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-ed3dadae1f5b50e1d54161f9ec050f98ecf961f47b8af3d230667403b85bcd4d | Two circles \( O \) and \( O' \) are internally tangent at point \( P \), with \( O \) being the larger circle. \( AB \) is a chord of circle \( O \) and a tangent to circle \( O' \) at point \( C \). If the radius of circle \( O \) is 10 and the radius of circle \( O' \) is 4, find the length of \( PC \). | [
"4\\sqrt{2}"
] | [
"Two circles \\( O \\) and \\( O' \\) are internally tangent at point \\( P \\), with \\( O \\) being the larger circle. \\( AB \\) is a chord of circle \\( O \\) and a tangent to circle \\( O' \\) at point \\( C \\). If the radius of circle \\( O \\) is 10 and the radius of circle \\( O' \\) is 4, find the length ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-44870936536c2cca625721d8fd25040612f5a3bd5804890a94d4bd77fc331bbd | Two circles \( O \) and \( O' \) intersect at points \( A \) and \( B \). Let \( CD \) be a chord of circle \( O \). Define \( E \) as the intersection of \( CA \) with circle \( O' \) and \( F \) as the intersection of \( AD \) with circle \( O' \). If circles \( O \) and \( O' \) are orthogonal, calculate the measure... | [
"45^\\circ"
] | [
"Two circles \\( O \\) and \\( O' \\) intersect at points \\( A \\) and \\( B \\). Let \\( CD \\) be a chord of circle \\( O \\). Define \\( E \\) as the intersection of \\( CA \\) with circle \\( O' \\) and \\( F \\) as the intersection of \\( AD \\) with circle \\( O' \\). If circles \\( O \\) and \\( O' \\) are ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-a5db123060901663a41170e4720c425289a4a3a1747eeb1d96c3ce8f73be2d59 | Two circles \( O \) and \( O' \) with the same radius intersect at points \( A \) and \( B \). Points \( P \) and \( Q \) are on circles \( O \) and \( O' \) respectively, and \( P \neq A, B \) and \( Q \neq A, B \). Point \( R \) is chosen such that \( PAQR \) forms a parallelogram. Given that points \( B, R, P, Q \) ... | [
"2\\sqrt{r^2 - \\left( \\dfrac{OO'}{2} \\right)^2}"
] | [
"Two circles \\( O \\) and \\( O' \\) with the same radius intersect at points \\( A \\) and \\( B \\). Points \\( P \\) and \\( Q \\) are on circles \\( O \\) and \\( O' \\) respectively, and \\( P \\neq A, B \\) and \\( Q \\neq A, B \\). Point \\( R \\) is chosen such that \\( PAQR \\) forms a parallelogram. Give... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-b5c3b6c279ac5f4cc18f2b848e29b32116584a127c1a3837332355dd269d3f11 | Two circles \( O \) and \( O' \) with the same radius intersect at points \( A \) and \( B \). Points \( P \) and \( Q \) are on circles \( O \) and \( O' \) respectively, and \( P \neq A, B \) and \( Q \neq A, B \). Point \( R \) is chosen such that \( PAQR \) is a parallelogram. Given that \( B, R, P, Q \) are cyclic... | [
"10"
] | [
"Two circles \\( O \\) and \\( O' \\) with the same radius intersect at points \\( A \\) and \\( B \\). Points \\( P \\) and \\( Q \\) are on circles \\( O \\) and \\( O' \\) respectively, and \\( P \\neq A, B \\) and \\( Q \\neq A, B \\). Point \\( R \\) is chosen such that \\( PAQR \\) is a parallelogram. Given t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-90fa0878fa67e65ac6b4f77afe97403a67bbf72356e919d090ee4fa5f8e8ee64 | Two circles \( O_1 \) and \( O_2 \) are externally tangent to each other and both are tangent to a line at points \( A \) and \( B \), respectively. Let \( AP \) be the diameter of circle \( O_1 \) and let \( Q \) be a point such that the line \( PQ \) is tangent to circle \( O_2 \). If the radius of circle \( O_1 \) i... | [
"6"
] | [
"Two circles \\( O_1 \\) and \\( O_2 \\) are externally tangent to each other and both are tangent to a line at points \\( A \\) and \\( B \\), respectively. Let \\( AP \\) be the diameter of circle \\( O_1 \\) and let \\( Q \\) be a point such that the line \\( PQ \\) is tangent to circle \\( O_2 \\). If the radiu... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-09f6d43d169036731203d9669e9e337ad51d305d6f61c7b167b667b79379d320 | Two circles \( P \) and \( Q \) with equal radii \( r \) pass through each other's centers and intersect at points \( A \) and \( B \). Find the area of the circle that is tangent to the quadrilateral \( PAQB \) and whose center lies on the midpoint of \( AB \) in terms of \( r \). | [
"\\(\\frac{3\\pi r^2}{16}\\)"
] | [
"Two circles \\( P \\) and \\( Q \\) with equal radii \\( r \\) pass through each other's centers and intersect at points \\( A \\) and \\( B \\). Find the area of the circle that is tangent to the quadrilateral \\( PAQB \\) and whose center lies on the midpoint of \\( AB \\) in terms of \\( r \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-090901a5e37125cbe1849c13dc6b92c939e5db3c9d0dacadc445b7802f4cea78 | Two circles \( R_1 \) and \( R_2 \) intersect at points \( X \) and \( Y \). Circle \( R \) is internally tangent to \( R_1 \) and \( R_2 \) at points \( M \) and \( N \), respectively, with \( R > R_1, R_2 \). Line \( XY \) intersects circle \( R \) at point \( A \). Line \( AM \) intersects \( R_1 \) at point \( C \)... | [
"no_valid_answer_found"
] | [
"Two circles \\( R_1 \\) and \\( R_2 \\) intersect at points \\( X \\) and \\( Y \\). Circle \\( R \\) is internally tangent to \\( R_1 \\) and \\( R_2 \\) at points \\( M \\) and \\( N \\), respectively, with \\( R > R_1, R_2 \\). Line \\( XY \\) intersects circle \\( R \\) at point \\( A \\). Line \\( AM \\) inte... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad36873da19573be5e89a3d8129d49da33dff34ed2816b564d52be79f32adf24 | Two circles \( S \) and \( T \) intersect at points \( A \) and \( B \). Choose any point \( P \) on the line segment \( AB \). Let \( CD \) be a chord of circle \( S \) passing through \( P \), with \( C \) and \( D \) on \( S \). Let \( EF \) be a chord of circle \( T \) passing through \( P \), with \( E \) and \( F... | [
"180^\\circ"
] | [
"Two circles \\( S \\) and \\( T \\) intersect at points \\( A \\) and \\( B \\). Choose any point \\( P \\) on the line segment \\( AB \\). Let \\( CD \\) be a chord of circle \\( S \\) passing through \\( P \\), with \\( C \\) and \\( D \\) on \\( S \\). Let \\( EF \\) be a chord of circle \\( T \\) passing throu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-e9ed8f248d4f89bf60a628f834e6ca52219a05e5cca7714eb87712052202ef66 | Two circles \( S \) and \( T \) intersect at points \( A \) and \( B \). Let \( P \) be any point on \( AB \). Consider a chord \( CD \) of circle \( S \) passing through \( P \) and a chord \( EF \) of circle \( T \) passing through \( P \). If the radius of circle \( S \) is 10 units and the radius of circle \( T \) ... | [
"\\dfrac{15\\sqrt{15}}{4}"
] | [
"Two circles \\( S \\) and \\( T \\) intersect at points \\( A \\) and \\( B \\). Let \\( P \\) be any point on \\( AB \\). Consider a chord \\( CD \\) of circle \\( S \\) passing through \\( P \\) and a chord \\( EF \\) of circle \\( T \\) passing through \\( P \\). If the radius of circle \\( S \\) is 10 units an... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a524c8333577d8034d1277dba2eefd92dfc87c0b6356e16d82d718288314e8ed | Two circles \( S \) and \( T \) touch at \( X \). They have a common tangent which meets \( S \) at \( A \) and \( T \) at \( B \). The points \( A \) and \( B \) are different. Let \( AP \) be a diameter of \( S \). If the radius of circle \( S \) is 5 and the radius of circle \( T \) is 3, calculate the length of seg... | [
"4\\sqrt{10}"
] | [
"Two circles \\( S \\) and \\( T \\) touch at \\( X \\). They have a common tangent which meets \\( S \\) at \\( A \\) and \\( T \\) at \\( B \\). The points \\( A \\) and \\( B \\) are different. Let \\( AP \\) be a diameter of \\( S \\). If the radius of circle \\( S \\) is 5 and the radius of circle \\( T \\) is... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-311580292480cd9020969dffe4ea8533038d5a162ed76a096daf440c830991a7 | Two circles \( S \) and \( T \) touch at point \( X \). They have a common tangent that meets circle \( S \) at point \( A \) and circle \( T \) at point \( B \). Point \( A \) and point \( B \) are different. Let \( AP \) be a diameter of circle \( S \). If the radius of circle \( S \) is 5 and the radius of circle \(... | [
"4\\sqrt{10}"
] | [
"Two circles \\( S \\) and \\( T \\) touch at point \\( X \\). They have a common tangent that meets circle \\( S \\) at point \\( A \\) and circle \\( T \\) at point \\( B \\). Point \\( A \\) and point \\( B \\) are different. Let \\( AP \\) be a diameter of circle \\( S \\). If the radius of circle \\( S \\) is ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a1b5c854fad10beb436b1d6a7cfd54f4a048c650c765ed8e3087f99162b23eba | Two circles \( S \) and \( T \) touch at point \( X \). They have a common tangent that meets circle \( S \) at point \( A \) and circle \( T \) at point \( B \). Point \( A \) and point \( B \) are different. Let \( AP \) be a diameter of circle \( S \). If the radius of circle \( S \) is 5 units and the radius of cir... | [
"4\\sqrt{10}"
] | [
