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-3f4746760522bbd49a8452e334b670988cb9e96e7d191370dff41dc539b8dc04 | Two circles $ w_1 $ and $ w_2 $ with centers at points $ O_1 $ and $ O_2 $, respectively, intersect at points $ A $ and $ B $. The line passing through the point $ B $ intersects the circles $ w_1 $ and $ w_2 $ at the points $ C $ and $ D $, respectively. The tangents to the circles $ w_1 $ and $ w_2 $, which are drawn... | [
"10"
] | [
"Two circles $ w_1 $ and $ w_2 $ with centers at points $ O_1 $ and $ O_2 $, respectively, intersect at points $ A $ and $ B $. The line passing through the point $ B $ intersects the circles $ w_1 $ and $ w_2 $ at the points $ C $ and $ D $, respectively. The tangents to the circles $ w_1 $ and $ w_2 $, which are ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-65da41376dd4687e89f0f43720b75d9720336d532d9d844b5307166f84599702 | Two circles $(O)$ and $(K)$ intersect at points $A$ and $B$ such that $K$ lies on $(O)$. The tangents at $A$ to $(O)$ meet $(K)$ again at $P$. The line $PB$ intersects $(O)$ again at $C$. A line through $P$ intersects $(O)$ at points $M$ and $N$. The perpendicular bisectors of $AM$ and $AN$ intersect $PA$ at points $Q$... | [
"\\dfrac{1}{2}"
] | [
"Two circles $(O)$ and $(K)$ intersect at points $A$ and $B$ such that $K$ lies on $(O)$. The tangents at $A$ to $(O)$ meet $(K)$ again at $P$. The line $PB$ intersects $(O)$ again at $C$. A line through $P$ intersects $(O)$ at points $M$ and $N$. The perpendicular bisectors of $AM$ and $AN$ intersect $PA$ at point... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cc511ee247a08e4bcd2b5d6e3e4f9f9745db6a995f1e904753ed4c8d850cfafc | Two circles $(O)$ and $(K)$ with radii $r$ and $R$ respectively, where $\frac{r}{R}=k$ (a constant ratio), intersect each other at points $M$ and $N$. Both circles are tangent to a fixed segment $AB=2a$. Find the locus of points $M$ and $N$. | [
"\\(\\left(Z, \\sqrt{ZA \\cdot ZB} \\right)\\)"
] | [
"Two circles $(O)$ and $(K)$ with radii $r$ and $R$ respectively, where $\\frac{r}{R}=k$ (a constant ratio), intersect each other at points $M$ and $N$. Both circles are tangent to a fixed segment $AB=2a$. Find the locus of points $M$ and $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a5d02d7561a2f0f0d89cc0125a3e115ed340e9b01d2419cdee950688beaf654b | Two circles $(O)$ and $(O')$ intersect at points $A$ and $B$. Points $P$ and $Q$ are taken on circles $(O)$ and $(O')$, respectively, such that $AP = AQ$. The line $PQ$ intersects circle $(O)$ at $M$ and circle $(O')$ at $N$. Let $E$ and $F$ be the centers of the arcs $BP$ and $BQ$ (not containing $A$), respectively. I... | [
"1"
] | [
"Two circles $(O)$ and $(O')$ intersect at points $A$ and $B$. Points $P$ and $Q$ are taken on circles $(O)$ and $(O')$, respectively, such that $AP = AQ$. The line $PQ$ intersects circle $(O)$ at $M$ and circle $(O')$ at $N$. Let $E$ and $F$ be the centers of the arcs $BP$ and $BQ$ (not containing $A$), respective... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-7e970521dc9e5a1d5848bac6a7e570334237bc072351e7fd8632dc9229ab8bfb | Two circles $(O,r)$ and $(K,r)$ are tangent externally at point $A$. From a point $P$ on circle $(O)$, tangents $PB$ and $PC$ are drawn to circle $(K)$, intersecting circle $(O)$ again at points $D$ and $E$ respectively. If $DB = 2EC$, calculate the ratio of the lengths $PA$ to $PD$. | [
"2"
] | [
"Two circles $(O,r)$ and $(K,r)$ are tangent externally at point $A$. From a point $P$ on circle $(O)$, tangents $PB$ and $PC$ are drawn to circle $(K)$, intersecting circle $(O)$ again at points $D$ and $E$ respectively. If $DB = 2EC$, calculate the ratio of the lengths $PA$ to $PD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6a15ba99c2c0ead3ab91f8a40ead4ed40eb24b722af6d1e0340d0f71a4d4a88c | Two circles $(O_1)$ and $(O_2)$ touch externally at point $C$ and internally at points $A$ and $B$ respectively with another circle $(O)$. Suppose that the common tangent of $(O_1)$ and $(O_2)$ at $C$ meets $(O)$ at $P$ such that $PA = PB$. If the radius of circle $(O)$ is $r$, and the distance from $P$ to line $AB$ is... | [
"\\dfrac{18}{25}"
] | [
"Two circles $(O_1)$ and $(O_2)$ touch externally at point $C$ and internally at points $A$ and $B$ respectively with another circle $(O)$. Suppose that the common tangent of $(O_1)$ and $(O_2)$ at $C$ meets $(O)$ at $P$ such that $PA = PB$. If the radius of circle $(O)$ is $r$, and the distance from $P$ to line $A... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-a655b8e9e78a492e9002b6c30a1859993696f3c08043b2be51ef44f7d6c254fd | Two circles $(O_1)$ and $(O_2)$ with respective centers $O_1$ and $O_2$ are given on a plane. Let $M_1$ and $M_2$ be points on $(O_1)$ and $(O_2)$ respectively, and let the lines $O_1M_1$ and $O_2M_2$ meet at $Q$. Starting simultaneously from these positions, the points $M_1$ and $M_2$ move clockwise on their own circl... | [
"Circle centered at the midpoint of \\( O_1O_2 \\) with a radius equal to the distance from this midpoint to the midpoint of \\( M_1M_2 \\)."
] | [
"Two circles $(O_1)$ and $(O_2)$ with respective centers $O_1$ and $O_2$ are given on a plane. Let $M_1$ and $M_2$ be points on $(O_1)$ and $(O_2)$ respectively, and let the lines $O_1M_1$ and $O_2M_2$ meet at $Q$. Starting simultaneously from these positions, the points $M_1$ and $M_2$ move clockwise on their own ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 4
} | 18 | 0 | 0 |
omr-ws-dfd0d74ded5dedd1a427354cb6d5e346074bad82f491f92c4d9a5767551f8c83 | Two circles $(O_1)$ and $(O_2)$ with respective centers $O_1$ and $O_2$ are given on a plane. Let $M_1$ and $M_2$ be points on $(O_1)$ and $(O_2)$ respectively, and let the lines $O_1M_1$ and $O_2M_2$ meet at $Q$. Starting simultaneously from these positions, the points $M_1$ and $M_2$ move clockwise on their own circl... | [
"\\dfrac{30}{7}"
] | [
"Two circles $(O_1)$ and $(O_2)$ with respective centers $O_1$ and $O_2$ are given on a plane. Let $M_1$ and $M_2$ be points on $(O_1)$ and $(O_2)$ respectively, and let the lines $O_1M_1$ and $O_2M_2$ meet at $Q$. Starting simultaneously from these positions, the points $M_1$ and $M_2$ move clockwise on their own ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-d9506d4e9b6f31f92f443b5b7831ab60c33ca9b713f5c30934574b143a7a220e | Two circles $(O_1, R_1)$ and $(O_2, R_2)$ intersect at points $A$ and $B$. A line $l$ passing through $B$ intersects $(O_1)$ at $P$ and $(O_2)$ at $Q$. Determine the position of line $l$ that maximizes the area of triangle $APQ$. | [
"\\( l \\) is parallel to \\( O_1O_2 \\)"
] | [
"Two circles $(O_1, R_1)$ and $(O_2, R_2)$ intersect at points $A$ and $B$. A line $l$ passing through $B$ intersects $(O_1)$ at $P$ and $(O_2)$ at $Q$. Determine the position of line $l$ that maximizes the area of triangle $APQ$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 10
} | 12 | 0 | 0 |
omr-ws-b6288e6022718ba0f51edbd1e8fbf1b53decbedd8a7a95157d6e37063b0bba0c | Two circles $(O_1, R_1)$ and $(O_2, R_2)$ intersect at points $A$ and $B$. A line $l$ passing through $B$ intersects $(O_1)$ at $P$ and $(O_2)$ at $Q$. Let $M$ be the midpoint of $PQ$. For a specific configuration where $R_1 = 5$, $R_2 = 12$, and $O_1O_2 = 13$, determine the distance from $M$ to the line $O_1O_2$ as $l... | [
"no_valid_answer_found"
] | [
"Two circles $(O_1, R_1)$ and $(O_2, R_2)$ intersect at points $A$ and $B$. A line $l$ passing through $B$ intersects $(O_1)$ at $P$ and $(O_2)$ at $Q$. Let $M$ be the midpoint of $PQ$. For a specific configuration where $R_1 = 5$, $R_2 = 12$, and $O_1O_2 = 13$, determine the distance from $M$ to the line $O_1O_2$ ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44665b9dc9c7cfc436dccf6f7202a81c04a4a1087cd39951fa68f11df046caf2 | Two circles $(U)$ and $(V)$ intersect at points $A$ and $B$. A line $d$ intersects $(U)$ at points $P$ and $Q$, and $(V)$ at points $R$ and $S$. Let $t_P, t_Q, t_R, t_S$ be the tangents to the circles at points $P, Q, R, S$ respectively. Another circle $(W)$ passes through points $A$ and $B$. If the circumcircle of the... | [
"1"
] | [
"Two circles $(U)$ and $(V)$ intersect at points $A$ and $B$. A line $d$ intersects $(U)$ at points $P$ and $Q$, and $(V)$ at points $R$ and $S$. Let $t_P, t_Q, t_R, t_S$ be the tangents to the circles at points $P, Q, R, S$ respectively. Another circle $(W)$ passes through points $A$ and $B$. If the circumcircle o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8610a7a82bc580162da26c740c629202d7cd6297ca1fed4f7dc6d77d0aab6af7 | Two circles $A$ and $B$ have radii $\sqrt{2}$ and $2$ units, respectively, and the distance between their centers is $2\sqrt{2}$ units. The circles intersect at points $P$ and $Q$. If $PR$ is a chord of circle $B$ that is bisected by circle $A$, find the length of $PR$. | [
"\\( 2\\sqrt{2} \\)"
] | [
