( A \in C_1 \) and \( B \in C_2 \) (both different from \( T \)) such that \( \angle ATB = 90^\circ \), find the locus of midpoints of all such segments \( AB \). | [
"The circle with diameter \\( O_1O_2 \\)."
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) with different radii touch each other externally at point \\( T \\). For any points \\( A \\in C_1 \\) and \\( B \\in C_2 \\) (both different from \\( T \\)) such that \\( \\angle ATB = 90^\\circ \\), find the locus of midpoints of all such segments \\( AB \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-f046d20d9fc94e2a09b5ce40f32781425930418731402eb6d148067c09ea23ac | Two circles \( C_1 \) and \( C_2 \) with different radii touch each other externally at point \( T \). For any points \( A \in C_1 \) and \( B \in C_2 \) (both different from \( T \)) such that \( \angle ATB = 90^\circ \), let \( P \) be the point of concurrency of all such lines \( AB \). If the radii of \( C_1 \) and... | [
"\\dfrac{120}{7}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) with different radii touch each other externally at point \\( T \\). For any points \\( A \\in C_1 \\) and \\( B \\in C_2 \\) (both different from \\( T \\)) such that \\( \\angle ATB = 90^\\circ \\), let \\( P \\) be the point of concurrency of all such lines \\( AB \\). If... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fbdb2aed0a037b853b5701e11ecd9626a4bfd8ef7e82af222ba5d2b87453cce5 | Two circles \( C_1 \) and \( C_2 \) with radii 3 and 6, respectively, are tangent externally. Both circles are also tangent internally to a larger circle \( C_3 \) with radius 9. If \( AB \) is a chord of \( C_3 \) that is tangent to both \( C_1 \) and \( C_2 \), find the length of \( AB \). | [
"\\( 6 \\sqrt{5} \\)"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) with radii 3 and 6, respectively, are tangent externally. Both circles are also tangent internally to a larger circle \\( C_3 \\) with radius 9. If \\( AB \\) is a chord of \\( C_3 \\) that is tangent to both \\( C_1 \\) and \\( C_2 \\), find the length of \\( AB \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e568ed3675191d7a16d1a59e5eb4cda813477049fdc046adc14be9297f30e642 | Two circles \( C_1 \) and \( C_2 \) with radii \( a \) and \( b \) touch each other externally and both touch a unit circle \( C \) internally. A circle \( C_3 \) with radius \( r \) is inscribed to touch \( C_1 \), \( C_2 \) externally and \( C \) internally. If \( a = \frac{1}{3} \) and \( b = \frac{1}{4} \), calcula... | [
"\\dfrac{3 - \\sqrt{5}}{8}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) with radii \\( a \\) and \\( b \\) touch each other externally and both touch a unit circle \\( C \\) internally. A circle \\( C_3 \\) with radius \\( r \\) is inscribed to touch \\( C_1 \\), \\( C_2 \\) externally and \\( C \\) internally. If \\( a = \\frac{1}{3} \\) and \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a391d6a564ee09ef4425d6c9f4cb8c934b59d86bd752fb0dc3d225d204fe5ec8 | Two circles \( C_1 \) and \( C_2 \) with radii \( r_1 = 9 \) and \( r_2 = 16 \) touch each other externally and both touch a line \( l \). A circle \( C_3 \) with radius \( r_3 \) is tangent to \( l \) and externally to \( C_1 \) and \( C_2 \). Calculate the value of \( \frac{1}{\sqrt{r_3}} \). | [
"\\dfrac{7}{12}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) with radii \\( r_1 = 9 \\) and \\( r_2 = 16 \\) touch each other externally and both touch a line \\( l \\). A circle \\( C_3 \\) with radius \\( r_3 \\) is tangent to \\( l \\) and externally to \\( C_1 \\) and \\( C_2 \\). Calculate the value of \\( \\frac{1}{\\sqrt{r_3}} ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e0e4f844fcf13cf41bbc4c78e410235a76d541346dbe435de2425e4c0a5f080 | Two circles \( C_1 \) and \( C_2 \) with radii \( r_1 \) and \( r_2 \) (\( r_1 < r_2 \)) are externally tangent. The line \( d_1 \) is a common tangent to both circles, intersecting \( C_1 \) at \( A \) and \( C_2 \) at \( D \). The line \( d_2 \) is parallel to \( d_1 \), tangent to \( C_1 \), and intersects \( C_2 \)... | [