"Two circles \\( S \\) and \\( T \\) touch at point \\( X \\). They have a common tangent that meets circle \\( S \\) at point \\( A \\) and circle \\( T \\) at point \\( B \\). Point \\( A \\) and point \\( B \\) are different. Let \\( AP \\) be a diameter of circle \\( S \\). If the radius of circle \\( S \\) is ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-199f8d5b241442bf41a9366aa65bb0076b06cf590ac77da2eb815b79d64b2450 | Two circles \( S \) and \( T \) touch at point \( X \). They have a common tangent that meets circle \( S \) at point \( A \) and circle \( T \) at point \( B \). Point \( AP \) is a diameter of circle \( S \). If the radius of circle \( S \) is \( r \) and the radius of circle \( T \) is \( 2r \), find the length of s... | [
"2\\sqrt{3}r"
] | [
"Two circles \\( S \\) and \\( T \\) touch at point \\( X \\). They have a common tangent that meets circle \\( S \\) at point \\( A \\) and circle \\( T \\) at point \\( B \\). Point \\( AP \\) is a diameter of circle \\( S \\). If the radius of circle \\( S \\) is \\( r \\) and the radius of circle \\( T \\) is \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cbcce114e6919479c2bceda8b5fde600add3129db0efc8ebdd1a8b45ec52dac4 | 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 \( B \). Let \( \overline{AP} \) be a diameter of circle \( S \). If the radius of circle \( S \) is 5 and the radius of circle \( T \) is 3, find the length of segm... | [
"4\\sqrt{10}"
] | [
"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 \\). Let \\( \\overline{AP} \\) be a diameter of circle \\( S \\). If the radius of circle \\( S \\) is 5 and the radius of circle \\( T \\) i... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb7a03ec81d337f0ebb1b2e029681c016c87e4fc16eefbc127f902e6846707fb | Two circles \( S_1 \) and \( S_2 \) are placed so that they do not overlap each other, neither completely nor partially. They have centers in \( O_1 \) and \( O_2 \), respectively. Further, \( L_1 \) and \( M_1 \) are different points on \( S_1 \) so that \( O_2L_1 \) and \( O_2M_1 \) are tangent to \( S_1 \), and simi... | [
"\\dfrac{15}{8}"
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\) are placed so that they do not overlap each other, neither completely nor partially. They have centers in \\( O_1 \\) and \\( O_2 \\), respectively. Further, \\( L_1 \\) and \\( M_1 \\) are different points on \\( S_1 \\) so that \\( O_2L_1 \\) and \\( O_2M_1 \\) are tangent... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b3d1c74d2e1ab497ac1ed42d4a0e7226d4585878649f1e5bb284248eb267b50 | Two circles \( S_1 \) and \( S_2 \) are tangent at point \( P \). A common tangent, not through \( P \), touches \( S_1 \) at \( A \) and \( S_2 \) at \( B \). Points \( C \) and \( D \), on \( S_1 \) and \( S_2 \) respectively, are such that \( P \) is on the line \( CD \). If the radii of \( S_1 \) and \( S_2 \) are ... | [
""
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\) are tangent at point \\( P \\). A common tangent, not through \\( P \\), touches \\( S_1 \\) at \\( A \\) and \\( S_2 \\) at \\( B \\). Points \\( C \\) and \\( D \\), on \\( S_1 \\) and \\( S_2 \\) respectively, are such that \\( P \\) is on the line \\( CD \\). If the radi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-402d93f93ad019db242114f10e5a63b749680d6d116c9ba5f8c272b7d0581496 | Two circles \( S_1 \) and \( S_2 \) each touch a straight line \( p \) at the same point \( P \). All points of \( S_2 \), except \( P \), are in the interior of \( S_1 \). A straight line \( q \) is perpendicular to \( p \), touches \( S_2 \) at \( R \), cuts \( p \) at \( L \), and cuts \( S_1 \) at \( N \) and \( M ... | [
"4\\sqrt{2}"
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\) each touch a straight line \\( p \\) at the same point \\( P \\). All points of \\( S_2 \\), except \\( P \\), are in the interior of \\( S_1 \\). A straight line \\( q \\) is perpendicular to \\( p \\), touches \\( S_2 \\) at \\( R \\), cuts \\( p \\) at \\( L \\), and cuts... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-416912385bdb45d730e983f71031c165a7dc963ed621c97d3c2ade806676dc36 | Two circles \( S_1 \) and \( S_2 \) have centers \( O_1 \) and \( O_2 \) respectively, and do not intersect nor contain each other. Line \( \ell_1 \) is tangent to \( S_1 \) at \( A \) and \( S_2 \) at \( B \), and does not intersect segment \( O_1O_2 \). Line \( \ell_2 \) is tangent to \( S_1 \) at \( C \) and \( S_2 ... | [
"\\dfrac{29 + 6\\sqrt{6}}{5}"
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\) have centers \\( O_1 \\) and \\( O_2 \\) respectively, and do not intersect nor contain each other. Line \\( \\ell_1 \\) is tangent to \\( S_1 \\) at \\( A \\) and \\( S_2 \\) at \\( B \\), and does not intersect segment \\( O_1O_2 \\). Line \\( \\ell_2 \\) is tangent to \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-b618ef7928bcab940eef3dcf2bdf998296595854350d08b755251d6d6ac1ee41 | Two circles \( S_1 \) and \( S_2 \) intersect at different points \( P \) and \( Q \). The arc of \( S_1 \) lying inside \( S_2 \) measures \( 2a \) and the arc of \( S_2 \) lying inside \( S_1 \) measures \( 2b \). Let \( T \) be any point on \( S_1 \). Let \( R \) and \( S \) be the points of intersection of \( S_2 \... | [
"PQ"
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\) intersect at different points \\( P \\) and \\( Q \\). The arc of \\( S_1 \\) lying inside \\( S_2 \\) measures \\( 2a \\) and the arc of \\( S_2 \\) lying inside \\( S_1 \\) measures \\( 2b \\). Let \\( T \\) be any point on \\( S_1 \\). Let \\( R \\) and \\( S \\) be the p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-2f8168da1ac9d965f0f3ac75efe75ddce6c504faf0bd99b935702de5dbdf0c72 | Two circles \( S_1 \) and \( S_2 \) intersect at points \( A \) and \( B \). Let \( Q \) be a point on \( S_1 \). The lines \( QA \) and \( QB \) meet the circle \( S_2 \) at points \( C \) and \( D \), respectively (apart from \( A \) and \( B \)), while the tangents to \( S_1 \) at \( A \) and \( B \) meet at \( P \)... | [
"8"
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\) intersect at points \\( A \\) and \\( B \\). Let \\( Q \\) be a point on \\( S_1 \\). The lines \\( QA \\) and \\( QB \\) meet the circle \\( S_2 \\) at points \\( C \\) and \\( D \\), respectively (apart from \\( A \\) and \\( B \\)), while the tangents to \\( S_1 \\) at \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5e32d0e855ddc43ab0ff52a4791e1c7682755c56f0de7dcdcb331ccd521dd0d0 | Two circles \( S_1 \) and \( S_2 \) intersect at points \( A \) and \( B \). Two equal segments \( CD \) and \( EF \) through \( B \) intersect \( S_1 \) and \( S_2 \) at \( C, E \) and \( D, F \) respectively. \( CF \) intersects \( S_1 \) and \( S_2 \) at \( Q \) and \( P \) respectively. \( M \) and \( N \) are the ... | [
"12.5"
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\) intersect at points \\( A \\) and \\( B \\). Two equal segments \\( CD \\) and \\( EF \\) through \\( B \\) intersect \\( S_1 \\) and \\( S_2 \\) at \\( C, E \\) and \\( D, F \\) respectively. \\( CF \\) intersects \\( S_1 \\) and \\( S_2 \\) at \\( Q \\) and \\( P \\) respe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-f42753c5a105611dd8152d6cf7dbb85dc3c4433ea7e0cc2e3d23c475e7bfe4df | Two circles \( S_1 \) and \( S_2 \) touch each other externally at \( K \); they also touch a circle \( S \) internally at \( A_1 \) and \( A_2 \) respectively. Let \( P \) be one point of intersection of \( S \) with the common tangent to \( S_1 \) and \( S_2 \) at \( K \). The line \( PA_1 \) meets \( S_1 \) again at... | [
"2\\sqrt{15}"
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\) touch each other externally at \\( K \\); they also touch a circle \\( S \\) internally at \\( A_1 \\) and \\( A_2 \\) respectively. Let \\( P \\) be one point of intersection of \\( S \\) with the common tangent to \\( S_1 \\) and \\( S_2 \\) at \\( K \\). The line \\( PA_1... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-4556529ef0edde8a7f90d72a5ccfecc1934ae3ae35ba2d41cfeb29b8fc84b0ed | Two circles \( S_1 \) and \( S_2 \) touch externally at point \( F \). Their external common tangent touches \( S_1 \) at \( A \) and \( S_2 \) at \( B \). A line parallel to \( AB \) and tangent to \( S_2 \) at \( C \) intersects \( S_1 \) at points \( D \) and \( E \). If the radius of \( S_1 \) is 4 units and the ra... | [
"\\dfrac{8\\sqrt{5}}{5}"
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\) touch externally at point \\( F \\). Their external common tangent touches \\( S_1 \\) at \\( A \\) and \\( S_2 \\) at \\( B \\). A line parallel to \\( AB \\) and tangent to \\( S_2 \\) at \\( C \\) intersects \\( S_1 \\) at points \\( D \\) and \\( E \\). If the radius of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19c4d4a025dac62b733dee6c560ad33718483b63e5d0764d7e419da0251ab4db | Two circles \( S_1 \) and \( S_2 \) touch externally at point \( F \). Their external common tangent touches \( S_1 \) at \( A \) and \( S_2 \) at \( B \). A line parallel to \( AB \) and tangent to \( S_2 \) at \( C \) intersects \( S_1 \) at points \( D \) and \( E \). Let \( P \) be the intersection point of the com... | [