"Two circles $A$ and $B$ have radii $\\sqrt{2}$ and $2$ units, respectively, and the distance between their centers is $2\\sqrt{2}$ units. The circles intersect at points $P$ and $Q$. If $PR$ is a chord of circle $B$ that is bisected by circle $A$, find the length of $PR$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 1
} | 23 | 0 | 0 |
omr-ws-bd23471d619716dcd07e358afb5294bc890c88718e2ffed48bd46ce087112064 | Two circles $A$ and $B$ with radii $a$ and $b$, respectively, intersect at points $P$ and $Q$. The distance between their centers is $d$. Segment $AB$ intersects the circles at points $X$ and $Y$. Find the length of $PQ$ in terms of $a$, $b$, and $d$. | [
"\\(2\\sqrt{a^2 - \\left(\\frac{a - b + d}{2}\\right)^2}\\)"
] | [
"Two circles $A$ and $B$ with radii $a$ and $b$, respectively, intersect at points $P$ and $Q$. The distance between their centers is $d$. Segment $AB$ intersects the circles at points $X$ and $Y$. Find the length of $PQ$ in terms of $a$, $b$, and $d$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df9fba0dc856c71851261b1d385bfe2a578c4d4da53b62db678e9cd373f6b943 | Two circles $A$ and $B$ with radii $a$ and $b$, respectively, intersect at points $P$ and $Q$. The distance between their centers is $d$. Segment $AB$ intersects the circles at points $X$ and $Y$. Find the length of $XY$. | [
"\\( a + b - d \\)"
] | [
"Two circles $A$ and $B$ with radii $a$ and $b$, respectively, intersect at points $P$ and $Q$. The distance between their centers is $d$. Segment $AB$ intersects the circles at points $X$ and $Y$. Find the length of $XY$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-c95fe57a8938d829d237b3a36a6379a3dc56928b8cb3fc2df946af18653a1cd9 | Two circles $A$ and $B$, each with radius 1, are tangent to each other and to a horizontal line. Circle $C_1$ with diameter $d_1$ is inscribed within circles $A$, $B$, and the horizontal line. Circle $C_2$ with diameter $d_2$ is inscribed within circles $A$, $B$, and $C_1$. Circle $C_3$ with diameter $d_3$ is inscribed... | [
"\\( \\frac{1}{2n(n+1)} \\)"
] | [
"Two circles $A$ and $B$, each with radius 1, are tangent to each other and to a horizontal line. Circle $C_1$ with diameter $d_1$ is inscribed within circles $A$, $B$, and the horizontal line. Circle $C_2$ with diameter $d_2$ is inscribed within circles $A$, $B$, and $C_1$. Circle $C_3$ with diameter $d_3$ is insc... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 6,
"tir": 2
} | 10 | 0 | 0 |
omr-ws-27c198dcd79d6473f677cf132a4adb79b8e9bd7f78e0a74ef4fe448f67302596 | 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 $QB = 1000$, calculate $AB$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4e2c1a76f83d14734bdc768741d1c99552860a44f694f7da38bc3245e4038aca | Two circles $C_1$ and $C_2$ with equations $x^2+y^2+ax+by+c=0$ and $x^2+y^2+kx+my+n=0$ respectively are known to be perpendicular. If $a=4$, $b=3$, $k=2$, and $m=5$, find the value of $c+n$. | [
"\\dfrac{23}{2}"
] | [
"Two circles $C_1$ and $C_2$ with equations $x^2+y^2+ax+by+c=0$ and $x^2+y^2+kx+my+n=0$ respectively are known to be perpendicular. If $a=4$, $b=3$, $k=2$, and $m=5$, find the value of $c+n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-36ac6471c45e22ab03a851105120b83f0d197c0c7ba2e50d5fdc450e72b20251 | 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 segment $CD$. The extension of $DB$ meets $K_1$ at another point $E$, and the extension of $CB$ meets $K_2$ at another point $F$. Let $l_1$ and... | [
"no_valid_answer_found"
] | [
"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 segment $CD$. The extension of $DB$ meets $K_1$ at another point $E$, and the extension of $CB$ meets $K_2$ at another point $F$. Let $l_1... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06377f29d58157f438fac25da22421ad94d62c925ca3510b5fa83139c501fa7a | Two circles $L_1$ and $L_2$ have radii $\sqrt{2}$ and $2$, respectively, and the distance between their centers is $2\sqrt{2}$. The circles intersect at points $P$ and $Q$. A chord $PR$ of circle $L_2$ is bisected by circle $L_1$. Find the length of chord $PR$. | [
"\\(\\sqrt{7}\\)"
] | [
"Two circles $L_1$ and $L_2$ have radii $\\sqrt{2}$ and $2$, respectively, and the distance between their centers is $2\\sqrt{2}$. The circles intersect at points $P$ and $Q$. A chord $PR$ of circle $L_2$ is bisected by circle $L_1$. Find the length of chord $PR$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 6,
"tir": 23
} | 60 | 0 | 0 |
omr-ws-011dd34064246b6bdaeedc36ad0433728ffb6480e9a53eb0cd604a028f62b231 | Two circles $O_1$ and $O_2$ touch each other externally at a point $T$. A quadrilateral $ABCD$ is inscribed in $O_1$, and the lines $DA$ and $CB$ are tangent to $O_2$ at points $E$ and $F$ respectively. Line $BN$ bisects $\angle ABF$ and meets segment $EF$ at $N$. Line $FT$ meets the arc $AT$ (not passing through the p... | [
"\\dfrac{r}{2}"
] | [
"Two circles $O_1$ and $O_2$ touch each other externally at a point $T$. A quadrilateral $ABCD$ is inscribed in $O_1$, and the lines $DA$ and $CB$ are tangent to $O_2$ at points $E$ and $F$ respectively. Line $BN$ bisects $\\angle ABF$ and meets segment $EF$ at $N$. Line $FT$ meets the arc $AT$ (not passing through... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c35becf4652ec4f726253c6409297625cd717119240d2947888e0af3be17b70 | Two circles $T_1$ and $T_2$ have radii $\sqrt{2}$ and $2$ units, respectively, and their centers are $2\sqrt{2}$ units apart. The circles intersect at points $P$ and $Q$. If $PR$ is a chord of $T_2$ that is bisected by $T_1$, find the length of $PR$. | [
"\\(\\sqrt{7}\\)"
] | [
"Two circles $T_1$ and $T_2$ have radii $\\sqrt{2}$ and $2$ units, respectively, and their centers are $2\\sqrt{2}$ units apart. The circles intersect at points $P$ and $Q$. If $PR$ is a chord of $T_2$ that is bisected by $T_1$, find the length of $PR$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 3,
"tir": 16
} | 35 | 0 | 0 |
omr-ws-2069971c62e47a9912e49bb512d71cbcca1a782acb55389b76c86153d2c734d4 | Two circles $\Gamma$ and $\Gamma'$ intersect at points $A$ and $D$. A line is tangent to $\Gamma$ at $E$ and to $\Gamma'$ at $F$. A line passing through point $D$ intersects $\Gamma$ at $B$ and $\Gamma'$ at $C$. Let $P$ be the second intersection point of the circumcircles of $\triangle BDE$ and $\triangle CDF$. If $AD... | [
"4"
] | [
"Two circles $\\Gamma$ and $\\Gamma'$ intersect at points $A$ and $D$. A line is tangent to $\\Gamma$ at $E$ and to $\\Gamma'$ at $F$. A line passing through point $D$ intersects $\\Gamma$ at $B$ and $\\Gamma'$ at $C$. Let $P$ be the second intersection point of the circumcircles of $\\triangle BDE$ and $\\triangle... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-32ee33880b150904eab3eecfc90890eddb82a609fa15018e528e1357b973a3f6 | Two circles $\Gamma$ and $\Omega$ intersect at points A and G. The line $t$ is tangent to $\Gamma$ at B and to $\Omega$ at C. If G is the barycenter (centroid) of triangle ABC, what is the most possible value of the angle $\angle BAC$? | [
"\\(60^\\circ\\)"
] | [
"Two circles $\\Gamma$ and $\\Omega$ intersect at points A and G. The line $t$ is tangent to $\\Gamma$ at B and to $\\Omega$ at C. If G is the barycenter (centroid) of triangle ABC, what is the most possible value of the angle $\\angle BAC$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 2,
"tir": 4
} | 16 | 0 | 0 |
omr-ws-465b35228b4d382fff8425ab601868ecbb78b8d3afd1fb3919ac5a1d191271c4 | Two circles $\Gamma$ and $\Omega$ intersect at two distinct points $A$ and $B$. Let $P$ be a point on $\Gamma$. The tangent at $P$ to $\Gamma$ meets $\Omega$ at the points $C$ and $D$, where $D$ lies between $P$ and $C$, and $ABCD$ is a convex quadrilateral. The lines $CA$ and $CB$ meet $\Gamma$ again at $E$ and $F$ re... | [
"r(3 + \\sqrt{3})"
] | [
"Two circles $\\Gamma$ and $\\Omega$ intersect at two distinct points $A$ and $B$. Let $P$ be a point on $\\Gamma$. The tangent at $P$ to $\\Gamma$ meets $\\Omega$ at the points $C$ and $D$, where $D$ lies between $P$ and $C$, and $ABCD$ is a convex quadrilateral. The lines $CA$ and $CB$ meet $\\Gamma$ again at $E$... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-25419cdb5482044bbfc3219c95a39277e34d93512dd72b6a210ea7d9ffbab8aa | Two circles $\Gamma$ and $\Sigma$ have centers $O$ and $O'$, respectively, with $O'$ lying on $\Gamma$. Let $A$ be a point on $\Sigma$, and let $M$ be the midpoint of $AO'$. Let $B$ be another point on $\Sigma$ such that $AB \parallel OM$. If the radius of $\Gamma$ is 10 and the radius of $\Sigma$ is 5, calculate the d... | [
"10"
] | [
"Two circles $\\Gamma$ and $\\Sigma$ have centers $O$ and $O'$, respectively, with $O'$ lying on $\\Gamma$. Let $A$ be a point on $\\Sigma$, and let $M$ be the midpoint of $AO'$. Let $B$ be another point on $\\Sigma$ such that $AB \\parallel OM$. If the radius of $\\Gamma$ is 10 and the radius of $\\Sigma$ is 5, ca... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-90165de6ec5e1158610284626fad39651ad1b2394740497dea1dc5a984a11437 | Two circles $\Gamma_1$ and $\Gamma_2$ are externally tangent to each other and are both internally tangent to another circle $\Gamma$ at points $P$ and $Q$ respectively. A chord $AB$ of $\Gamma$ touches $\Gamma_1$ and $\Gamma_2$ at points $M$ and $N$ respectively. Let $l$ be the common tangent of $\Gamma_1$ and $\Gamma... | [