"4\\sqrt{5}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) with radii \\( r_1 \\) and \\( r_2 \\) (\\( r_1 < r_2 \\)) are externally tangent. The line \\( d_1 \\) is a common tangent to both circles, intersecting \\( C_1 \\) at \\( A \\) and \\( C_2 \\) at \\( D \\). The line \\( d_2 \\) is parallel to \\( d_1 \\), tangent to \\( C_... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72dfe86ae18e2d7774e82d5cb54f64b813db7ec1289c761ae65bac8debd4d016 | Two circles \( C_1 \) and \( C_2 \) with radii \( r_1 \) and \( r_2 \) respectively touch each other externally. Points \( A \) and \( B \) lie on \( C_1 \), and points \( C \) and \( D \) lie on \( C_2 \) such that \( AD \) and \( BC \) are common external tangents to the two circles. Calculate the radius of the circl... | [
"\\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 \\) lie on \\( C_1 \\), and points \\( C \\) and \\( D \\) lie on \\( C_2 \\) such that \\( AD \\) and \\( BC \\) are common external tangents to the two circles. Calc... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-c4e579ffc8af5fcddcc6a40bac2fa641ee8a23114f56def1ca4af7456fde9a4f | Two circles \( C_1 \) and \( C_2 \) with respective radii \( r_1 \) and \( r_2 \) intersect at points \( A \) and \( B \). A variable line \( r \) through \( B \) meets \( C_1 \) and \( C_2 \) again at points \( P_r \) and \( Q_r \) respectively. Let \( M \) be the point such that the perpendicular bisector of each seg... | [
"\\dfrac{119}{13}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\) with respective radii \\( r_1 \\) and \\( r_2 \\) intersect at points \\( A \\) and \\( B \\). A variable line \\( r \\) through \\( B \\) meets \\( C_1 \\) and \\( C_2 \\) again at points \\( P_r \\) and \\( Q_r \\) respectively. Let \\( M \\) be the point such that the per... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-7ac3520644ba1b8f17274dc10e7f22ecb31d9847e77d10dd938a07c5ee4331d7 | Two circles \( C_1 \) and \( C_2 \), each of unit radius, have centers \( A_1 \) and \( A_2 \) such that \( A_1A_2 = 6 \). Let \( P \) be the midpoint of \( A_1A_2 \) and let \( C_3 \) be a circle externally tangent to both \( C_1 \) and \( C_2 \). \( C_1 \) and \( C_3 \) have a common tangent that passes through \( P ... | [
"8"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\), each of unit radius, have centers \\( A_1 \\) and \\( A_2 \\) such that \\( A_1A_2 = 6 \\). Let \\( P \\) be the midpoint of \\( A_1A_2 \\) and let \\( C_3 \\) be a circle externally tangent to both \\( C_1 \\) and \\( C_2 \\). \\( C_1 \\) and \\( C_3 \\) have a common tang... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-96c875a987c29cc6194eb7b280c6f5eb39a188b4fb4730479617277e576848c0 | Two circles \( C_1 \) and \( C_2 \), with centers \( O_1 \) and \( O_2 \) respectively, intersect at points \( A \) and \( B \). The center \( O_1 \) lies on circle \( C_2 \). A line \( l \) passes through \( O_1 \) but does not pass through \( O_2 \). Let \( P \) and \( Q \) be the projections of \( A \) and \( B \) r... | [
"\\dfrac{24}{5}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\), with centers \\( O_1 \\) and \\( O_2 \\) respectively, intersect at points \\( A \\) and \\( B \\). The center \\( O_1 \\) lies on circle \\( C_2 \\). A line \\( l \\) passes through \\( O_1 \\) but does not pass through \\( O_2 \\). Let \\( P \\) and \\( Q \\) be the proje... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b8057bed39af0cc1addd1fe9126c4fc6ba2f16faf8a13b21a34f5d780aa3a3a | Two circles \( C_1 \) and \( C_2 \), with centers \( O_1 \) and \( O_2 \) respectively, intersect at two points \( A \) and \( B \). Let \( X \) and \( Y \) be points on \( C_1 \), different from \( A \) and \( B \). Lines \( XA \) and \( YA \) intersect \( C_2 \) again at \( Z \) and \( W \) respectively. Let \( M \) ... | [
"1"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\), with centers \\( O_1 \\) and \\( O_2 \\) respectively, intersect at two points \\( A \\) and \\( B \\). Let \\( X \\) and \\( Y \\) be points on \\( C_1 \\), different from \\( A \\) and \\( B \\). Lines \\( XA \\) and \\( YA \\) intersect \\( C_2 \\) again at \\( Z \\) and... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-9d23d1d6e03c698e148d91eb7c20c333ec203e441ba9d297f8a410a29b5c4bf3 | Two circles \( C_1 \) and \( C_2 \), with centers at \( D \) and \( E \) respectively, touch at \( B \). The circle having \( DE \) as diameter intersects \( C_1 \) at \( H \) and \( C_2 \) at \( K \). The points \( H \) and \( K \) both lie on the same side of the line \( DE \). \( HK \) extended in both directions me... | [