"\\dfrac{3\\sqrt{10}}{2}"
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\) touch externally at point \\( F \\). Their external common tangent touches \\( S_1 \\) at \\( A \\) and \\( S_2 \\) at \\( B \\). A line parallel to \\( AB \\) and tangent to \\( S_2 \\) at \\( C \\) intersects \\( S_1 \\) at points \\( D \\) and \\( E \\). Let \\( P \\) be ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-73a2969c65cd2bd4622ed52de859646b7fcd9051aabf5deba275c1ff5729ac1a | Two circles \( S_1 \) and \( S_2 \) with centers \( O_1 \) and \( O_2 \) respectively intersect at points \( A \) and \( B \). The tangents at \( A \) to \( S_1 \) and \( S_2 \) meet segments \( BO_2 \) and \( BO_1 \) at points \( K \) and \( L \) respectively. If the length of \( O_1O_2 \) is 10 units, find the length... | [
"10"
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\) with centers \\( O_1 \\) and \\( O_2 \\) respectively intersect at points \\( A \\) and \\( B \\). The tangents at \\( A \\) to \\( S_1 \\) and \\( S_2 \\) meet segments \\( BO_2 \\) and \\( BO_1 \\) at points \\( K \\) and \\( L \\) respectively. If the length of \\( O_1O_2... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-b28a8896ccb32b3fe0b8149976fca6ed9bec3abf1ca8cb0b719670f4ef4c2694 | Two circles \( S_1 \) and \( S_2 \) with equal radii intersect at two points. A line \( l \) intersects \( S_1 \) at \( B \) and \( D \) and \( S_2 \) at \( A \) and \( C \) (in the order \( A, B, C, D \)). Two circles \( W_1 \) and \( W_2 \) are drawn such that both are tangent externally to \( S_1 \) and internally t... | [
"1"
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\) with equal radii intersect at two points. A line \\( l \\) intersects \\( S_1 \\) at \\( B \\) and \\( D \\) and \\( S_2 \\) at \\( A \\) and \\( C \\) (in the order \\( A, B, C, D \\)). Two circles \\( W_1 \\) and \\( W_2 \\) are drawn such that both are tangent externally ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-830dc6558bf54583a84e9cd0eb0522af77eb0b9a9ac4bb14c48c39c7f8754990 | Two circles \( S_1 \) and \( S_2 \) with equal radius intersect at two points. A line \( l \) intersects \( S_1 \) at points \( B \) and \( D \) and \( S_2 \) at points \( A \) and \( C \) (in the order \( A, B, C, D \) on the line). Two circles \( W_1 \) and \( W_2 \) are drawn such that both are tangent externally to... | [
"no_valid_answer_found"
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\) with equal radius intersect at two points. A line \\( l \\) intersects \\( S_1 \\) at points \\( B \\) and \\( D \\) and \\( S_2 \\) at points \\( A \\) and \\( C \\) (in the order \\( A, B, C, D \\) on the line). Two circles \\( W_1 \\) and \\( W_2 \\) are drawn such that b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f454ab75a6cce0b35a4036bdc19a19456757aaad25c9aec73d106dfd2b4fc33a | Two circles \( S_1 \) and \( S_2 \), with radii 6 units and 3 units respectively, are tangent to each other externally. Let \( AC \) and \( BD \) be their direct common tangents with \( A \) and \( B \) on \( S_1 \), and \( C \) and \( D \) on \( S_2 \). Find the area of quadrilateral \( ABDC \) to the nearest integer. | [
"68"
] | [
"Two circles \\( S_1 \\) and \\( S_2 \\), with radii 6 units and 3 units respectively, are tangent to each other externally. Let \\( AC \\) and \\( BD \\) be their direct common tangents with \\( A \\) and \\( B \\) on \\( S_1 \\), and \\( C \\) and \\( D \\) on \\( S_2 \\). Find the area of quadrilateral \\( ABDC ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-42df0182d2f66a5207f6a69dafe7b6a4040d96f4f1a950183d276e5f83e306fd | Two circles \( S_{1} \) and \( S_{2} \) touch each other externally at \( K \); they also touch a circle \( S \) internally at \( A_{1} \) and \( A_{2} \) respectively. Let \( P \) be one point of intersection of \( S \) with the common tangent to \( S_{1} \) and \( S_{2} \) at \( K \). The line \( PA_{1} \) meets \( S... | [
"8"
] | [
"Two circles \\( S_{1} \\) and \\( S_{2} \\) touch each other externally at \\( K \\); they also touch a circle \\( S \\) internally at \\( A_{1} \\) and \\( A_{2} \\) respectively. Let \\( P \\) be one point of intersection of \\( S \\) with the common tangent to \\( S_{1} \\) and \\( S_{2} \\) at \\( K \\). The l... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-a8c79925511498781932325744261a5f385340653acdebf6017c9561b9d57a04 | Two circles \( S_{1} \) and \( S_{2} \) with centers \( O_{1} \) and \( O_{2} \), respectively, intersect at two points \( A \) and \( B \). The tangents to the circles \( S_{1} \) and \( S_{2} \) at the point \( A \) intersect the segments \( BO_{2} \) and \( BO_{1} \) at the points \( K \) and \( L \), respectively. ... | [
"\\dfrac{55}{9}"
] | [
"Two circles \\( S_{1} \\) and \\( S_{2} \\) with centers \\( O_{1} \\) and \\( O_{2} \\), respectively, intersect at two points \\( A \\) and \\( B \\). The tangents to the circles \\( S_{1} \\) and \\( S_{2} \\) at the point \\( A \\) intersect the segments \\( BO_{2} \\) and \\( BO_{1} \\) at the points \\( K \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-184f84f5969d4a41c9eaf281b554bae2f43eb7e59be1b7ad65069dea19f35f20 | Two circles \( T \) and \( S \) touch each other externally at point \( K \). They have a common tangent that touches \( T \) at \( A \) and \( S \) at \( B \). Let \( AP \) be a diameter of \( T \), and let \( Q \) be one of the two possible points from which the tangent from \( P \) touches \( S \). If the radius of ... | [
"10"
] | [
"Two circles \\( T \\) and \\( S \\) touch each other externally at point \\( K \\). They have a common tangent that touches \\( T \\) at \\( A \\) and \\( S \\) at \\( B \\). Let \\( AP \\) be a diameter of \\( T \\), and let \\( Q \\) be one of the two possible points from which the tangent from \\( P \\) touches... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f082d8e99fe82bb7fdf5fdc02b7472ccbc069fa39567b73fc60ca12d6ab1adc | Two circles \( U \) and \( V \) have distinct radii and are tangent to each other externally at point \( T \). Points \( A \) and \( B \) are on circles \( U \) and \( V \) respectively, both distinct from \( T \), such that \( \angle ATB = 90^\circ \). Find the locus of the midpoint of \( AB \). | [
"The locus of the midpoint of \\( AB \\) is a circle with center at the midpoint of the line segment joining the centers of the two circles \\( U \\) and \\( V \\), and radius \\( \\frac{1}{2}(AC + BD) \\)."
] | [
"Two circles \\( U \\) and \\( V \\) have distinct radii and are tangent to each other externally at point \\( T \\). Points \\( A \\) and \\( B \\) are on circles \\( U \\) and \\( V \\) respectively, both distinct from \\( T \\), such that \\( \\angle ATB = 90^\\circ \\). Find the locus of the midpoint of \\( AB ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 10
} | 30 | 0 | 0 |
omr-ws-e3a0263518e1f25f17604f216e53fa5771871fbcb64223fd7fb9bd9970d7ff18 | Two circles \( U \) and \( V \) have radii \( r \) and \( R \) respectively, and are tangent to each other externally at point \( T \). Points \( A \) and \( B \) are on circles \( U \) and \( V \) respectively, both distinct from \( T \), such that \( \angle ATB = 90^\circ \). If the fixed point through which line \( ... | [
"\\dfrac{r(r + R)}{r - R}"
] | [
"Two circles \\( U \\) and \\( V \\) have radii \\( r \\) and \\( R \\) respectively, and are tangent to each other externally at point \\( T \\). Points \\( A \\) and \\( B \\) are on circles \\( U \\) and \\( V \\) respectively, both distinct from \\( T \\), such that \\( \\angle ATB = 90^\\circ \\). If the fixed... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-b6694da942831c9206452c8fd8c4c5fb59ccb48f04d89993d790022b49a09d06 | Two circles \( W \) and \( W' \) with the same radius intersect at points \( A \) and \( B \). The midpoint of \( AB \) is \( O \). A chord \( DC \) of circle \( W \) passes through \( O \) and intersects circle \( W' \) at point \( P \). A chord \( EF \) of circle \( W' \) passes through \( O \) and intersects circle ... | [
"\\dfrac{r}{2}"
] | [
"Two circles \\( W \\) and \\( W' \\) with the same radius intersect at points \\( A \\) and \\( B \\). The midpoint of \\( AB \\) is \\( O \\). A chord \\( DC \\) of circle \\( W \\) passes through \\( O \\) and intersects circle \\( W' \\) at point \\( P \\). A chord \\( EF \\) of circle \\( W' \\) passes through... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-78c1c608cf11e2d6569b4f1e6e2bd63bc00c47cf209eddc28f4d68dfeb9bb53e | Two circles \( W \) and \( Z \) with centers \( O_1 \) and \( O_2 \) respectively intersect at points \( A \) and \( B \). A line through \( B \) intersects \( W \) at \( C \) and \( Z \) at \( D \). The tangent lines to \( W \) at \( C \) and to \( Z \) at \( D \) intersect at point \( E \). Let \( T \) be the circumc... | [
"2R"
] | [