"2\\sqrt{6}"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ are externally tangent to each other and are both internally tangent to another circle $\\Gamma$ at points $P$ and $Q$ respectively. A chord $AB$ of $\\Gamma$ touches $\\Gamma_1$ and $\\Gamma_2$ at points $M$ and $N$ respectively. Let $l$ be the common tangent of $\\Gamma_1$... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-1b85c763b0a67df15a8ba0762b9579a1502577ccd2e5a9fc42912272bd54cb4e | Two circles $\Gamma_1$ and $\Gamma_2$ are tangent at point $S$. Triangle $ABC$ is inscribed in $\Gamma_1$ and line $CB$ is tangent to $\Gamma_2$ at point $M$. Point $D$ is on line $CB$ such that $AD$ is tangent to $\Gamma_2$ at point $N$. Let line $MS$ meet $\Gamma_1$ at point $T$. If the circumradius of $\triangle ABC... | [
"\\dfrac{1}{2}"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ are tangent at point $S$. Triangle $ABC$ is inscribed in $\\Gamma_1$ and line $CB$ is tangent to $\\Gamma_2$ at point $M$. Point $D$ is on line $CB$ such that $AD$ is tangent to $\\Gamma_2$ at point $N$. Let line $MS$ meet $\\Gamma_1$ at point $T$. If the circumradius of $\\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-4dff0cd0f6fbd868c751d446084754dcb6905eb521154591b43495973f8a5f7d | Two circles $\Gamma_1$ and $\Gamma_2$ have common external tangents $\ell_1$ and $\ell_2$ meeting at $T$. Suppose $\ell_1$ touches $\Gamma_1$ at $A$ and $\ell_2$ touches $\Gamma_2$ at $B$. A circle $\Omega$ through $A$ and $B$ intersects $\Gamma_1$ again at $C$ and $\Gamma_2$ again at $D$, such that quadrilateral $ABCD... | [
"8"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ have common external tangents $\\ell_1$ and $\\ell_2$ meeting at $T$. Suppose $\\ell_1$ touches $\\Gamma_1$ at $A$ and $\\ell_2$ touches $\\Gamma_2$ at $B$. A circle $\\Omega$ through $A$ and $B$ intersects $\\Gamma_1$ again at $C$ and $\\Gamma_2$ again at $D$, such that qua... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22ae3316f3e8c163f3910ca36e19d4e7f0c0ce4c086f8213309631f1431fec68 | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$ with $AB = 10$. The line segment joining the centers of the two circles intersects $\Gamma_1$ and $\Gamma_2$ at $P$ and $Q$, respectively. If $PQ = 3$ and the radius of $\Gamma_1$ is 13, find the radius of $\Gamma_2$. | [
"\\(\\frac{29}{4}\\)"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $A$ and $B$ with $AB = 10$. The line segment joining the centers of the two circles intersects $\\Gamma_1$ and $\\Gamma_2$ at $P$ and $Q$, respectively. If $PQ = 3$ and the radius of $\\Gamma_1$ is 13, find the radius of $\\Gamma_2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-2e1f9c45b158c7ff9ccf4a5d25d65b08e84f35a9af6a50d80959980e3bf8d202 | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$. A line $\ell$ passing through $B$ intersects $\Gamma_1$ at $K$ and $\Gamma_2$ at $M$ (with $K, M \neq B$). A line $\ell_1$ parallel to $AM$ is tangent to $\Gamma_1$ at $Q$. The line $QA$ intersects $\Gamma_2$ at $R$ (with $R \neq A$). A line $\ell_2... | [
"\\theta"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $A$ and $B$. A line $\\ell$ passing through $B$ intersects $\\Gamma_1$ at $K$ and $\\Gamma_2$ at $M$ (with $K, M \\neq B$). A line $\\ell_1$ parallel to $AM$ is tangent to $\\Gamma_1$ at $Q$. The line $QA$ intersects $\\Gamma_2$ at $R$ (with $R \\neq A$).... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2871f19b4574c1eb4f3dd14ef9d7ed80f167a322d89aadeb8e7b8dd2f1790de | 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 $D$, with $B$ 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 from $AD$. The line $EA$ intersects $\Gamm... | [
"10"
] | [
"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 $D$, with $B$ 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 from $AD$. The line $EA$ interse... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f177899b06bb16902ca2cc9ff5d6b260b40926b63f8f7eb7aa927f8bd187fe28 | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ again at points $C$ and $D$ respectively, with $A$ between $C$ and $D$. The tangent at $A$ to $\Gamma_2$ intersects $\Gamma_1$ again at $E$. Let $F$ be a point on $\Gamma_2$ such that $F$ and $... | [
"\\angle ABC"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $A$ and $B$. A line through $A$ intersects $\\Gamma_1$ and $\\Gamma_2$ again at points $C$ and $D$ respectively, with $A$ between $C$ and $D$. The tangent at $A$ to $\\Gamma_2$ intersects $\\Gamma_1$ again at $E$. Let $F$ be a point on $\\Gamma_2$ such th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5084e19a09bdfeda55f079c9742ffa564657f4c03e19645329affd832ec0f751 | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ at $C$ and $\Gamma_2$ at $D$. The tangent line to $\Gamma_1$ at $C$ intersects $AB$ at $E$. The circumcircle of $\triangle CDE$ intersects $\Gamma_1$ and $\Gamma_2$ again at $F$ and $G$ respectively. Line $EF... | [
"2"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $A$ and $B$. A line through $A$ intersects $\\Gamma_1$ at $C$ and $\\Gamma_2$ at $D$. The tangent line to $\\Gamma_1$ at $C$ intersects $AB$ at $E$. The circumcircle of $\\triangle CDE$ intersects $\\Gamma_1$ and $\\Gamma_2$ again at $F$ and $G$ respectiv... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-800bc43660249b565f17fd80ec2f3d923d24ab006d4fb711b4c2a156771c9a4e | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$. A line through $B$ intersects $\Gamma_1$ and $\Gamma_2$ again at points $C$ and $D$, respectively. Another line through $B$ intersects $\Gamma_1$ and $\Gamma_2$ again at points $E$ and $F$, respectively. Line $CF$ intersects $\Gamma_1$ and $\Gamma_2... | [
"\\dfrac{1}{2}"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $A$ and $B$. A line through $B$ intersects $\\Gamma_1$ and $\\Gamma_2$ again at points $C$ and $D$, respectively. Another line through $B$ intersects $\\Gamma_1$ and $\\Gamma_2$ again at points $E$ and $F$, respectively. Line $CF$ intersects $\\Gamma_1$ a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-2a1574f21c7e3914d8376f2b3d8dc58f93820059e4390768dacdd776c2423c31 | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$. Point $P$ is not on the line $AB$. Line $AP$ intersects $\Gamma_1$ and $\Gamma_2$ again at points $K$ and $L$ respectively, and line $BP$ intersects $\Gamma_1$ and $\Gamma_2$ again at points $M$ and $N$ respectively. The centers of the circles circu... | [
"\\text{undefined}"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $A$ and $B$. Point $P$ is not on the line $AB$. Line $AP$ intersects $\\Gamma_1$ and $\\Gamma_2$ again at points $K$ and $L$ respectively, and line $BP$ intersects $\\Gamma_1$ and $\\Gamma_2$ again at points $M$ and $N$ respectively. The centers of the ci... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-ff46f6d5e77431e100005a22ea19225572228417dc0a31e477bc99f5040b832f | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$. Points $C$ and $D$ lie on $\Gamma_1$, and points $E$ and $F$ lie on $\Gamma_2$ such that $A$ and $B$ lie on segments $CE$ and $DF$ respectively, and segments $CE$ and $DF$ do not intersect. Let $CF$ meet $\Gamma_1$ and $\Gamma_2$ again at points $K$... | [
"1"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $A$ and $B$. Points $C$ and $D$ lie on $\\Gamma_1$, and points $E$ and $F$ lie on $\\Gamma_2$ such that $A$ and $B$ lie on segments $CE$ and $DF$ respectively, and segments $CE$ and $DF$ do not intersect. Let $CF$ meet $\\Gamma_1$ and $\\Gamma_2$ again at... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1d0fca90c4508475e0526e33da11aa5941175689f5857ed8c1b5bcff85a293d | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$. Rays $\ell_1$ and $\ell_2$ passing through $A$ intersect $\Gamma_1$ at points $D$ and $E$ respectively, and $\Gamma_2$ at points $F$ and $C$ respectively, with $E$ and $F$ lying on line segments $AC$ and $AD$ respectively. The lines $AB$, $CF$, and ... | [
"6"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $A$ and $B$. Rays $\\ell_1$ and $\\ell_2$ passing through $A$ intersect $\\Gamma_1$ at points $D$ and $E$ respectively, and $\\Gamma_2$ at points $F$ and $C$ respectively, with $E$ and $F$ lying on line segments $AC$ and $AD$ respectively. The lines $AB$,... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-ba7bfdfdc31a7b01a1cdad8174fc9da93b35cc3ac7ac7be46c322c4ef6b4d8be | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $Z$ (with $A \neq Z$). Let $B$ be the center of $\Gamma_1$ and $C$ be the center of $\Gamma_2$. The exterior angle bisector of $\angle BAC$ intersects $\Gamma_1$ again at $X$ and $\Gamma_2$ again at $Y$. Let $O$ be the circumcenter of $\triangle XYZ$. If... | [