"\\dfrac{\\sqrt{2145} - 15}{8}"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\), with centers at \\( D \\) and \\( E \\) respectively, touch at \\( B \\). The circle having \\( DE \\) as diameter intersects \\( C_1 \\) at \\( H \\) and \\( C_2 \\) at \\( K \\). The points \\( H \\) and \\( K \\) both lie on the same side of the line \\( DE \\). \\( HK \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7ac62e636754020d392808b8cddee54cff8ab39b26f40ae7ba136485b4d44c80 | Two circles \( C_1 \) and \( C_2 \), with centers at \( D \) and \( E \) respectively, touch at \( B \). The circle having \( DE \) as diameter intersects \( C_1 \) at \( H \) and \( C_2 \) at \( K \). The points \( H \) and \( K \) both lie on the same side of the line \( DE \). \( HK \) extended in both directions me... | [
"no_valid_answer_found"
] | [
"Two circles \\( C_1 \\) and \\( C_2 \\), with centers at \\( D \\) and \\( E \\) respectively, touch at \\( B \\). The circle having \\( DE \\) as diameter intersects \\( C_1 \\) at \\( H \\) and \\( C_2 \\) at \\( K \\). The points \\( H \\) and \\( K \\) both lie on the same side of the line \\( DE \\). \\( HK \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-455a52f6662582fcc5eef43eef62a0ef1fb62aa9ac038049011a82035579be6c | Two circles \( C_1(O_1) \) and \( C_2(O_2) \) with distinct radii intersect at points \( A \) and \( B \). The tangent from \( A \) to \( C_1 \) intersects the tangent from \( B \) to \( C_2 \) at point \( M \). If the radii of \( C_1 \) and \( C_2 \) are \( r_1 = 5 \) and \( r_2 = 12 \) respectively, and the distance ... | [
"\\dfrac{119}{120}"
] | [
"Two circles \\( C_1(O_1) \\) and \\( C_2(O_2) \\) with distinct radii intersect at points \\( A \\) and \\( B \\). The tangent from \\( A \\) to \\( C_1 \\) intersects the tangent from \\( B \\) to \\( C_2 \\) at point \\( M \\). If the radii of \\( C_1 \\) and \\( C_2 \\) are \\( r_1 = 5 \\) and \\( r_2 = 12 \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b7de1e6a5610bca2edd867288a3bd23fe97edcf070ca3330483ce0dd649c8d2 | Two circles \( C_{1} \) and \( C_{2} \) intersect at points \( A \) and \( B \). A line through \( B \) intersects \( C_{1} \) at \( C \) and \( C_{2} \) at \( D \). Another line through \( B \) intersects \( C_{1} \) at \( E \) and \( C_{2} \) at \( F \). The line \( CF \) intersects \( C_{1} \) at \( P \) and \( C_{2... | [
"90"
] | [
"Two circles \\( C_{1} \\) and \\( C_{2} \\) intersect at points \\( A \\) and \\( B \\). A line through \\( B \\) intersects \\( C_{1} \\) at \\( C \\) and \\( C_{2} \\) at \\( D \\). Another line through \\( B \\) intersects \\( C_{1} \\) at \\( E \\) and \\( C_{2} \\) at \\( F \\). The line \\( CF \\) intersects... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-2062e7d94e4e90b61946890a2a5335917410ba7121e7b68b5c3b41ab0ee84d03 | Two circles \( C_{1} \) and \( C_{2} \) touch externally at point \( A \). \( BC \) is a common tangent, touching \( C_{1} \) at \( B \) and \( C_{2} \) at \( C \). Let \( D \) be the point where the common tangent at \( A \) intersects \( BC \). If the radii of \( C_{1} \) and \( C_{2} \) are 4 and 9 respectively, fin... | [
"6"
] | [
"Two circles \\( C_{1} \\) and \\( C_{2} \\) touch externally at point \\( A \\). \\( BC \\) is a common tangent, touching \\( C_{1} \\) at \\( B \\) and \\( C_{2} \\) at \\( C \\). Let \\( D \\) be the point where the common tangent at \\( A \\) intersects \\( BC \\). If the radii of \\( C_{1} \\) and \\( C_{2} \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-8a5446c79e656b505db234bdc4d237b8b91931fee0363acf288ade56c630bcaa | Two circles \( G1 \) and \( G2 \) intersect at points \( M \) and \( N \). Let \( AB \) be the line tangent to these circles at points \( A \) and \( B \), respectively, such that \( M \) lies closer to \( AB \) than \( N \). Let \( CD \) be the line parallel to \( AB \) and passing through \( M \), with \( C \) on \( ... | [