"Two circles \\( W \\) and \\( Z \\) with centers \\( O_1 \\) and \\( O_2 \\) respectively intersect at points \\( A \\) and \\( B \\). A line through \\( B \\) intersects \\( W \\) at \\( C \\) and \\( Z \\) at \\( D \\). The tangent lines to \\( W \\) at \\( C \\) and to \\( Z \\) at \\( D \\) intersect at point ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-193b50f31cbb313b4deff080c1e14da8603df658de729311fbedd45af407b634 | Two circles \( W(1) \) and \( W(2) \) intersect at points \( X \) and \( Y \). Both circles \( W(1) \) and \( W(2) \) are inside a larger circle \( W \) and touch \( W \) at points \( A \) and \( B \), respectively. Line \( AB \) intersects \( W(1) \) and \( W(2) \) at points \( A(1) \) and \( A(2) \), respectively. A ... | [
"1"
] | [
"Two circles \\( W(1) \\) and \\( W(2) \\) intersect at points \\( X \\) and \\( Y \\). Both circles \\( W(1) \\) and \\( W(2) \\) are inside a larger circle \\( W \\) and touch \\( W \\) at points \\( A \\) and \\( B \\), respectively. Line \\( AB \\) intersects \\( W(1) \\) and \\( W(2) \\) at points \\( A(1) \\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-feadc2d3e0676d616eb669c9f2b934fc3d533f0d11cff358bdd70bd5eace0fbd | Two circles \( W1 \) and \( W2 \) intersect at points \( A \) and \( B \). A line \( l \) passing through \( A \) intersects \( W1 \) and \( W2 \) at points \( P \) and \( Q \) respectively. Let \( K \) be the midpoint of arc \( PB \) not containing \( A \), and \( S \) be the midpoint of arc \( BQ \) not containing \(... | [
"\\dfrac{\\sqrt{2}}{2}"
] | [
"Two circles \\( W1 \\) and \\( W2 \\) intersect at points \\( A \\) and \\( B \\). A line \\( l \\) passing through \\( A \\) intersects \\( W1 \\) and \\( W2 \\) at points \\( P \\) and \\( Q \\) respectively. Let \\( K \\) be the midpoint of arc \\( PB \\) not containing \\( A \\), and \\( S \\) be the midpoint ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-a4b4616824c588ae9da8145b4a679c5fa62e7b1ac959dd7aeff3d308ad07d9d3 | Two circles \( W1 \) and \( W2 \) intersect at points \( P \) and \( Q \). Line \( L1 \) is tangent to \( W1 \) at \( A \) and to \( W2 \) at \( B \). Line \( L2 \) is tangent to \( W1 \) at \( C \) and to \( W2 \) at \( D \). Line \( L \) passes through the centers \( O1 \) and \( O2 \) of the circles \( W1 \) and \( ... | [
"no_valid_answer_found"
] | [
"Two circles \\( W1 \\) and \\( W2 \\) intersect at points \\( P \\) and \\( Q \\). Line \\( L1 \\) is tangent to \\( W1 \\) at \\( A \\) and to \\( W2 \\) at \\( B \\). Line \\( L2 \\) is tangent to \\( W1 \\) at \\( C \\) and to \\( W2 \\) at \\( D \\). Line \\( L \\) passes through the centers \\( O1 \\) and \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2a883092d2b04e2a477ccc00126719aa86ec6bd9a4a5803903b624b3c10dd7b1 | Two circles \( W_1 \) and \( W_2 \) intersect at point \( A \). A variable circle \( W \) is tangent externally to both \( W_1 \) and \( W_2 \). Let the tangents from \( A \) to \( W \) meet \( W \) at points \( X \) and \( Y \). If the coordinates of \( A \) are \((0,0)\), the center of \( W_1 \) is \((1,0)\) with rad... | [
"(0, 2)"
] | [
"Two circles \\( W_1 \\) and \\( W_2 \\) intersect at point \\( A \\). A variable circle \\( W \\) is tangent externally to both \\( W_1 \\) and \\( W_2 \\). Let the tangents from \\( A \\) to \\( W \\) meet \\( W \\) at points \\( X \\) and \\( Y \\). If the coordinates of \\( A \\) are \\((0,0)\\), the center of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-df7e0a52196b6bf8c7185c508d64e12f46651face61b38e9eb55a10c23114459 | Two circles \( W_1 \) and \( W_2 \) intersect at points \( A \) and \( B \). Let the tangent lines from \( A \) and \( B \) to \( W_1 \) and \( W_2 \) intersect at point \( P \). Let the second tangent lines from \( P \) to \( W_1 \) and \( W_2 \) meet the circles at points \( C \) and \( D \), respectively. If \( \ang... | [
"0"
] | [
"Two circles \\( W_1 \\) and \\( W_2 \\) intersect at points \\( A \\) and \\( B \\). Let the tangent lines from \\( A \\) and \\( B \\) to \\( W_1 \\) and \\( W_2 \\) intersect at point \\( P \\). Let the second tangent lines from \\( P \\) to \\( W_1 \\) and \\( W_2 \\) meet the circles at points \\( C \\) and \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d813d17288cbe4f53eaceea02cdf187853cf01c46454c3543649e138b0047578 | Two circles \( W_1 \) and \( W_2 \) intersect at points \( A \) and \( B \). \( P \) is the intersection point of the tangents to \( W_1 \) at \( A \) and \( B \). Let \( Y \) be an arbitrary point on \( W_2 \). The extension of line \( YA \) intersects \( W_1 \) again at \( Q \). \( QP \) also cuts \( W_1 \) again at ... | [
"1"
] | [
"Two circles \\( W_1 \\) and \\( W_2 \\) intersect at points \\( A \\) and \\( B \\). \\( P \\) is the intersection point of the tangents to \\( W_1 \\) at \\( A \\) and \\( B \\). Let \\( Y \\) be an arbitrary point on \\( W_2 \\). The extension of line \\( YA \\) intersects \\( W_1 \\) again at \\( Q \\). \\( QP ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d8dbfbd5f307d079d95ebbddd1fea71425b310e1458f3aed95c6fc2cb5be84c3 | Two circles \( Y_1 \) and \( Y_2 \) intersect at points \( A \) and \( B \). A point \( C \) lies on \( Y_1 \) and points \( D \) and \( E \) lie on \( Y_2 \) such that \( CAD \) and \( CBE \) are straight lines. Points \( P \) and \( Q \) are constructed on \( AB \) (produced or otherwise) such that \( \angle QDA = \a... | [
"1"
] | [
"Two circles \\( Y_1 \\) and \\( Y_2 \\) intersect at points \\( A \\) and \\( B \\). A point \\( C \\) lies on \\( Y_1 \\) and points \\( D \\) and \\( E \\) lie on \\( Y_2 \\) such that \\( CAD \\) and \\( CBE \\) are straight lines. Points \\( P \\) and \\( Q \\) are constructed on \\( AB \\) (produced or otherw... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-e4426f7790f055a494846b91c1981571d3e5cf6c81ef30b363ecdf1db25b9930 | Two circles \( \Gamma \) and \( \Gamma' \) intersect at point \( A \). A line is tangent to \( \Gamma \) and \( \Gamma' \) at points \( E \) and \( F \), respectively. Let \( BC \) be a line parallel to the common tangent \( EF \) of \( \Gamma \) and \( \Gamma' \). Let \( P \) be the second intersection point of the ci... | [
"5"
] | [
"Two circles \\( \\Gamma \\) and \\( \\Gamma' \\) intersect at point \\( A \\). A line is tangent to \\( \\Gamma \\) and \\( \\Gamma' \\) at points \\( E \\) and \\( F \\), respectively. Let \\( BC \\) be a line parallel to the common tangent \\( EF \\) of \\( \\Gamma \\) and \\( \\Gamma' \\). Let \\( P \\) be the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-8bcbc82177414425c4b821e73310d10b26c555d7cdf3bd181784a9f5a29143e6 | Two circles \( \Gamma \) and \( \Gamma' \) intersect at points \( A \) and \( D \). A line is tangent to \( \Gamma \) at \( E \) and to \( \Gamma' \) at \( F \). The line \( BC \) contains point \( D \) and is parallel to the line \( EF \), with \( C \) on \( \Gamma \) and \( B \) on \( \Gamma' \). Let \( P \) be the s... | [
"16"
] | [
"Two circles \\( \\Gamma \\) and \\( \\Gamma' \\) intersect at points \\( A \\) and \\( D \\). A line is tangent to \\( \\Gamma \\) at \\( E \\) and to \\( \\Gamma' \\) at \\( F \\). The line \\( BC \\) contains point \\( D \\) and is parallel to the line \\( EF \\), with \\( C \\) on \\( \\Gamma \\) and \\( B \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0aed801a81df6afe27a69b17662f6c6388f6a215caa1fb6e121fb5b15a41515e | Two circles \( \Gamma \) and \( \Omega \) intersect at \( A \) and \( G \). The line \( t \) is tangent to \( \Gamma \) at \( B \) and \( \Omega \) at \( C \). If \( G \) is the centroid of \( ABC \), what is the largest possible value of the angle \( \angle BAC \)? | [
"60^\\circ"
] | [
"Two circles \\( \\Gamma \\) and \\( \\Omega \\) intersect at \\( A \\) and \\( G \\). The line \\( t \\) is tangent to \\( \\Gamma \\) at \\( B \\) and \\( \\Omega \\) at \\( C \\). If \\( G \\) is the centroid of \\( ABC \\), what is the largest possible value of the angle \\( \\angle BAC \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-61673678c2ef941432d78fd2139c084b01e96b94c56888a91595df8bcc4fdd9d | 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 \( \el... | [
"20"
] | [
"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 \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c41d8f0f6a529afb3b0e7890438b8254820d7a75f09e6f4124176198845e9eb0 | Two circles \( \Gamma_1 \) and \( \Gamma_2 \) intersect at points \( A \) and \( B \). Consider a circle \( \Gamma \) contained in \( \Gamma_1 \) and \( \Gamma_2 \), which is tangent to both of them at \( D \) and \( E \) respectively. Let \( C \) be one of the intersection points of line \( AB \) with \( \Gamma \), \(... | [
"no_valid_answer_found"
] | [
"Two circles \\( \\Gamma_1 \\) and \\( \\Gamma_2 \\) intersect at points \\( A \\) and \\( B \\). Consider a circle \\( \\Gamma \\) contained in \\( \\Gamma_1 \\) and \\( \\Gamma_2 \\), which is tangent to both of them at \\( D \\) and \\( E \\) respectively. Let \\( C \\) be one of the intersection points of line ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1e9af595d54ca0f1888c1bff9a9a8b2c9677622e7f648a26cab85aee50da47d | Two circles \( \Gamma_1 \) and \( \Gamma_2 \) intersect at points \( A \) and \( B \). Let \( r_1 \) be the tangent to \( \Gamma_1 \) at \( A \) and \( r_2 \) be the tangent to \( \Gamma_2 \) at \( B \). Let \( r_1 \cap r_2 = C \). Let \( T \) be the intersection of \( r_1 \) and \( \Gamma_2 \) (with \( T \neq A \)). C... | [