"1"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $A$ and $Z$ (with $A \\neq Z$). Let $B$ be the center of $\\Gamma_1$ and $C$ be the center of $\\Gamma_2$. The exterior angle bisector of $\\angle BAC$ intersects $\\Gamma_1$ again at $X$ and $\\Gamma_2$ again at $Y$. Let $O$ be the circumcenter of $\\tri... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-71be7e784f968d8e629eddbad0eaddffd92983842be29aff05935db036a38514 | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $C$ and $D$. Line $L$ passes through $D$ and intersects $\Gamma_1$ and $\Gamma_2$ at points $A$ and $B$, respectively. Points $P$ and $Q$ are located on $\Gamma_1$ and $\Gamma_2$, respectively. Lines $PD$ and $AC$ intersect at $M$, and lines $QD$ and $BC$ inters... | [
"1"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $C$ and $D$. Line $L$ passes through $D$ and intersects $\\Gamma_1$ and $\\Gamma_2$ at points $A$ and $B$, respectively. Points $P$ and $Q$ are located on $\\Gamma_1$ and $\\Gamma_2$, respectively. Lines $PD$ and $AC$ intersect at $M$, and lines $QD$ and ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-d01cda3789dd88a79a5914ed1a4aa2dcabf5634ffcd93ad8f301e95da4a458ad | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $M$ and $N$. A line $\ell$ is tangent to $\Gamma_1$ at $A$ and to $\Gamma_2$ at $B$. The lines passing through $A$ and $B$ and perpendicular to $\ell$ intersect $MN$ at $C$ and $D$, respectively. If the length of $AB$ is $10$ units and the length of $CD$ is $14$... | [
"140"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $M$ and $N$. A line $\\ell$ is tangent to $\\Gamma_1$ at $A$ and to $\\Gamma_2$ at $B$. The lines passing through $A$ and $B$ and perpendicular to $\\ell$ intersect $MN$ at $C$ and $D$, respectively. If the length of $AB$ is $10$ units and the length of $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-5e0a90d6753715650425e2b278ffb4ff9fbd0f7727ac11ca045a363e07714eb4 | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $P$ and $Q$. A straight line passing through point $Q$ intersects circles $\Gamma_1$ and $\Gamma_2$ at points $A$ and $B$ respectively. Another straight line intersects the circles $\Gamma_1$ and $\Gamma_2$ at points $C$ and $D$ respectively. The angle bisector ... | [
"(5, 6)"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $P$ and $Q$. A straight line passing through point $Q$ intersects circles $\\Gamma_1$ and $\\Gamma_2$ at points $A$ and $B$ respectively. Another straight line intersects the circles $\\Gamma_1$ and $\\Gamma_2$ at points $C$ and $D$ respectively. The angl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dfe5c4850dfd077f5e59a5ab34e40390ef91d053c84a29e18e095bd4383e7bbe | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $P$ and $Q$. Points $M$ on $\Gamma_1$ and $N$ on $\Gamma_2$ are such that $P$ is the midpoint of $\overline{MN}$. The tangents to $\Gamma_1$ and $\Gamma_2$ at $M$ and $N$ respectively meet at a point $X$. If $PM = 12$, calculate the value of $PX \cdot PQ$. | [
"144"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $P$ and $Q$. Points $M$ on $\\Gamma_1$ and $N$ on $\\Gamma_2$ are such that $P$ is the midpoint of $\\overline{MN}$. The tangents to $\\Gamma_1$ and $\\Gamma_2$ at $M$ and $N$ respectively meet at a point $X$. If $PM = 12$, calculate the value of $PX \\cd... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8cb49ea2de9aa92ae6cba0665ce148d4d29292dd082c11c33e3a135173f4ceca | Two circles $\Gamma_1$ and $\Gamma_2$ meet at $A$ and $B$. A line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ respectively. Another line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $E$ and $F$ respectively. Line $CF$ meets $\Gamma_1$ and $\Gamma_2$ again at $P$ and $Q$ respectively. $M$ an... | [
"1"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ meet at $A$ and $B$. A line through $B$ meets $\\Gamma_1$ and $\\Gamma_2$ again at $C$ and $D$ respectively. Another line through $B$ meets $\\Gamma_1$ and $\\Gamma_2$ again at $E$ and $F$ respectively. Line $CF$ meets $\\Gamma_1$ and $\\Gamma_2$ again at $P$ and $Q$ respect... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-12ba592b111d2ca72d010262f2fd7d649c0ab04d826ec97223b8320e2a2f8fac | Two circles $\Gamma_1$ and $\Gamma_2$ touch a circle $\omega(O, R)$ externally and internally at $P_1$ and $P_2$ respectively. $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. Given $OA=a$, $OB=b$, and $AB=d$, find the length of $P_1P_2$. | [
"\\sqrt{a^2 + b^2 - d^2}"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ touch a circle $\\omega(O, R)$ externally and internally at $P_1$ and $P_2$ respectively. $\\Gamma_1$ and $\\Gamma_2$ intersect at $A$ and $B$. Given $OA=a$, $OB=b$, and $AB=d$, find the length of $P_1P_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-00a13777af9a5eaac4380c1aedfaf4ad4de37ec10c321c9392d92dddb5180aa3 | Two circles $\Gamma_1$ and $\Gamma_2$ with radii $r_1$ and $r_2$ ($r_2 > r_1$), respectively, are externally tangent. The straight line $t_1$ is tangent to the circles $\Gamma_1$ and $\Gamma_2$ at points $A$ and $D$, respectively. The parallel line $t_2$ to the line $t_1$ is tangent to the circle $\Gamma_1$ and interse... | [
"no_valid_answer_found"
] | [
"Two circles $\\Gamma_1$ and $\\Gamma_2$ with radii $r_1$ and $r_2$ ($r_2 > r_1$), respectively, are externally tangent. The straight line $t_1$ is tangent to the circles $\\Gamma_1$ and $\\Gamma_2$ at points $A$ and $D$, respectively. The parallel line $t_2$ to the line $t_1$ is tangent to the circle $\\Gamma_1$ a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a4c87e2ca0cfabfbf7b00995f08a462094ac33167163f7a6be6644f3686fc1a | Two circles $\Gamma_{1}$ and $\Gamma_2$ with radii $r_1$ and $r_2$ ($r_2 > r_1$), respectively, are externally tangent. The straight line $t_1$ is tangent to the circles $\Gamma_1$ and $\Gamma_2$ at points $A$ and $D$, respectively. The parallel line $t_2$ to the line $t_1$ is tangent to the circle $\Gamma_1$ and inter... | [
"2"
] | [
"Two circles $\\Gamma_{1}$ and $\\Gamma_2$ with radii $r_1$ and $r_2$ ($r_2 > r_1$), respectively, are externally tangent. The straight line $t_1$ is tangent to the circles $\\Gamma_1$ and $\\Gamma_2$ at points $A$ and $D$, respectively. The parallel line $t_2$ to the line $t_1$ is tangent to the circle $\\Gamma_1$... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80f58d74bda8310585362383d124495e5b4931dbcabc4318f1eaec47218ea569 | Two circles $\Gamma_{1}$ and $\Gamma_{2}$ intersect at points $D$ and $E$. Points $B$ and $C$ are on $\Gamma_{1}$ and $\Gamma_{2}$, respectively, such that $D$ is the midpoint of $BC$. The tangents to $\Gamma_{1}$ at $B$ and to $\Gamma_{2}$ at $C$ meet at point $A$. If $DC = 5$, calculate the value of $DA \cdot DE$. | [
"25"
] | [
"Two circles $\\Gamma_{1}$ and $\\Gamma_{2}$ intersect at points $D$ and $E$. Points $B$ and $C$ are on $\\Gamma_{1}$ and $\\Gamma_{2}$, respectively, such that $D$ is the midpoint of $BC$. The tangents to $\\Gamma_{1}$ at $B$ and to $\\Gamma_{2}$ at $C$ meet at point $A$. If $DC = 5$, calculate the value of $DA \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b6337e564c967482e9d6b615e5e880bb8dce0d38f3f24e192f5f29b32eb9d62 | Two circles $\Omega$ and $\Gamma$ are internally tangent at the point $B$. The chord $AC$ of $\Gamma$ is tangent to $\Omega$ at the point $L$, and the segments $AB$ and $BC$ intersect $\Omega$ at the points $M$ and $N$. Let $M_1$ and $N_1$ be the reflections of $M$ and $N$ about the line $BL$; and let $M_2$ and $N_2$ b... | [
"12"
] | [
"Two circles $\\Omega$ and $\\Gamma$ are internally tangent at the point $B$. The chord $AC$ of $\\Gamma$ is tangent to $\\Omega$ at the point $L$, and the segments $AB$ and $BC$ intersect $\\Omega$ at the points $M$ and $N$. Let $M_1$ and $N_1$ be the reflections of $M$ and $N$ about the line $BL$; and let $M_2$ a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aab907fbdb516179ac8052afc7ec585fae11a23f1e42aa8e17866ddcfbd8166b | Two circles $\Omega$ and $\omega$ are internally tangent at point $P$, with $\omega$ inside $\Omega$. Let $AB$ be a chord of $\Omega$ that is tangent to $\omega$ at $C$. The line $CP$ meets $\Omega$ again at point $Q$. The tangents from $Q$ to $\omega$ meet $\Omega$ again at points $R$ and $S$. Let $I$, $X$, and $Y$ be... | [
"1"
] | [
"Two circles $\\Omega$ and $\\omega$ are internally tangent at point $P$, with $\\omega$ inside $\\Omega$. Let $AB$ be a chord of $\\Omega$ that is tangent to $\\omega$ at $C$. The line $CP$ meets $\\Omega$ again at point $Q$. The tangents from $Q$ to $\\omega$ meet $\\Omega$ again at points $R$ and $S$. Let $I$, $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-f636c631db6c76888353129fbfbcfa9352c14718d1574173aa4a6a3f2df49f30 | Two circles $\Omega_1$ and $\Omega_2$ intersect at points $X$ and $Y$, with centers $O_1$ and $O_2$ respectively. Point $A$ lies on $\Omega_1$ and point $B$ lies on $\Omega_2$, such that $AB$ is a common tangent of $\Omega_1$ and $\Omega_2$. The tangents to $\Omega_1$ and $\Omega_2$ at their common point $X$ intersect ... | [