"12"
] | [
"Two circles \\( G1 \\) and \\( G2 \\) intersect at points \\( M \\) and \\( N \\). Let \\( AB \\) be the line tangent to these circles at points \\( A \\) and \\( B \\), respectively, such that \\( M \\) lies closer to \\( AB \\) than \\( N \\). Let \\( CD \\) be the line parallel to \\( AB \\) and passing through... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-d39beec6fe9ca0e65279846186709b68f2a081465da863c4fe0e0638538fa823 | Two circles \( G_1 \) and \( G_2 \) 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 \( G_1 \) ... | [
"1"
] | [
"Two circles \\( G_1 \\) and \\( G_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 ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4ba3a86279e61ac52d42dbf82beae9659649f0e4315301677f682c74dac963e | Two circles \( G_1 \) and \( G_2 \) 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 \( G_1 \) ... | [
"10"
] | [
"Two circles \\( G_1 \\) and \\( G_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 ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-35cc09ba5caf2da10902608cb402827ade7d50bbd8d6e24eb90f578058f252e1 | Two circles \( K \) and \( K_1 \) touch each other externally. An equilateral triangle \( ABC \) is inscribed in circle \( K \), and points \( A_1 \), \( B_1 \), \( C_1 \) lie on circle \( K_1 \) such that \( AA_1 \), \( BB_1 \), and \( CC_1 \) are tangent to \( K_1 \). If the radius of circle \( K \) is \( 6 \) and th... | [
"8\\sqrt{3}"
] | [
"Two circles \\( K \\) and \\( K_1 \\) touch each other externally. An equilateral triangle \\( ABC \\) is inscribed in circle \\( K \\), and points \\( A_1 \\), \\( B_1 \\), \\( C_1 \\) lie on circle \\( K_1 \\) such that \\( AA_1 \\), \\( BB_1 \\), and \\( CC_1 \\) are tangent to \\( K_1 \\). If the radius of cir... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-36e09ff76b8e61dcc42d6b5e903f956cca4b8fe6d89b23be383c3ad16fbc0944 | Two circles \( K_1 \) and \( K_2 \) intersect at points \( A \) and \( B \). Points \( C \) and \( D \) are on \( K_1 \) and \( K_2 \) respectively, such that \( A \) is the midpoint of segment \( CD \). The extension of \( DB \) meets \( K_1 \) at point \( E \), and the extension of \( CB \) meets \( K_2 \) at point \... | [
"(2, 2)"
] | [
"Two circles \\( K_1 \\) and \\( K_2 \\) intersect at points \\( A \\) and \\( B \\). Points \\( C \\) and \\( D \\) are on \\( K_1 \\) and \\( K_2 \\) respectively, such that \\( A \\) is the midpoint of segment \\( CD \\). The extension of \\( DB \\) meets \\( K_1 \\) at point \\( E \\), and the extension of \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-d412af4a073713e679d96ffe5ea9d2ca74029565dbd50b418dc54ceda7edc66f | Two circles \( K_1 \) and \( K_2 \) of different radii intersect at two points \( A \) and \( B \). Let \( C \) and \( D \) be points on \( K_1 \) and \( K_2 \), respectively, such that \( A \) is the midpoint of the segment \( CD \). The extension of \( DB \) meets \( K_1 \) at another point \( E \), and the extension... | [
"(2, 2)"
] | [
"Two circles \\( K_1 \\) and \\( K_2 \\) of different radii intersect at two points \\( A \\) and \\( B \\). Let \\( C \\) and \\( D \\) be points on \\( K_1 \\) and \\( K_2 \\), respectively, such that \\( A \\) is the midpoint of the segment \\( CD \\). The extension of \\( DB \\) meets \\( K_1 \\) at another poi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-431e2fdeab1e1d4dc5f21e80a0be6eaac98a3e20fed504d56253566e256f09b6 | Two circles \( K_1 \) and \( K_2 \), centered at \( O_1 \) and \( O_2 \) with radii \( 1 \) and \( \sqrt{2} \) respectively, intersect at points \( A \) and \( B \). Let \( C \) be a point on \( K_2 \) such that the midpoint of \( AC \) lies on \( K_1 \). Given that \( O_1O_2 = 2 \), find the length of the segment \( A... | [
"\\(\\sqrt{14}\\)"
] | [
"Two circles \\( K_1 \\) and \\( K_2 \\), centered at \\( O_1 \\) and \\( O_2 \\) with radii \\( 1 \\) and \\( \\sqrt{2} \\) respectively, intersect at points \\( A \\) and \\( B \\). Let \\( C \\) be a point on \\( K_2 \\) such that the midpoint of \\( AC \\) lies on \\( K_1 \\). Given that \\( O_1O_2 = 2 \\), fin... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.