"10"
] | [
"Two circles \\( \\Gamma_1 \\) and \\( \\Gamma_2 \\) intersect at points \\( A \\) and \\( B \\). Let \\( r_1 \\) be the tangent to \\( \\Gamma_1 \\) at \\( A \\) and \\( r_2 \\) be the tangent to \\( \\Gamma_2 \\) at \\( B \\). Let \\( r_1 \\cap r_2 = C \\). Let \\( T \\) be the intersection of \\( r_1 \\) and \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f03909064f5f45ee2fa6f39245437bedf4624bd2c0bb30df7e554938dd8c8bfb | Two circles \( \Gamma_{1} \) and \( \Gamma_{2} \) share the same center, with radii \( R_{1} \) and \( R_{2} \) respectively, where \( R_{1} > R_{2} \). Triangle \( \triangle ABC \) is inscribed in \( \Gamma_{2} \), and the rays \( AB \), \( BC \), and \( CA \) intersect \( \Gamma_{1} \) at points \( B_{1} \), \( C_{1}... | [
"4"
] | [
"Two circles \\( \\Gamma_{1} \\) and \\( \\Gamma_{2} \\) share the same center, with radii \\( R_{1} \\) and \\( R_{2} \\) respectively, where \\( R_{1} > R_{2} \\). Triangle \\( \\triangle ABC \\) is inscribed in \\( \\Gamma_{2} \\), and the rays \\( AB \\), \\( BC \\), and \\( CA \\) intersect \\( \\Gamma_{1} \\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-91c67893e30f9bb967cd3bc6ffcd3325dfc6f036de74973d7b4b568e52dc8ab6 | Two circles \( \Omega_{1} \) and \( \Omega_{2} \) are externally tangent at point \( T \). A common tangent to \( \Omega_{1} \) and \( \Omega_{2} \) that does not pass through \( T \) intersects \( \Omega_{1} \) and \( \Omega_{2} \) at points \( A \) and \( B \), respectively. Circle \( \Omega_{3} \) is tangent to \( \... | [
"no_valid_answer_found"
] | [
"Two circles \\( \\Omega_{1} \\) and \\( \\Omega_{2} \\) are externally tangent at point \\( T \\). A common tangent to \\( \\Omega_{1} \\) and \\( \\Omega_{2} \\) that does not pass through \\( T \\) intersects \\( \\Omega_{1} \\) and \\( \\Omega_{2} \\) at points \\( A \\) and \\( B \\), respectively. Circle \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2781fc1b2f80574aa95fb48c7820810d76f07f881e04aefde849af3b878308ab | Two circles \( \Omega_{1} \) and \( \Omega_{2} \) are externally tangent to each other at a point \( I \), and both are tangent to a third circle \( \Omega \) which encloses them. The common tangent to \( \Omega_{1} \) and \( \Omega_{2} \) at \( I \) meets \( \Omega \) at point \( A \). A common tangent to \( \Omega_{1... | [
"\\dfrac{40\\sqrt{3} - 12\\sqrt{10}}{7}"
] | [
"Two circles \\( \\Omega_{1} \\) and \\( \\Omega_{2} \\) are externally tangent to each other at a point \\( I \\), and both are tangent to a third circle \\( \\Omega \\) which encloses them. The common tangent to \\( \\Omega_{1} \\) and \\( \\Omega_{2} \\) at \\( I \\) meets \\( \\Omega \\) at point \\( A \\). A c... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-e0f2e99bbbf635ce62065b26306f3f7161a0c7b802d56b7f0d91d53bf2ec79a7 | Two circles \( \gamma_1 \) and \( \gamma_2 \) meet at points \( X \) and \( Y \). Consider the parallel through \( Y \) to the nearest common tangent of the circles. This parallel meets again \( \gamma_1 \) and \( \gamma_2 \) at \( A \) and \( B \) respectively. Let \( O \) be the center of the circle tangent to \( \ga... | [
"60^\\circ"
] | [
"Two circles \\( \\gamma_1 \\) and \\( \\gamma_2 \\) meet at points \\( X \\) and \\( Y \\). Consider the parallel through \\( Y \\) to the nearest common tangent of the circles. This parallel meets again \\( \\gamma_1 \\) and \\( \\gamma_2 \\) at \\( A \\) and \\( B \\) respectively. Let \\( O \\) be the center of... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-22c0f92be4a1be7196a9911b9ed51beea8ee14531d905e711f06938e8ff575de | Two circles \( \gamma_{1} \) and \( \gamma_{2} \) intersect at points \( X \) and \( Y \). A line through \( Y \) parallel to the nearest common tangent of the circles intersects \( \gamma_{1} \) again at \( A \) and \( \gamma_{2} \) again at \( B \). Let \( O \) be the center of the circle tangent to \( \gamma_{1} \),... | [
"60^\\circ"
] | [
"Two circles \\( \\gamma_{1} \\) and \\( \\gamma_{2} \\) intersect at points \\( X \\) and \\( Y \\). A line through \\( Y \\) parallel to the nearest common tangent of the circles intersects \\( \\gamma_{1} \\) again at \\( A \\) and \\( \\gamma_{2} \\) again at \\( B \\). Let \\( O \\) be the center of the circle... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-5df0837d73e0da2f42974f073dcd7c1702266630cf4f2be843017f4989f130f6 | Two circles \( \left(\Omega_1\right) \) and \( \left(\Omega_2\right) \) touch internally at point \( T \). Points \( M \) and \( N \) are on \( \left(\Omega_1\right) \) and different from \( T \). Points \( A, B, C, D \) are on \( \left(\Omega_2\right) \) such that chords \( AB \) and \( CD \) pass through \( M \) and ... | [
"1"
] | [
"Two circles \\( \\left(\\Omega_1\\right) \\) and \\( \\left(\\Omega_2\\right) \\) touch internally at point \\( T \\). Points \\( M \\) and \\( N \\) are on \\( \\left(\\Omega_1\\right) \\) and different from \\( T \\). Points \\( A, B, C, D \\) are on \\( \\left(\\Omega_2\\right) \\) such that chords \\( AB \\) a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-9c71e2cb59aeb3c3ae6c2589eb33fe34c1d64b09aec1ba57ee394edafade35b2 | Two circles \( \omega_1 \) and \( \omega_2 \) are tangent internally at point \( P \). Points \( A \) and \( B \) are chosen on their common tangent such that \( P \) lies between \( A \) and \( B \). Let \( C \) and \( D \) be the intersection points of the tangents from \( A \) to \( \omega_1 \) and from \( B \) to \... | [
"0"
] | [
"Two circles \\( \\omega_1 \\) and \\( \\omega_2 \\) are tangent internally at point \\( P \\). Points \\( A \\) and \\( B \\) are chosen on their common tangent such that \\( P \\) lies between \\( A \\) and \\( B \\). Let \\( C \\) and \\( D \\) be the intersection points of the tangents from \\( A \\) to \\( \\o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-b1892833ecb22a5a4e87e37ef576706491d9ceb73fcb7182947d5a80fd205eb3 | Two circles \( \omega_1 \) and \( \omega_2 \) have centers \( O_1 \) and \( O_2 \) respectively and intersect at \( A \) and \( B \), where the radius of \( \omega_2 \) is twice the radius of \( \omega_1 \). Let \( C \) be the image of \( B \) after rotating 180 degrees about \( O_2 \). A point \( F \) is chosen on the... | [
"128"
] | [
"Two circles \\( \\omega_1 \\) and \\( \\omega_2 \\) have centers \\( O_1 \\) and \\( O_2 \\) respectively and intersect at \\( A \\) and \\( B \\), where the radius of \\( \\omega_2 \\) is twice the radius of \\( \\omega_1 \\). Let \\( C \\) be the image of \\( B \\) after rotating 180 degrees about \\( O_2 \\). A... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-214e8253b0da92f9577addc2712e3d14eb96f099c3e3e40c331a554086cd14c3 | 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... | [
"1"
] | [
"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 \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-9e42f1872fb7947f8613dab4c8ed5c05d7aa1e084189b2648736a7de34f7fd04 | Two circles \( \omega_{1} \) and \( \omega_{2} \) intersect at points \( A \) and \( B \). Segments \( PQ \) and \( RS \) are common tangents to these circles, with points \( P \) and \( R \) on \( \omega_{1} \) and points \( Q \) and \( S \) on \( \omega_{2} \). It is given that \( RB \parallel PQ \). Ray \( RB \) int... | [
"\\(\\frac{1}{3}\\)"
] | [
"Two circles \\( \\omega_{1} \\) and \\( \\omega_{2} \\) intersect at points \\( A \\) and \\( B \\). Segments \\( PQ \\) and \\( RS \\) are common tangents to these circles, with points \\( P \\) and \\( R \\) on \\( \\omega_{1} \\) and points \\( Q \\) and \\( S \\) on \\( \\omega_{2} \\). It is given that \\( RB... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-ab274c69b1621ea0831a0ce609ee12f24eb5f03223776c5e24d5c9596c6a9313 | Two circles \( c_1 \) and \( c_2 \) intersect at points \( A \) and \( B \), and \( c_1 \) passes through the center \( O \) of \( c_2 \). A line intersects \( c_1 \) at points \( K \) and \( O \), and \( c_2 \) at points \( L \) and \( M \) such that \( L \) lies on the segment \( KO \). Point \( P \) is on \( AB \) s... | [
"1"
] | [
"Two circles \\( c_1 \\) and \\( c_2 \\) intersect at points \\( A \\) and \\( B \\), and \\( c_1 \\) passes through the center \\( O \\) of \\( c_2 \\). A line intersects \\( c_1 \\) at points \\( K \\) and \\( O \\), and \\( c_2 \\) at points \\( L \\) and \\( M \\) such that \\( L \\) lies on the segment \\( KO ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-96f6f46d7e5c8d6a78f4e28d98b6eee1506c888681b337c4724d311fe71a9960 | Two circles \( k1 \) and \( k2 \) intersect at points \( A \) and \( B \). Points \( C \) and \( D \) lie on \( k1 \), while points \( E \) and \( F \) lie on \( k2 \) such that \( A, C, E \) are collinear and \( B, D, F \) are collinear. Points \( G \) and \( H \) are on lines \( ACE \) and \( BDF \), respectively. Th... | [
"\\dfrac{10}{3}"
] | [