"(2, 1)"
] | [
"Two circles $\\Omega_1$ and $\\Omega_2$ intersect at points $X$ and $Y$, with centers $O_1$ and $O_2$ respectively. Point $A$ lies on $\\Omega_1$ and point $B$ lies on $\\Omega_2$, such that $AB$ is a common tangent of $\\Omega_1$ and $\\Omega_2$. The tangents to $\\Omega_1$ and $\\Omega_2$ at their common point $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-4bac61d5b3954c5be566b4558d88dc8a224e6f81199af767020457e109f258e4 | Two circles $\Omega_{1}$ and $\Omega_{2}$ touch internally the circle $\Omega$ at points $M$ and $N$, respectively, and the center of $\Omega_{2}$ lies on $\Omega_{1}$. The common chord of the circles $\Omega_{1}$ and $\Omega_{2}$ intersects $\Omega$ at points $A$ and $B$. The lines $MA$ and $MB$ intersect $\Omega_{1}$... | [
"\\dfrac{\\sqrt{3}}{2} r"
] | [
"Two circles $\\Omega_{1}$ and $\\Omega_{2}$ touch internally the circle $\\Omega$ at points $M$ and $N$, respectively, and the center of $\\Omega_{2}$ lies on $\\Omega_{1}$. The common chord of the circles $\\Omega_{1}$ and $\\Omega_{2}$ intersects $\\Omega$ at points $A$ and $B$. The lines $MA$ and $MB$ intersect... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-2160a27004d77153498b86ba3c7cc45cabf202697b589c2758d28226f34d4b91 | Two circles $\alpha$ and $\beta$ centered at $A$ and $B$ respectively intersect at $C$ and $D$. The segment $AB$ intersects the circles $\alpha$ and $\beta$ at $K$ and $L$ respectively. The ray $DK$ meets the circle $\beta$ again at a point $N$ and the ray $DL$ meets the circle $\alpha$ again at a point $M$. Let $I$ be... | [
"(2, 2\\sqrt{2} - 2)"
] | [
"Two circles $\\alpha$ and $\\beta$ centered at $A$ and $B$ respectively intersect at $C$ and $D$. The segment $AB$ intersects the circles $\\alpha$ and $\\beta$ at $K$ and $L$ respectively. The ray $DK$ meets the circle $\\beta$ again at a point $N$ and the ray $DL$ meets the circle $\\alpha$ again at a point $M$.... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b23d76918a12f2f66f5e7fb6ecef36ceb54a4e1a6aa49917af029e9e126df352 | Two circles $\alpha$ and $\beta$ intersect at points $P$ and $Q$. Let $R$ be a point on $\beta$, and let the lines connecting $R$ with $P$ and $Q$ intersect $\alpha$ again at $S$ and $T$ respectively. If the tangent to $\beta$ at $R$ intersects $ST$ at point $U$, calculate the ratio $\frac{RU}{UT}$ when $PQ$ is the dia... | [
"1"
] | [
"Two circles $\\alpha$ and $\\beta$ intersect at points $P$ and $Q$. Let $R$ be a point on $\\beta$, and let the lines connecting $R$ with $P$ and $Q$ intersect $\\alpha$ again at $S$ and $T$ respectively. If the tangent to $\\beta$ at $R$ intersects $ST$ at point $U$, calculate the ratio $\\frac{RU}{UT}$ when $PQ$... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-df43d49bc16e7808ac14f0475fe91ea80d2e89701a7b4beb3cd8c71ba2177ce4 | Two circles $\alpha$ and $\beta$ intersect at two points and are both tangent to a segment $AB$. The points $X$, $Y$, $Z$, and $W$ are the inverse points of $A$ and $B$ through $\alpha$ and $\beta$, respectively. It is given that $XYZW$ is inscribed in a circle with center $O$. If the intersection points of $\alpha$ an... | [
"74"
] | [
"Two circles $\\alpha$ and $\\beta$ intersect at two points and are both tangent to a segment $AB$. The points $X$, $Y$, $Z$, and $W$ are the inverse points of $A$ and $B$ through $\\alpha$ and $\\beta$, respectively. It is given that $XYZW$ is inscribed in a circle with center $O$. If the intersection points of $\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-7592c3a794755cc154ad9a6c15c044706aa0ff50409b1135a390be8e033530a1 | Two circles $\alpha$ and $\beta$ touch externally at the point $X$. Let $A$ and $P$ be two distinct points on $\alpha$ different from $X$. Let $AX$ and $PX$ meet $\beta$ again at points $B$ and $Q$ respectively. If the measure of $\angle APX$ is $30^\circ$, find the measure of $\angle BQX$. | [
"30^\\circ"
] | [
"Two circles $\\alpha$ and $\\beta$ touch externally at the point $X$. Let $A$ and $P$ be two distinct points on $\\alpha$ different from $X$. Let $AX$ and $PX$ meet $\\beta$ again at points $B$ and $Q$ respectively. If the measure of $\\angle APX$ is $30^\\circ$, find the measure of $\\angle BQX$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-63e0b3f1e638b3a82b5a308c65905402da2062b66ae06435a6db8eac63d6f6de | Two circles $\gamma$ and $\gamma'$ intersect at points $A$ and $B$. The tangent to $\gamma'$ at $A$ meets $\gamma$ again at $C$, and the tangent to $\gamma$ at $A$ meets $\gamma'$ again at $C'$. The line $CC'$ separates points $A$ and $B$. Let $\Gamma$ be the circle externally tangent to $\gamma$, externally tangent to... | [
"1"
] | [
"Two circles $\\gamma$ and $\\gamma'$ intersect at points $A$ and $B$. The tangent to $\\gamma'$ at $A$ meets $\\gamma$ again at $C$, and the tangent to $\\gamma$ at $A$ meets $\\gamma'$ again at $C'$. The line $CC'$ separates points $A$ and $B$. Let $\\Gamma$ be the circle externally tangent to $\\gamma$, external... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-3bb3aefc30509a4c97c78c70ff2c7418c665bd78f7028317ce84d767655be53b | Two circles $\gamma_1$ and $\gamma_2$ are contained inside the circle $\gamma$ and are tangent to $\gamma$ at the distinct points $M$ and $N$, respectively. $\gamma_1$ passes through the center of $\gamma_2$. The line passing through the two points of intersection of $\gamma_1$ and $\gamma$ meets at points $A$ and $B$.... | [
"6"
] | [
"Two circles $\\gamma_1$ and $\\gamma_2$ are contained inside the circle $\\gamma$ and are tangent to $\\gamma$ at the distinct points $M$ and $N$, respectively. $\\gamma_1$ passes through the center of $\\gamma_2$. The line passing through the two points of intersection of $\\gamma_1$ and $\\gamma$ meets at points... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-af411bd777c72b40d947c012871ebdd71bd800d7f80b9f52158b878d5488896d | Two circles $\gamma_1$ and $\gamma_2$ intersect at points $A$ and $B$. A line $r$ through $B$ meets $\gamma_1$ at $C$ and $\gamma_2$ at $D$ such that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $\gamma_1$ at $E$, at the smaller distance from $AD$. Line $EA$ meets $\gamma_2$ at $... | [
"\\theta"
] | [
"Two circles $\\gamma_1$ and $\\gamma_2$ intersect at points $A$ and $B$. A line $r$ through $B$ meets $\\gamma_1$ at $C$ and $\\gamma_2$ at $D$ such that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $\\gamma_1$ at $E$, at the smaller distance from $AD$. Line $EA$ meets $\\ga... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-9e55bd87296e0400e70a6d48aa4704fc2a5a8fcf941d1d8ccd5fef46e1dac9cf | Two circles $\gamma_1$ and $\gamma_2$ intersect at points $A$ and $B$. A line $r$ through $B$ meets $\gamma_1$ at $C$ and $\gamma_2$ at $D$ such that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $\gamma_1$ at $E$, at the smaller distance from $AD$. Line $EA$ meets $\gamma_2$ at $... | [
"\\dfrac{5 - \\sqrt{7}}{2}"
] | [
"Two circles $\\gamma_1$ and $\\gamma_2$ intersect at points $A$ and $B$. A line $r$ through $B$ meets $\\gamma_1$ at $C$ and $\\gamma_2$ at $D$ such that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $\\gamma_1$ at $E$, at the smaller distance from $AD$. Line $EA$ meets $\\ga... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95ea0e9cd25f9216be8ca5eaa3080d1904d51700c9211958371f441ec319861e | Two circles $\gamma_1$ and $\gamma_2$ intersect at points $A$ and $B$. Their centers are $O_1$ and $O_2$, respectively. Point $C$ is on $\gamma_1$, and segment $CO_2$ intersects $\gamma_2$ at point $X$. The circumcircles of triangles $AXO_1$, $BXO_1$, and $CXO_1$ intersect $\gamma_1$ again at points $D$, $E$, and $F$, ... | [
"1"
] | [
"Two circles $\\gamma_1$ and $\\gamma_2$ intersect at points $A$ and $B$. Their centers are $O_1$ and $O_2$, respectively. Point $C$ is on $\\gamma_1$, and segment $CO_2$ intersects $\\gamma_2$ at point $X$. The circumcircles of triangles $AXO_1$, $BXO_1$, and $CXO_1$ intersect $\\gamma_1$ again at points $D$, $E$,... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-3454a820d06d5935a962fe399a02b9651783d377087aa30ecc6ab8a9fbab91fd | Two circles $\gamma_1$ and $\gamma_2$ intersect at two points, one of which is $A$. The tangent to $\gamma_1$ at $A$ meets $\gamma_2$ again at $B$, and the tangent to $\gamma_2$ at $A$ meets $\gamma_1$ again at $C$. The line $BC$ meets $\gamma_1$ and $\gamma_2$ again at $D_1$ and $D_2$, respectively. Points $E_1$ and $... | [
"no_valid_answer_found"
] | [
"Two circles $\\gamma_1$ and $\\gamma_2$ intersect at two points, one of which is $A$. The tangent to $\\gamma_1$ at $A$ meets $\\gamma_2$ again at $B$, and the tangent to $\\gamma_2$ at $A$ meets $\\gamma_1$ again at $C$. The line $BC$ meets $\\gamma_1$ and $\\gamma_2$ again at $D_1$ and $D_2$, respectively. Point... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75fd825cd26a372dac89a0179989302e66ced2d167ea50b90aa3bd4301febe2f | Two circles $\gamma_1$ and $\gamma_2$ with centers $A$ and $B$ respectively intersect at points $C$ and $D$. The circumcircle of $\triangle ABC$ intersects $\gamma_1$ at $E$ and $\gamma_2$ at $F$, where the arc $EF$ not containing $C$ lies outside both $\gamma_1$ and $\gamma_2$. Let $M$ be the midpoint of the arc $EF$ ... | [