"Two circles \\( k1 \\) and \\( k2 \\) intersect at points \\( A \\) and \\( B \\). Points \\( C \\) and \\( D \\) lie on \\( k1 \\), while points \\( E \\) and \\( F \\) lie on \\( k2 \\) such that \\( A, C, E \\) are collinear and \\( B, D, F \\) are collinear. Points \\( G \\) and \\( H \\) are on lines \\( ACE ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b875bffdf5d50bacd52832bccee177d8aea4ae29c245e14d86d82434f01ee17 | Two circles \( k_1 \) and \( k_2 \) are externally tangent at point \( T \). A line tangent to \( k_2 \) at \( X \) intersects \( k_1 \) at points \( A \) and \( B \), and \( XT \) intersects \( k_1 \) at \( S \) (where \( S \neq T \)). Point \( C \) is on the arc \( \widehat{TS} \) not containing \( A \) and \( B \). ... | [
"\\dfrac{15}{4}"
] | [
"Two circles \\( k_1 \\) and \\( k_2 \\) are externally tangent at point \\( T \\). A line tangent to \\( k_2 \\) at \\( X \\) intersects \\( k_1 \\) at points \\( A \\) and \\( B \\), and \\( XT \\) intersects \\( k_1 \\) at \\( S \\) (where \\( S \\neq T \\)). Point \\( C \\) is on the arc \\( \\widehat{TS} \\) n... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-4cbd029ec75eba865e3a74b92a2aeb3becbace69c54135510b2d6cd1943f092b | Two circles \( k_1 \) and \( k_2 \) are given with centers \( P \) and \( R \) respectively, touching externally at point \( A \). Let \( p \) be their common tangent line which does not pass through \( A \) and touches \( k_1 \) at \( B \) and \( k_2 \) at \( C \). The line \( PR \) cuts \( BC \) at point \( E \) and ... | [
"\\(\\frac{3}{2}\\)"
] | [
"Two circles \\( k_1 \\) and \\( k_2 \\) are given with centers \\( P \\) and \\( R \\) respectively, touching externally at point \\( A \\). Let \\( p \\) be their common tangent line which does not pass through \\( A \\) and touches \\( k_1 \\) at \\( B \\) and \\( k_2 \\) at \\( C \\). The line \\( PR \\) cuts \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2a18f1e68d551cc139462e97ea9360f9b8912fe0099d209b3485868bb78ab1f1 | Two circles \( k_1 \) and \( k_2 \) intersect at points \( A \) and \( B \), with \( k_1 \) passing through the center \( O \) of \( k_2 \). A line \( p \) intersects \( k_1 \) at points \( K \) and \( O \), and \( k_2 \) at points \( L \) and \( M \) such that \( L \) is between \( K \) and \( O \). Let \( P \) be the... | [
"1"
] | [
"Two circles \\( k_1 \\) and \\( k_2 \\) intersect at points \\( A \\) and \\( B \\), with \\( k_1 \\) passing through the center \\( O \\) of \\( k_2 \\). A line \\( p \\) intersects \\( k_1 \\) at points \\( K \\) and \\( O \\), and \\( k_2 \\) at points \\( L \\) and \\( M \\) such that \\( L \\) is between \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-076cae9c8ea5d3197bfa6c32764206200c4755c64478ff47a7180eee218fcaea | Two circles \( k_1 \) and \( k_2 \) intersect at points \( A \) and \( B \). A line \( r \) passes through \( B \), intersecting \( k_1 \) at \( C \) and \( k_2 \) at \( D \) such that \( B \) is between \( C \) and \( D \). A line \( s \) parallel to \( AD \) is tangent to \( k_1 \) at \( E \) and is at the minimum di... | [
"30^\\circ"
] | [
"Two circles \\( k_1 \\) and \\( k_2 \\) intersect at points \\( A \\) and \\( B \\). A line \\( r \\) passes through \\( B \\), intersecting \\( k_1 \\) at \\( C \\) and \\( k_2 \\) at \\( D \\) such that \\( B \\) is between \\( C \\) and \\( D \\). A line \\( s \\) parallel to \\( AD \\) is tangent to \\( k_1 \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a845d200caf324ea027c7fcba5e2672c9a39e97fdceab053012c2bc359d4f186 | Two circles \( k_1 \) and \( k_2 \) intersect at points \( A \) and \( B \). A line through \( B \) intersects \( k_1 \) at \( C \) and \( k_2 \) at \( E \). Another line through \( B \) intersects \( k_1 \) at \( D \) and \( k_2 \) at \( F \). Assume \( B \) lies between \( C \) and \( E \), and between \( D \) and \(... | [
"75"
] | [
"Two circles \\( k_1 \\) and \\( k_2 \\) intersect at points \\( A \\) and \\( B \\). A line through \\( B \\) intersects \\( k_1 \\) at \\( C \\) and \\( k_2 \\) at \\( E \\). Another line through \\( B \\) intersects \\( k_1 \\) at \\( D \\) and \\( k_2 \\) at \\( F \\). Assume \\( B \\) lies between \\( C \\) an... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-687d3edbd82b338e699f6d2e7bc50bd6187768bacfdb2d400d25cb3e717528d3 | Two circles \( k_1 \) and \( k_2 \) intersect at points \( A \) and \( B \). The line tangent to \( k_1 \) at \( A \) intersects \( k_2 \) at point \( C \), and the line tangent to \( k_2 \) at \( B \) intersects \( k_1 \) at point \( D \). If \( BC = 12 \) and \( AD = 18 \), find the value of \( \frac{AC^2}{BD^2} \). | [
"\\dfrac{9}{4}"
] | [
"Two circles \\( k_1 \\) and \\( k_2 \\) intersect at points \\( A \\) and \\( B \\). The line tangent to \\( k_1 \\) at \\( A \\) intersects \\( k_2 \\) at point \\( C \\), and the line tangent to \\( k_2 \\) at \\( B \\) intersects \\( k_1 \\) at point \\( D \\). If \\( BC = 12 \\) and \\( AD = 18 \\), find the v... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-3212651fc8e24b79d7d8fbeac9e9e1821b30d80b4879a06251cb513cebb53697 | Two circles \( k_1 \) and \( k_2 \) intersect at points \( P \) and \( Q \). A straight line intersects circle \( k_1 \) at points \( A \) and \( C \) and circle \( k_2 \) at points \( B \) and \( D \), with point \( B \) between points \( A \) and \( C \) and point \( C \) lies between points \( B \) and \( D \). If t... | [
"1"
] | [
"Two circles \\( k_1 \\) and \\( k_2 \\) intersect at points \\( P \\) and \\( Q \\). A straight line intersects circle \\( k_1 \\) at points \\( A \\) and \\( C \\) and circle \\( k_2 \\) at points \\( B \\) and \\( D \\), with point \\( B \\) between points \\( A \\) and \\( C \\) and point \\( C \\) lies between... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-c4cdc4d9933da759d2c9bb0cd19808e4684ee759bd0a7b3a2b6675b19363d69a | Two circles \( k_1 \) and \( k_2 \) touch each other externally at \( K \). They also touch a circle \( \omega \) internally at \( A_1 \) and \( A_2 \), respectively. Let \( P \) be one point of intersection of \( \omega \) with the common tangent to \( k_1 \) and \( k_2 \) at \( K \). The line \( PA_1 \) meets \( k_1 ... | [
"2\\sqrt{r_1 r_2}"
] | [
"Two circles \\( k_1 \\) and \\( k_2 \\) touch each other externally at \\( K \\). They also touch a circle \\( \\omega \\) internally at \\( A_1 \\) and \\( A_2 \\), respectively. Let \\( P \\) be one point of intersection of \\( \\omega \\) with the common tangent to \\( k_1 \\) and \\( k_2 \\) at \\( K \\). The ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-6b200f880a4493bae41a07bd94107e094c5f010f485f7bfbc7c974b319085f03 | Two circles \( k_1 \) and \( k_2 \) with centers \( O_1 \) and \( O_2 \) are tangent to each other externally at point \( O \). Points \( X \) and \( Y \) on \( k_1 \) and \( k_2 \) respectively are such that rays \( O_1X \) and \( O_2Y \) are parallel and codirectional. Let \( T_1 \) and \( T_2 \) be the points where ... | [
"no_valid_answer_found"
] | [
"Two circles \\( k_1 \\) and \\( k_2 \\) with centers \\( O_1 \\) and \\( O_2 \\) are tangent to each other externally at point \\( O \\). Points \\( X \\) and \\( Y \\) on \\( k_1 \\) and \\( k_2 \\) respectively are such that rays \\( O_1X \\) and \\( O_2Y \\) are parallel and codirectional. Let \\( T_1 \\) and \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac0287c8270b00d1896fe5ea65d4fdbc9fca39dd0b9c45476c83eb9328662010 | Two circles \( k_1 \) and \( k_2 \) with centers \( O_1 \) and \( O_2 \) respectively are externally tangent at point \( P \). A circle \( k_3 \) is tangent to \( k_1 \) at \( Q \) and to \( k_2 \) at \( R \). The lines \( PQ \) and \( PR \) meet \( k_3 \) again at points \( A \) and \( B \) respectively. If \( AO_2 \)... | [
"90"
] | [
"Two circles \\( k_1 \\) and \\( k_2 \\) with centers \\( O_1 \\) and \\( O_2 \\) respectively are externally tangent at point \\( P \\). A circle \\( k_3 \\) is tangent to \\( k_1 \\) at \\( Q \\) and to \\( k_2 \\) at \\( R \\). The lines \\( PQ \\) and \\( PR \\) meet \\( k_3 \\) again at points \\( A \\) and \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ec9648df141e1fc2b3a51030f440e9a0c86c1b5195b352739d143e8a9f463021 | Two circles \( k_1 \) and \( k_2 \) with radii \( r_1 \) and \( r_2 \) touch each other externally at point \( Q \). The other endpoints of the diameters through \( Q \) are \( P \) on \( k_1 \) and \( R \) on \( k_2 \). Points \( A \) and \( B \) are chosen on the arcs \( PQ \) of \( k_1 \) such that \( PBQA \) is a c... | [
"(0, 0)"
] | [
"Two circles \\( k_1 \\) and \\( k_2 \\) with radii \\( r_1 \\) and \\( r_2 \\) touch each other externally at point \\( Q \\). The other endpoints of the diameters through \\( Q \\) are \\( P \\) on \\( k_1 \\) and \\( R \\) on \\( k_2 \\). Points \\( A \\) and \\( B \\) are chosen on the arcs \\( PQ \\) of \\( k_... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-764546cefccd538ad3eebbe492a31485de66de6ac70eb75de3c56338eb290d02 | Two circles \( k_1 \) and \( k_2 \) with the same radius intersect at points \( B \) and \( C \). The center \( O_1 \) of \( k_1 \) lies on \( k_2 \) and the center \( O_2 \) of \( k_2 \) lies on \( k_1 \). \( AB \) is a diameter of \( k_1 \) and \( AO_2 \) intersects \( k_2 \) at points \( M \) and \( N \), with \( M ... | [
"\\(\\sqrt{3}:1\\)"
] | [
"Two circles \\( k_1 \\) and \\( k_2 \\) with the same radius intersect at points \\( B \\) and \\( C \\). The center \\( O_1 \\) of \\( k_1 \\) lies on \\( k_2 \\) and the center \\( O_2 \\) of \\( k_2 \\) lies on \\( k_1 \\). \\( AB \\) is a diameter of \\( k_1 \\) and \\( AO_2 \\) intersects \\( k_2 \\) at point... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 17
} | 30 | 0 | 0 |
omr-ws-6ecea05fcd6173e5a5263748d91db1c350728c295d7a35528d0fdafb237cf976 | Two circles \( k_1 \) and \( k_2 \) with the same radius intersect at points \( B \) and \( C \). The center \( O_1 \) of \( k_1 \) lies on \( k_2 \) and the center \( O_2 \) of \( k_2 \) lies on \( k_1 \). \( AB \) is a diameter of \( k_1 \), and \( AO_2 \) intersects \( k_2 \) at points \( M \) and \( N \), with \( M... | [
"\\(\\sqrt{3}\\)"
] | [
"Two circles \\( k_1 \\) and \\( k_2 \\) with the same radius intersect at points \\( B \\) and \\( C \\). The center \\( O_1 \\) of \\( k_1 \\) lies on \\( k_2 \\) and the center \\( O_2 \\) of \\( k_2 \\) lies on \\( k_1 \\). \\( AB \\) is a diameter of \\( k_1 \\), and \\( AO_2 \\) intersects \\( k_2 \\) at poin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 5,
"tir": 20
} | 70 | 0 | 0 |
omr-ws-fc5070f7d4753d9c197252e644aadce6854a80484ce41289b916824ea92a3e28 | Two circles \( k_1(O_1, R) \) and \( k_2(O_2, r) \) are given in the plane such that \( R \ge \sqrt{2} r \) and \( O_1O_2 = \sqrt{R^2 + r^2 - r\sqrt{4R^2 + r^2}} \). Let \( A \) be an arbitrary point on \( k_1 \). The tangents from \( A \) to \( k_2 \) touch \( k_2 \) at \( B \) and \( C \) and intersect \( k_1 \) agai... | [
"1"
] | [
"Two circles \\( k_1(O_1, R) \\) and \\( k_2(O_2, r) \\) are given in the plane such that \\( R \\ge \\sqrt{2} r \\) and \\( O_1O_2 = \\sqrt{R^2 + r^2 - r\\sqrt{4R^2 + r^2}} \\). Let \\( A \\) be an arbitrary point on \\( k_1 \\). The tangents from \\( A \\) to \\( k_2 \\) touch \\( k_2 \\) at \\( B \\) and \\( C \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-bb29bd9a4166af30a3adf071d1aee07c5122b70f4e55003c48b9e39cb80df1c9 | Two circles \( k_{1} \) and \( k_{2} \) with centers \( O_{1} \) and \( O_{2} \) (radius \( k_{2} \) > radius \( k_{1} \)) are externally tangent at point \( M \). For a point \( A \) on \( k_{2} \) not on line \( O_{1}O_{2} \), let \( AB \) and \( AC \) be the two tangents to \( k_{1} \) (with \( B \) and \( C \) on \... | [
"5"
] | [
"Two circles \\( k_{1} \\) and \\( k_{2} \\) with centers \\( O_{1} \\) and \\( O_{2} \\) (radius \\( k_{2} \\) > radius \\( k_{1} \\)) are externally tangent at point \\( M \\). For a point \\( A \\) on \\( k_{2} \\) not on line \\( O_{1}O_{2} \\), let \\( AB \\) and \\( AC \\) be the two tangents to \\( k_{1} \\)... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-73f5e52cf7b2e3d933fdae444562fbba61561ec4399d44ac88c454b048506a3b | Two circles \( s_1 \) and \( s_2 \) with centers \( O_1 \) and \( O_2 \) intersect at points \( A \) and \( B \). Let \( M \) be an arbitrary point on circle \( s_1 \), \( MA \) intersects \( s_2 \) at point \( P \), and \( MB \) intersects \( s_2 \) at point \( Q \). If the quadrilateral \( AO_1BO_2 \) can be inscribe... | [
"2"
] | [
"Two circles \\( s_1 \\) and \\( s_2 \\) with centers \\( O_1 \\) and \\( O_2 \\) intersect at points \\( A \\) and \\( B \\). Let \\( M \\) be an arbitrary point on circle \\( s_1 \\), \\( MA \\) intersects \\( s_2 \\) at point \\( P \\), and \\( MB \\) intersects \\( s_2 \\) at point \\( Q \\). If the quadrilater... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-9d06a7100ed5d1cf64198cdc927e4eebcb44b0cf1a8e1f1c23d871b9820e95b3 | Two circles \( w1 \) and \( w2 \) intersect at points \( A \) and \( B \). Their common external tangent touches \( w1 \) at \( X \) and \( w2 \) at \( Y \). A point \( Z \) is chosen on line \( XY \). Circle \( w3 \), passing through \( A \), \( B \), and \( Z \), intersects line \( XY \) at point \( W \) (not equal t... | [
"2"
] | [
"Two circles \\( w1 \\) and \\( w2 \\) intersect at points \\( A \\) and \\( B \\). Their common external tangent touches \\( w1 \\) at \\( X \\) and \\( w2 \\) at \\( Y \\). A point \\( Z \\) is chosen on line \\( XY \\). Circle \\( w3 \\), passing through \\( A \\), \\( B \\), and \\( Z \\), intersects line \\( X... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5ab2e2f295292b83d5464dfed1ccc8abbd0f50a5a9edb2f3406a215f24baa46e | Two circles \( w1 \) and \( w2 \) with centers \( P \) and \( Q \) intersect at points \( A \) and \( B \). Point \( C \) is on circle \( w1 \). Line \( BC \) intersects circle \( w2 \) at point \( D \). Line \( AD \) intersects circle \( w1 \) again at point \( E \). Line \( CE \) intersects line \( AB \) at point \( ... | [
"10"
] | [
"Two circles \\( w1 \\) and \\( w2 \\) with centers \\( P \\) and \\( Q \\) intersect at points \\( A \\) and \\( B \\). Point \\( C \\) is on circle \\( w1 \\). Line \\( BC \\) intersects circle \\( w2 \\) at point \\( D \\). Line \\( AD \\) intersects circle \\( w1 \\) again at point \\( E \\). Line \\( CE \\) in... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-6f6c2c8464445ec9fb6b5498f55248e575619345783e240d087b4788e5c19d23 | Two circles \( w_1 \) and \( w_2 \) are tangent at point \( T \). Two lines \( a \) and \( b \) through \( T \) intersect the circle \( w_1 \) again at \( A \) and \( B \) respectively, and intersect \( w_2 \) again at \( A_1 \) and \( B_1 \) respectively. Let \( X \) be a point in the plane, not lying on either circle... | [
"1"
] | [
"Two circles \\( w_1 \\) and \\( w_2 \\) are tangent at point \\( T \\). Two lines \\( a \\) and \\( b \\) through \\( T \\) intersect the circle \\( w_1 \\) again at \\( A \\) and \\( B \\) respectively, and intersect \\( w_2 \\) again at \\( A_1 \\) and \\( B_1 \\) respectively. Let \\( X \\) be a point in the pl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-1b2a7ece5d6a6106d72cc1b728ca2bde288c77702870c3fea5c688eb3c2f7611 | Two circles \( w_1 \) and \( w_2 \) in the plane are centered at \( A \) and \( B \) respectively and intersect at points \( C \) and \( D \). The circumcircle of \( \triangle ABC \) intersects \( w_1 \) at \( E \) and \( w_2 \) at \( F \), where the arc \( EF \) not containing \( C \) lies outside \( w_1 \) and \( w_2... | [
"\\dfrac{1}{2}"
] | [
"Two circles \\( w_1 \\) and \\( w_2 \\) in the plane are centered at \\( A \\) and \\( B \\) respectively and intersect at points \\( C \\) and \\( D \\). The circumcircle of \\( \\triangle ABC \\) intersects \\( w_1 \\) at \\( E \\) and \\( w_2 \\) at \\( F \\), where the arc \\( EF \\) not containing \\( C \\) l... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-5e4f49f4485afb8b3772cbe720194e4e41e37596cfcb349c6e4705807103d17c | Two circles \( w_1 \) and \( w_2 \) intersect at points \( A \) and \( B \). A common tangent line \( l \) to these circles, closer to \( A \) than \( B \), touches \( w_1 \) at \( C \) and \( w_2 \) at \( D \). The lines \( CA \) and \( DA \) intersect \( w_2 \) and \( w_1 \) at \( E \) and \( F \) respectively. A lin... | [
"1"
] | [
"Two circles \\( w_1 \\) and \\( w_2 \\) intersect at points \\( A \\) and \\( B \\). A common tangent line \\( l \\) to these circles, closer to \\( A \\) than \\( B \\), touches \\( w_1 \\) at \\( C \\) and \\( w_2 \\) at \\( D \\). The lines \\( CA \\) and \\( DA \\) intersect \\( w_2 \\) and \\( w_1 \\) at \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ec39fde64a99d876d8926abaae7db1ed1f7642e1f7c6ca425a52dacfd7f82f88 | Two circles \( w_1 \) and \( w_2 \) intersect at points \( A \) and \( B \). Tangent lines at points \( A \) and \( B \) of circle \( w_1 \) intersect at \( X \), and tangent lines at points \( A \) and \( B \) of circle \( w_2 \) intersect at \( Y \). Point \( C \) is an arbitrary point on segment \( AB \). Segment \(... | [
"no_valid_answer_found"
] | [
"Two circles \\( w_1 \\) and \\( w_2 \\) intersect at points \\( A \\) and \\( B \\). Tangent lines at points \\( A \\) and \\( B \\) of circle \\( w_1 \\) intersect at \\( X \\), and tangent lines at points \\( A \\) and \\( B \\) of circle \\( w_2 \\) intersect at \\( Y \\). Point \\( C \\) is an arbitrary point ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41720a90f9d4ac572c4eae1660fe615413e426c5328a2eafb97b5b5fcab6ceec | Two circles \( w_1 \) and \( w_2 \) intersect at points \( A \) and \( B \). Tangents \( \ell_1 \) and \( \ell_2 \) are drawn to \( w_1 \) and \( w_2 \) respectively through point \( A \). The perpendiculars dropped from point \( B \) to \( \ell_2 \) and \( \ell_1 \) intersect the circles \( w_1 \) and \( w_2 \) at poi... | [
"\\dfrac{266}{25}"
] | [
"Two circles \\( w_1 \\) and \\( w_2 \\) intersect at points \\( A \\) and \\( B \\). Tangents \\( \\ell_1 \\) and \\( \\ell_2 \\) are drawn to \\( w_1 \\) and \\( w_2 \\) respectively through point \\( A \\). The perpendiculars dropped from point \\( B \\) to \\( \\ell_2 \\) and \\( \\ell_1 \\) intersect the circl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-3ec65b0b84fac2cb29e2545032c00327107b33527ebe5f93c9b0995697f5a847 | Two circles \( w_1 \) and \( w_2 \) intersect at points \( P \) and \( Q \). Point \( C \) is any point on \( w_1 \) different from \( P \) and \( Q \). Line \( CP \) intersects \( w_2 \) at point \( A \), and line \( CQ \) intersects \( w_2 \) at point \( B \). Determine the locus of the circumcenter of triangle \( AB... | [