"90"
] | [
"Two circles $\\gamma_1$ and $\\gamma_2$ with centers $A$ and $B$ respectively intersect at points $C$ and $D$. The circumcircle of $\\triangle ABC$ intersects $\\gamma_1$ at $E$ and $\\gamma_2$ at $F$, where the arc $EF$ not containing $C$ lies outside both $\\gamma_1$ and $\\gamma_2$. Let $M$ be the midpoint of t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-66aa4ca84b8aeda5d1a9156ac2b01c0a6bd3cbcd7e3d823f19b15f5fb3b3f84c | Two circles $\gamma_1$ and $\gamma_2$ with centers at points $O_1$ and $O_2$ and radii $4$ and $9$ respectively touch externally at point $B$. A common external tangent to these circles touches the larger circle at point $N$ and intersects the common interior tangent at the point $K$. Find the radius of the circle insc... | [
"\\dfrac{18}{5}"
] | [
"Two circles $\\gamma_1$ and $\\gamma_2$ with centers at points $O_1$ and $O_2$ and radii $4$ and $9$ respectively touch externally at point $B$. A common external tangent to these circles touches the larger circle at point $N$ and intersects the common interior tangent at the point $K$. Find the radius of the circ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-aec74308722dfb23c60bc0242902f62954bc15345c9e692b5587ac900701ef28 | Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ intersect at points $M$ and $N$. A common tangent line touches $\mathcal{C}_1$ at $P$ and $\mathcal{C}_2$ at $Q$, with the tangent line being closer to $N$ than to $M$. The line $PN$ intersects circle $\mathcal{C}_2$ again at point $R$. If the coordinates of $M$, $N$, $P$... | [
"\\left( -\\dfrac{2}{17}, -\\dfrac{26}{17} \\right)"
] | [
"Two circles $\\mathcal{C}_1$ and $\\mathcal{C}_2$ intersect at points $M$ and $N$. A common tangent line touches $\\mathcal{C}_1$ at $P$ and $\\mathcal{C}_2$ at $Q$, with the tangent line being closer to $N$ than to $M$. The line $PN$ intersects circle $\\mathcal{C}_2$ again at point $R$. If the coordinates of $M$... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-a2fc7c60500e49e93352a91f8eeca71341c1592f54f084e168a060080bfb1855 | Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ intersect at points $P$ and $Q$. A line through $P$ intersects $\mathcal{C}_1$ and $\mathcal{C}_2$ again at points $A$ and $B$, respectively, and $X$ is the midpoint of $AB$. The line through $Q$ and $X$ intersects $\mathcal{C}_1$ and $\mathcal{C}_2$ again at points $Y$ a... | [
"\\dfrac{24\\sqrt{69}}{13}"
] | [
"Two circles $\\mathcal{C}_1$ and $\\mathcal{C}_2$ intersect at points $P$ and $Q$. A line through $P$ intersects $\\mathcal{C}_1$ and $\\mathcal{C}_2$ again at points $A$ and $B$, respectively, and $X$ is the midpoint of $AB$. The line through $Q$ and $X$ intersects $\\mathcal{C}_1$ and $\\mathcal{C}_2$ again at p... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-edbe6921402612e47a98f52db39688612dc34274b36a00f86a6c70f8019cdfe6 | Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ with centers $(1, 1)$ and $(4, 5)$ and radii $r_1 < r_2$, respectively, are drawn on the coordinate plane. The product of the slopes of the two common external tangents of $\mathcal{C}_1$ and $\mathcal{C}_2$ is $3$. If the value of $(r_2 - r_1)^2$ can be expressed as a co... | [
"13"
] | [
"Two circles $\\mathcal{C}_1$ and $\\mathcal{C}_2$ with centers $(1, 1)$ and $(4, 5)$ and radii $r_1 < r_2$, respectively, are drawn on the coordinate plane. The product of the slopes of the two common external tangents of $\\mathcal{C}_1$ and $\\mathcal{C}_2$ is $3$. If the value of $(r_2 - r_1)^2$ can be expresse... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 30,
"tir": 3
} | 33 | 0 | 0 |
omr-ws-f5d32a12aae3ef364b9bac8bddd269d4acfd25435a8ab73cbdbfc279ae3b1187 | Two circles $\omega A$ and $\omega B$ intersect at points $M$ and $N$. Lines $l_1$ and $l_2$ intersect $\omega A$ at points $S, T$ and $U, V$, respectively, and intersect $\omega B$ at points $X, Y$ and $Z, W$, respectively. The lines $SV$ and $TU$ intersect $XZ$ and $YW$ at points $A, B$ and $C, D$, respectively. Calc... | [
"1"
] | [
"Two circles $\\omega A$ and $\\omega B$ intersect at points $M$ and $N$. Lines $l_1$ and $l_2$ intersect $\\omega A$ at points $S, T$ and $U, V$, respectively, and intersect $\\omega B$ at points $X, Y$ and $Z, W$, respectively. The lines $SV$ and $TU$ intersect $XZ$ and $YW$ at points $A, B$ and $C, D$, respectiv... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-be58c9e448abe3dd8691f57227036dd6e569695b72e927a91c4be4a17c717109 | Two circles $\omega$ and $\Gamma$ (with the same diameter) intersect at points $X$ and $Y$. Consider a circle $r$ that is externally tangent to $\omega$ at point $T$ and internally tangent to $\Gamma$ at point $K$. Let $P$ be the intersection point of lines $XT$ and $YK$. If the radius of $\Gamma$ is $R$, find the leng... | [
"R"
] | [
"Two circles $\\omega$ and $\\Gamma$ (with the same diameter) intersect at points $X$ and $Y$. Consider a circle $r$ that is externally tangent to $\\omega$ at point $T$ and internally tangent to $\\Gamma$ at point $K$. Let $P$ be the intersection point of lines $XT$ and $YK$. If the radius of $\\Gamma$ is $R$, fin... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-691fb7fe028b4e73d6557c2d2a6ccfba7fbae254c1e9fb449b6e4727e16273f7 | Two circles $\omega$ and $\Omega$ intersect at distinct points $A$ and $B$. A line through $B$ intersects $\omega$ and $\Omega$ at points $C$ and $D$, respectively, such that $B$ lies between $C$ and $D$. Another line through $B$ intersects $\omega$ and $\Omega$ at points $E$ and $F$, respectively, such that $E$ lies b... | [
"60"
] | [
"Two circles $\\omega$ and $\\Omega$ intersect at distinct points $A$ and $B$. A line through $B$ intersects $\\omega$ and $\\Omega$ at points $C$ and $D$, respectively, such that $B$ lies between $C$ and $D$. Another line through $B$ intersects $\\omega$ and $\\Omega$ at points $E$ and $F$, respectively, such that... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-b908756ba69c996b33d9d578b8c0a2c78e9d6bbdc111b2aa9f4fab7aa978ebe6 | Two circles $\omega$ and $\Omega$ intersect at points $A$ and $B$. The tangent line to $\Omega$ at $A$ intersects $\omega$ at $C$ (where $C \neq A$). The tangent line to $\omega$ at $A$ intersects $BC$ at $D$. On the line $AB$, choose $E$ such that $DE$ is perpendicular to $AC$. Choose the point $F$ on the line $AD$ su... | [
"r"
] | [
"Two circles $\\omega$ and $\\Omega$ intersect at points $A$ and $B$. The tangent line to $\\Omega$ at $A$ intersects $\\omega$ at $C$ (where $C \\neq A$). The tangent line to $\\omega$ at $A$ intersects $BC$ at $D$. On the line $AB$, choose $E$ such that $DE$ is perpendicular to $AC$. Choose the point $F$ on the l... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-63587a4f40d8695cd4c9ac401138912dbf58e5270d5e568dcdd7555bcb122505 | Two circles $\omega$ and $\gamma$ have the same center, and $\gamma$ lies inside $\omega$. Let $O$ be an arbitrary point on $\omega$. $OA$ and $OB$ are the tangent lines through $O$ to $\gamma$. Circle with center $O$ and radius $OA$ meets $\omega$ at points $C$ and $D$. If the length of the midline of triangle $OAB$ p... | [
"2"
] | [
"Two circles $\\omega$ and $\\gamma$ have the same center, and $\\gamma$ lies inside $\\omega$. Let $O$ be an arbitrary point on $\\omega$. $OA$ and $OB$ are the tangent lines through $O$ to $\\gamma$. Circle with center $O$ and radius $OA$ meets $\\omega$ at points $C$ and $D$. If the length of the midline of tria... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-74438801adc0d2b42918bb1d9f5c75ee61c15296533203b37e1a8851da89fafb | Two circles $\omega_1$ and $\omega_2$ are drawn on a plane with points $A$ and $B$ on them such that the segment $AB$ has the greatest possible length. $AR$ and $AS$ are tangents from point $A$ to circle $\omega_2$, and $BP$ and $BQ$ are tangents from point $B$ to circle $\omega_1$. The circle $\alpha_1$ touches circle... | [
"1"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ are drawn on a plane with points $A$ and $B$ on them such that the segment $AB$ has the greatest possible length. $AR$ and $AS$ are tangents from point $A$ to circle $\\omega_2$, and $BP$ and $BQ$ are tangents from point $B$ to circle $\\omega_1$. The circle $\\alpha_1$ touc... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-fa2c5d4239e74ba39cbfab02559c37415bfe4c308edb3b3fa91c4bfb47246f24 | Two circles $\omega_1$ and $\omega_2$ are externally tangent at point $K$, with $\omega_1$ having a larger radius than $\omega_2$. 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 $P$, and ray $AK$ ... | [
"**\n - $m + n = 184 + 1 = 185$."
] | [
"Two circles $\\omega_1$ and $\\omega_2$ are externally tangent at point $K$, with $\\omega_1$ having a larger radius than $\\omega_2$. 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 $P$, a... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9d686223b7b3f22b0ac9846163b6784cea8a0c5527d19b4301d9f884b723b6ad | Two circles $\omega_1$ and $\omega_2$ are tangent at a point $A$ and lie inside a circle $\Omega$. Circle $\omega_1$ is tangent to $\Omega$ at point $B$, and circle $\omega_2$ is tangent to $\Omega$ at point $C$. Line $AC$ intersects $\omega_1$ again at point $D$. If $A$, $B$, and $C$ are not collinear and triangle $DB... | [
"no_valid_answer_found"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ are tangent at a point $A$ and lie inside a circle $\\Omega$. Circle $\\omega_1$ is tangent to $\\Omega$ at point $B$, and circle $\\omega_2$ is tangent to $\\Omega$ at point $C$. Line $AC$ intersects $\\omega_1$ again at point $D$. If $A$, $B$, and $C$ are not collinear and... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4eb1719ee6742cd9ae16c9406626d4b6a6d5e93c514baf9193a87f9462793280 | Two circles $\omega_1$ and $\omega_2$ are tangent externally at the point $P$. A tangent to $\omega_1$ at point $A$ intersects $\omega_2$ at points $B$ and $C$. Line $CP$ intersects $\omega_1$ again at point $D$. If the measure of angle $DPB$ is $2\theta$, find the value of $\tan(\theta)$ given that the radii of $\omeg... | [
"\\sqrt{3}"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ are tangent externally at the point $P$. A tangent to $\\omega_1$ at point $A$ intersects $\\omega_2$ at points $B$ and $C$. Line $CP$ intersects $\\omega_1$ again at point $D$. If the measure of angle $DPB$ is $2\\theta$, find the value of $\\tan(\\theta)$ given that the ra... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-50968c0d9d6338f51e096a12bfe8b3c30509b911e57e30dc7b6c874500721cd2 | Two circles $\omega_1$ and $\omega_2$ are tangent to line $\ell$ at points $A$ and $B$ respectively, and are externally tangent to each other at point $D$. Choose a point $E$ on the smaller arc $BD$ of circle $\omega_2$. Line $DE$ intersects circle $\omega_1$ again at point $C$. If $\angle ABE = \theta$, find the value... | [
"1"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ are tangent to line $\\ell$ at points $A$ and $B$ respectively, and are externally tangent to each other at point $D$. Choose a point $E$ on the smaller arc $BD$ of circle $\\omega_2$. Line $DE$ intersects circle $\\omega_1$ again at point $C$. If $\\angle ABE = \\theta$, fi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-cbb598f73424202fb2285b576edd0f06c4902638b35fd3d0bd201d4ea18275a8 | Two circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ intersect at points $A$ and $B$. Let $X$ be a point on $\omega_1$ and $Y$ be a point on $\omega_2$. Let $XY$ intersect $O_1O_2$ at $Q$. Let the circles $(QXO_1)$ and $(QYO_2)$ intersect at $R$. If the common external tangents of $\omega_1$ and $\omega_2$... | [
"90"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ centered at $O_1$ and $O_2$ intersect at points $A$ and $B$. Let $X$ be a point on $\\omega_1$ and $Y$ be a point on $\\omega_2$. Let $XY$ intersect $O_1O_2$ at $Q$. Let the circles $(QXO_1)$ and $(QYO_2)$ intersect at $R$. If the common external tangents of $\\omega_1$ and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-757d5d5c18b09cf9a715e84ad8bc70218a9c4f60bddab57cbbe73162f452f268 | Two circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$ respectively, and intersect at two points $X$ and $Y$. Let $A$ and $P$ be points outside and inside both circles, respectively. Let $D$ be the foot from $A$ to $O_1O_2$, and suppose line $AP$ intersects $\omega_1$ at $Q$ and $R$ and $\omega_2$ at $S$ an... | [
"2"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ have centers $O_1$ and $O_2$ respectively, and intersect at two points $X$ and $Y$. Let $A$ and $P$ be points outside and inside both circles, respectively. Let $D$ be the foot from $A$ to $O_1O_2$, and suppose line $AP$ intersects $\\omega_1$ at $Q$ and $R$ and $\\omega_2$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-880b4dcb56b334b71089dd137089512bc01d2b44493189d330645cd9a5a5cf94 | Two circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$ with radii $25$ and $39$ 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 $S_1$ and $T_1$, and tangents from $O_1$ to $\omega_2$ meet $\omega_2$ at $... | [
"787"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ have centers $O_1$ and $O_2$ with radii $25$ and $39$ 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 $S_1$ and $T_1$, and tangents from $O_1$ to $\\omega_2$ meet $\\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-6d245e2148588799c0d40867d70f8f62cbb4746823ea50d6ff013690c2898c9e | Two circles $\omega_1$ and $\omega_2$ have radii $2$ and $5$, respectively, and are internally tangent at a point $A$. Let $B$ be a point on $\omega_1$ such that the tangent at $B$ to $\omega_1$, denoted by $\ell$, intersects $\omega_2$ at a point diametrically opposite $A$. Let $C$ be the reflection of $A$ over $B$, a... | [
"15"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ have radii $2$ and $5$, respectively, and are internally tangent at a point $A$. Let $B$ be a point on $\\omega_1$ such that the tangent at $B$ to $\\omega_1$, denoted by $\\ell$, intersects $\\omega_2$ at a point diametrically opposite $A$. Let $C$ be the reflection of $A$ ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 1
} | 20 | 0 | 0 |
omr-ws-75ed0f17e91645ea2510c7462b8c9e33b838ade85f64d5f4f5375a591b7795bd | Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A common external tangent line of these circles intersects $\omega_1$ at $C$ and $\omega_2$ at $D$, with $CD$ closer to $B$ than $A$. $BC$ meets $AD$ at $E$, $BD$ meets $AC$ at $F$, and $EF$ meets $AB$ at $N$. Let $K$ be the foot of the perpendicula... | [
"30"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. A common external tangent line of these circles intersects $\\omega_1$ at $C$ and $\\omega_2$ at $D$, with $CD$ closer to $B$ than $A$. $BC$ meets $AD$ at $E$, $BD$ meets $AC$ at $F$, and $EF$ meets $AB$ at $N$. Let $K$ be the foot of the per... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-22997d3c18525c1525d20f6ced69bb45ff330365f22041d2955d03884369ac09 | Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A common external tangent line of these circles intersects $\omega_1$ at $C$ and $\omega_2$ at $D$, with $CD$ closer to $B$ than $A$. $BC$ meets $AD$ at $E$, $BD$ meets $AC$ at $F$, and $EF$ meets $AB$ at $N$. Let $K$ be the foot of the perpendicula... | [
"20"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. A common external tangent line of these circles intersects $\\omega_1$ at $C$ and $\\omega_2$ at $D$, with $CD$ closer to $B$ than $A$. $BC$ meets $AD$ at $E$, $BD$ meets $AC$ at $F$, and $EF$ meets $AB$ at $N$. Let $K$ be the foot of the per... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-9f71f084cf4345b4e4daa5880f3f55392996561e7ff089046b8e86eac21119c4 | Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A fixed line $\ell_1$ passes through $A$ and intersects $\omega_1$ at $P$ and $\omega_2$ at $Q$. A variable line $\ell_2$ also passes through $A$ and intersects $\omega_1$ at $T$ and $\omega_2$ at $S$. Points $M$ and $N$ are the midpoints of segment... | [
"no_valid_answer_found"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. A fixed line $\\ell_1$ passes through $A$ and intersects $\\omega_1$ at $P$ and $\\omega_2$ at $Q$. A variable line $\\ell_2$ also passes through $A$ and intersects $\\omega_1$ at $T$ and $\\omega_2$ at $S$. Points $M$ and $N$ are the midpoin... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-821031c40b9f762af25fc262e3df2a8650fb5e817e8110ccd04b6425a3e5c087 | Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line $\ell_1$ through $A$ intersects $\omega_1$ and $\omega_2$ again at $C$ and $E$, respectively, and a point $G$ is chosen on this line between $A$ and $E$. A line $\ell_2$ through $B$ intersects $\omega_1$ and $\omega_2$ again at $D$ and $F$, r... | [
"1"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. A line $\\ell_1$ through $A$ intersects $\\omega_1$ and $\\omega_2$ again at $C$ and $E$, respectively, and a point $G$ is chosen on this line between $A$ and $E$. A line $\\ell_2$ through $B$ intersects $\\omega_1$ and $\\omega_2$ again at $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ae3242b98f7513f8bc81a543656463905f99da76288ab2be17dac9deff3c831 | Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line $l$ intersects $\omega_1$ at points $C$ and $D$ and $\omega_2$ at points $E$ and $F$. Let $P = (ACE) \cap (ADF)$, $Q = (BCE) \cap (BDF)$, $R = (ACF) \cap (ADE)$, and $S = (BCF) \cap (BDE)$. If $PRQS$ is an isosceles trapezoid, find the value ... | [
"2"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. A line $l$ intersects $\\omega_1$ at points $C$ and $D$ and $\\omega_2$ at points $E$ and $F$. Let $P = (ACE) \\cap (ADF)$, $Q = (BCE) \\cap (BDF)$, $R = (ACF) \\cap (ADE)$, and $S = (BCF) \\cap (BDE)$. If $PRQS$ is an isosceles trapezoid, fi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-34ee921fa0e9399a1d8e4863df384c5ef03fb6ae1787d8c31fd7a2fe66daf46d | Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A random line passing through $B$ intersects $\omega_1$ at $C$ and $\omega_2$ at $D$. Points $E$ and $Z$ are chosen on $\omega_1$ and $\omega_2$ respectively such that $CE = CB$ and $BD = DZ$. Suppose the line $BZ$ intersects $\omega_1$ again at $P$... | [
"1"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. A random line passing through $B$ intersects $\\omega_1$ at $C$ and $\\omega_2$ at $D$. Points $E$ and $Z$ are chosen on $\\omega_1$ and $\\omega_2$ respectively such that $CE = CB$ and $BD = DZ$. Suppose the line $BZ$ intersects $\\omega_1$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-8ba8b626180d25ba1c6fe857818f713f5da08144d1289fcd3a0750128f86eee0 | Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. An arbitrary line through $B$ meets $\omega_1$ at $C$ and $\omega_2$ at $D$. Points $E$ and $F$ are chosen on $\omega_1$ and $\omega_2$ respectively such that $CE = CB$ and $BD = DF$. Suppose that $BF$ meets $\omega_1$ at $P$, and $BE$ meets $\omega... | [
"-27"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. An arbitrary line through $B$ meets $\\omega_1$ at $C$ and $\\omega_2$ at $D$. Points $E$ and $F$ are chosen on $\\omega_1$ and $\\omega_2$ respectively such that $CE = CB$ and $BD = DF$. Suppose that $BF$ meets $\\omega_1$ at $P$, and $BE$ m... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-fa4efbeaa431d0867703311cf5cd1a63fdf8947785ee16d7f68e0604c55d0d81 | Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Let $PQ$ be a common tangent line of these circles with $P \in \omega_1$ and $Q \in \omega_2$. An arbitrary point $X$ lies on $\omega_1$. Line $AX$ intersects $\omega_2$ at $Y$ (other than $A$). Point $Y' \neq Y$ lies on $\omega_2$ such that $QY = Q... | [
"10"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. Let $PQ$ be a common tangent line of these circles with $P \\in \\omega_1$ and $Q \\in \\omega_2$. An arbitrary point $X$ lies on $\\omega_1$. Line $AX$ intersects $\\omega_2$ at $Y$ (other than $A$). Point $Y' \\neq Y$ lies on $\\omega_2$ su... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-aa66e0681a183d4f9baa7a41149f4c6c6cffb3a381bfd18b19ed0ea35cf0b5c6 | Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Line $\ell$ is tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$ so that $A$ is closer to $\ell$ than $B$. Let $X$ and $Y$ be points on major arcs $\overarc{PA}$ (on $\omega_1$) and $AQ$ (on $\omega_2$), respectively, such that $AX/PX = AY/QY = ... | [
"2"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. Line $\\ell$ is tangent to $\\omega_1$ at $P$ and to $\\omega_2$ at $Q$ so that $A$ is closer to $\\ell$ than $B$. Let $X$ and $Y$ be points on major arcs $\\overarc{PA}$ (on $\\omega_1$) and $AQ$ (on $\\omega_2$), respectively, such that $AX... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-340f81a8321234883201df9b3bfcb74bad3dc7943f94db71bbde6899e011578e | Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. The diameter $BK$ of $\omega_1$ intersects $\omega_2$ at point $D$ other than $B$, and the diameter $BP$ of $\omega_2$ intersects $\omega_1$ at point $C$ other than $B$, with $C$ and $D$ on opposite sides of the line $AB$. The line $AD$ intersects $... | [
"10"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. The diameter $BK$ of $\\omega_1$ intersects $\\omega_2$ at point $D$ other than $B$, and the diameter $BP$ of $\\omega_2$ intersects $\\omega_1$ at point $C$ other than $B$, with $C$ and $D$ on opposite sides of the line $AB$. The line $AD$ i... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-c1714f5e7185dff1d2c110bcdd2a4b1a1ec7339f69004c21c7f93ec3747c90e7 | Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Their common external tangent touches $\omega_1$ at $X$ and $\omega_2$ at $Y$. A point $Z$ is chosen on line $XY$. Circle $\Omega$, passing through $A$, $B$, and $Z$, intersects line $XY$ at point $W \ne Z$. Let $M$ be the midpoint of segment $ZW$. ... | [
"1"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. Their common external tangent touches $\\omega_1$ at $X$ and $\\omega_2$ at $Y$. A point $Z$ is chosen on line $XY$. Circle $\\Omega$, passing through $A$, $B$, and $Z$, intersects line $XY$ at point $W \\ne Z$. Let $M$ be the midpoint of seg... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-f362c9411ba8565224d953e4a7b4ffa07210dbc9b5dee57e35fd5a5943237429 | Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $G$. One of the common tangents of the circles intersects $\omega_1$ at $B$ and $\omega_2$ at $C$. Suppose $G$ is the barycenter (centroid) of triangle $ABC$. What is the greatest value that $\angle BAC$ can assume? | [
"\\(90^\\circ\\)"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $G$. One of the common tangents of the circles intersects $\\omega_1$ at $B$ and $\\omega_2$ at $C$. Suppose $G$ is the barycenter (centroid) of triangle $ABC$. What is the greatest value that $\\angle BAC$ can assume?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-639027ef4849018d502355795cc4e2c282f51cacd35ca87823560e5b4d3a0313 | 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 $\omega_1$ diametrically opposite to $Q$. $PA$ interse... | [
"2"
] | [
"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 $\\omega_1$ diametrically opposite to $Q$. ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-a11508db372cbd17ea7023cf44605d808de599515889544ba190ba03a60a4356 | Two circles $\omega_1$ and $\omega_2$ intersect at points $C$ and $D$. A line through $D$ intersects $\omega_1$ at $A$ and $\omega_2$ at $B$. Points $P$ and $Q$ are chosen on arcs $ADC$ and $BDC$, respectively. Lines $AC$ and $PD$ intersect at $M$, and lines $BC$ and $QD$ intersect at $N$. If $O$ is the circumcenter of... | [
"90"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $C$ and $D$. A line through $D$ intersects $\\omega_1$ at $A$ and $\\omega_2$ at $B$. Points $P$ and $Q$ are chosen on arcs $ADC$ and $BDC$, respectively. Lines $AC$ and $PD$ intersect at $M$, and lines $BC$ and $QD$ intersect at $N$. If $O$ is the circum... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-f87d7b8ec1b3bdb0ab477468c0ae679a38c12db2d009df5a0215c248e4dc75ac | Two circles $\omega_1$ and $\omega_2$ intersect at points $C$ and $D$. Let $M$ be the midpoint of $CD$. Points $P$ and $Q$ lie on $\omega_1$ and $\omega_2$, respectively, such that $PQ$ is a common external tangent to the two circles. Let $B$ be the foot of the perpendicular from $C$ to $PQ$, and let $A$ be the reflect... | [
"1"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $C$ and $D$. Let $M$ be the midpoint of $CD$. Points $P$ and $Q$ lie on $\\omega_1$ and $\\omega_2$, respectively, such that $PQ$ is a common external tangent to the two circles. Let $B$ be the foot of the perpendicular from $C$ to $PQ$, and let $A$ be th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f1c5997b4258cf7cf592dc2b0f4f039b3ab42679307db5faac31160135bccb6c | Two circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q$. A line $l$ intersects $\omega_1$ at $A$ and $B$, and $\omega_2$ at $C$ and $D$. The line $PA$ intersects $\omega_2$ again at $S \neq P$, and the line $PC$ intersects $\omega_1$ again at $T \neq P$. Let $O$ be the circumcenter of triangle $\triangle ... | [
"-96"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $P$ and $Q$. A line $l$ intersects $\\omega_1$ at $A$ and $B$, and $\\omega_2$ at $C$ and $D$. The line $PA$ intersects $\\omega_2$ again at $S \\neq P$, and the line $PC$ intersects $\\omega_1$ again at $T \\neq P$. Let $O$ be the circumcenter of triangl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-2463b1e35aafb835c72f34e8443a93421956501dc8fbdd84a1f4c17fb7634688 | Two circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q$. An arbitrary line passing through $P$ intersects $\omega_1$ and $\omega_2$ at points $A$ and $B$, respectively. Another line parallel to $AB$ intersects $\omega_1$ at points $D$ and $F$ and $\omega_2$ at points $E$ and $C$ such that $E$ and $F$ lie ... | [
"1"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $P$ and $Q$. An arbitrary line passing through $P$ intersects $\\omega_1$ and $\\omega_2$ at points $A$ and $B$, respectively. Another line parallel to $AB$ intersects $\\omega_1$ at points $D$ and $F$ and $\\omega_2$ at points $E$ and $C$ such that $E$ a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-70e67290cf944f6e49f45cc7a46753abf288d229fcea4f9183971b3df335dda6 | Two circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q$. The common tangent to $\omega_1$ and $\omega_2$ which is closer to $P$ touches $\omega_1$ and $\omega_2$ at $A$ and $B$ respectively. The tangent line to $\omega_1$ at $P$ intersects $\omega_2$ again at $C$. Lines $AP$ and $BC$ intersect at $R$. Let... | [
"\\dfrac{r_2}{r_1}"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $P$ and $Q$. The common tangent to $\\omega_1$ and $\\omega_2$ which is closer to $P$ touches $\\omega_1$ and $\\omega_2$ at $A$ and $B$ respectively. The tangent line to $\\omega_1$ at $P$ intersects $\\omega_2$ again at $C$. Lines $AP$ and $BC$ intersec... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-979f384ac38c6348a4d2a28f9221976c727672084bb347532842f28686171e38 | Two circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$ and are internally tangent to a larger circle $\Omega$ at points $A$ and $B$, respectively. The line $AB$ intersects $\omega_1$ and $\omega_2$ again at points $C$ and $D$, respectively. The circle inscribed in the curved triangle $CDX$ touches the si... | [
"\\theta"
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at points $X$ and $Y$ and are internally tangent to a larger circle $\\Omega$ at points $A$ and $B$, respectively. The line $AB$ intersects $\\omega_1$ and $\\omega_2$ again at points $C$ and $D$, respectively. The circle inscribed in the curved triangle $CDX$ touc... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6f8c4fd13c1fe6bec5a4820cf1f8c2057a58b6938482ef64789d921abf7294ed | Two circles $\omega_1$ and $\omega_2$ intersect at two points and are tangent to line $g$ at points $A$ and $B$ respectively. Let the centers of $\omega_1$ and $\omega_2$ be $O_1$ and $O_2$ respectively. Line $O_1O_2$ intersects $\omega_1$ at point $C$ outside $\omega_2$ and intersects $\omega_2$ at point $D$ inside $\... | [
""
] | [
"Two circles $\\omega_1$ and $\\omega_2$ intersect at two points and are tangent to line $g$ at points $A$ and $B$ respectively. Let the centers of $\\omega_1$ and $\\omega_2$ be $O_1$ and $O_2$ respectively. Line $O_1O_2$ intersects $\\omega_1$ at point $C$ outside $\\omega_2$ and intersects $\\omega_2$ at point $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
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