"\\text{The perpendicular bisector of segment } PQ"
] | [
"Two circles \\( w_1 \\) and \\( w_2 \\) intersect at points \\( P \\) and \\( Q \\). Point \\( C \\) is any point on \\( w_1 \\) different from \\( P \\) and \\( Q \\). Line \\( CP \\) intersects \\( w_2 \\) at point \\( A \\), and line \\( CQ \\) intersects \\( w_2 \\) at point \\( B \\). Determine the locus of t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-c501ed7ca0133e1daa87be255ac94391ff4f817516dce6cd4cfbb8722788ac40 | Two circles \( w_1 \) and \( w_2 \) intersect at points \( P \) and \( Q \). The common tangent to \( w_1 \) and \( w_2 \), which is closer to \( P \) than to \( Q \), touches these circles at points \( A \) and \( B \), respectively. The tangent to \( w_1 \) at \( P \) intersects \( w_2 \) at \( E \) (different from \... | [
"12"
] | [
"Two circles \\( w_1 \\) and \\( w_2 \\) intersect at points \\( P \\) and \\( Q \\). The common tangent to \\( w_1 \\) and \\( w_2 \\), which is closer to \\( P \\) than to \\( Q \\), touches these circles at points \\( A \\) and \\( B \\), respectively. The tangent to \\( w_1 \\) at \\( P \\) intersects \\( w_2 \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-0316f638b1333c69c7df5df1f8a8cd32d141fd898d112bd085bd52b23f7901a0 | Two circles \( w_1 \) and \( w_2 \) intersect at points \( P \) and \( Q \). The common tangent to \( w_1 \) and \( w_2 \), which is closer to point \( P \) than to \( Q \), touches these circles at points \( A \) and \( B \), respectively. The tangent to \( w_1 \) at point \( P \) intersects \( w_2 \) at point \( E \)... | [
"0"
] | [
"Two circles \\( w_1 \\) and \\( w_2 \\) intersect at points \\( P \\) and \\( Q \\). The common tangent to \\( w_1 \\) and \\( w_2 \\), which is closer to point \\( P \\) than to \\( Q \\), touches these circles at points \\( A \\) and \\( B \\), respectively. The tangent to \\( w_1 \\) at point \\( P \\) intersec... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-e97edd6253644f1bd83f1da16079afa4f1c4bc0128bcaee1c1f2bb988f9cc40c | Two circles \( w_1 \) and \( w_2 \) of different radii touch externally at \( L \). A line touches \( w_1 \) at \( A \) and \( w_2 \) at \( B \) (the points \( A \) and \( B \) are different from \( L \)). A point \( X \) is chosen in the plane. \( Y \) and \( Z \) are the second points of intersection of the lines \( ... | [
"24"
] | [
"Two circles \\( w_1 \\) and \\( w_2 \\) of different radii touch externally at \\( L \\). A line touches \\( w_1 \\) at \\( A \\) and \\( w_2 \\) at \\( B \\) (the points \\( A \\) and \\( B \\) are different from \\( L \\)). A point \\( X \\) is chosen in the plane. \\( Y \\) and \\( Z \\) are the second points o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c7989cdecb8bd25f7f0ee64f2c8375c26a251373efd29510efa9d718f1d481b4 | Two circles \((A)\) and \((C)\) are outside each other. Their internal tangent touches \((A)\) and \((C)\) at points \(H\) and \(E\), respectively. Their external tangent touches \((A)\) and \((C)\) at points \(G\) and \(F\), respectively. If the coordinates of \(A\) are \((0,0)\) and \(C\) are \((10,0)\), and the radi... | [
"3"
] | [
"Two circles \\((A)\\) and \\((C)\\) are outside each other. Their internal tangent touches \\((A)\\) and \\((C)\\) at points \\(H\\) and \\(E\\), respectively. Their external tangent touches \\((A)\\) and \\((C)\\) at points \\(G\\) and \\(F\\), respectively. If the coordinates of \\(A\\) are \\((0,0)\\) and \\(C\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-cda07a9c090d232adf3d837bf9d56329c1d18257778e7f4ac181722dec87b245 | Two circles \((C)\) and \((C')\) intersect at points \(M\) and \(N\), with the distance between their centers being \(l\). Let \(A\) be a point on circle \((C)\) (where \(A \neq M\) and \(A \neq N\)). The lines \(AM\) and \(AN\) intersect circle \((C')\) at points \(B\) and \(C\), respectively. Calculate the radius of ... | [
"l"
] | [
"Two circles \\((C)\\) and \\((C')\\) intersect at points \\(M\\) and \\(N\\), with the distance between their centers being \\(l\\). Let \\(A\\) be a point on circle \\((C)\\) (where \\(A \\neq M\\) and \\(A \\neq N\\)). The lines \\(AM\\) and \\(AN\\) intersect circle \\((C')\\) at points \\(B\\) and \\(C\\), res... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-7cbd44a562f466d9ea38bc0b11341b0493df557c19f63759523f2d7e91993e94 | Two circles \((O)\) and \((O')\) intersect at points \(A\) and \(B\). Let \(M\) be the midpoint of \(AB\). Let \(P\) be a point on \((O)\) and \(Q\) be a point on \((O')\) such that \(OP\) is parallel to \(O'Q\). The circle with diameter \(PQ\) intersects \((O)\) and \((O')\) at points \(K\) and \(L\) respectively. Let... | [
"0"
] | [
"Two circles \\((O)\\) and \\((O')\\) intersect at points \\(A\\) and \\(B\\). Let \\(M\\) be the midpoint of \\(AB\\). Let \\(P\\) be a point on \\((O)\\) and \\(Q\\) be a point on \\((O')\\) such that \\(OP\\) is parallel to \\(O'Q\\). The circle with diameter \\(PQ\\) intersects \\((O)\\) and \\((O')\\) at point... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-2e6f09339fa71277489869a55d7864771edae6f2b6b6dbaea9aa83fd52cffcb5 | Two circles \((O)\) and \((O')\) intersect at points \(A\) and \(B\). Through a point \(C\) on the opposite ray of the ray \(BA\), draw the tangents \(CD\) and \(CE\) to circle \((O)\). The line segment \(DE\) intersects circle \((O')\) at point \(F\). The tangent to circle \((O')\) at \(F\) intersects \(CD\) and \(CE\... | [
"70"
] | [
"Two circles \\((O)\\) and \\((O')\\) intersect at points \\(A\\) and \\(B\\). Through a point \\(C\\) on the opposite ray of the ray \\(BA\\), draw the tangents \\(CD\\) and \\(CE\\) to circle \\((O)\\). The line segment \\(DE\\) intersects circle \\((O')\\) at point \\(F\\). The tangent to circle \\((O')\\) at \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-a2d1e8a8c45daf504c5f1f21230a0a73c1a43dd9bd4b99b561b91f36fbfdd3a6 | Two circles \((O)\) and \(\Omega_{1}\) intersect at points \(A\) and \(B\) such that \(OA\) and \(OB\) are tangent to \(\Omega_{1}\). Let circle \(\Omega_{2}\) touch internally the circle \((O)\) and the midpoint of \(AB\) is on \(\Omega_{2}\). Suppose that the circles \(\Omega_{1}\) and \(\Omega_{2}\) intersect at poi... | [
"10"
] | [
"Two circles \\((O)\\) and \\(\\Omega_{1}\\) intersect at points \\(A\\) and \\(B\\) such that \\(OA\\) and \\(OB\\) are tangent to \\(\\Omega_{1}\\). Let circle \\(\\Omega_{2}\\) touch internally the circle \\((O)\\) and the midpoint of \\(AB\\) is on \\(\\Omega_{2}\\). Suppose that the circles \\(\\Omega_{1}\\) a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-a22d75a09d8f54505d24a45c55a63a7557ea6014a601677fe2b98e3f4d51850a | Two circles \((O; R)\) and \((O'; R')\) intersect at points \(A\) and \(B\). A point \(M\) varies on the opposite ray of the ray \(AB\). From \(M\), draw two tangent lines to the circle \((O'; R')\) at points \(C\) and \(D\), with \(D\) inside the circle \((O; R)\). The lines \(AD\) and \(AC\) intersect the circle \((O... | [
"\\dfrac{2dR^2}{d^2 + R^2 - R'^2}"
] | [
"Two circles \\((O; R)\\) and \\((O'; R')\\) intersect at points \\(A\\) and \\(B\\). A point \\(M\\) varies on the opposite ray of the ray \\(AB\\). From \\(M\\), draw two tangent lines to the circle \\((O'; R')\\) at points \\(C\\) and \\(D\\), with \\(D\\) inside the circle \\((O; R)\\). The lines \\(AD\\) and \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-978aa5e3813819f885726b38371311fd976e7b03dbc0e6ceaa48c84c7901a470 | Two circles \((O; R)\) and \((O'; r)\) are tangent internally at a point \(A\) (\(R > r\)). A line tangent to \((O')\) at a point \(D\) intersects \((O)\) at points \(B\) and \(C\). \((I)\) is the inscribed circle of triangle \(ABC\). Find the locus of \(I\). | [
"\\text{The circle with diameter } AO'"
] | [
"Two circles \\((O; R)\\) and \\((O'; r)\\) are tangent internally at a point \\(A\\) (\\(R > r\\)). A line tangent to \\((O')\\) at a point \\(D\\) intersects \\((O)\\) at points \\(B\\) and \\(C\\). \\((I)\\) is the inscribed circle of triangle \\(ABC\\). Find the locus of \\(I\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-913f80ab7d53d1e5e374f0eb1c1841ec7bec514821dfa9e276d7077dea0fbf60 | Two circles \((O_1)\) and \((O_2)\) are tangent at point \(A\). A common tangent to both circles touches them at points \(B_1\) and \(B_2\) respectively. Let \(C\) be the foot of the perpendicular from \(A\) to the line \(B_1B_2\). If the radii of the circles are \(r_1\) and \(r_2\) respectively, find the value of \(\f... | [
"4"
] | [
"Two circles \\((O_1)\\) and \\((O_2)\\) are tangent at point \\(A\\). A common tangent to both circles touches them at points \\(B_1\\) and \\(B_2\\) respectively. Let \\(C\\) be the foot of the perpendicular from \\(A\\) to the line \\(B_1B_2\\). If the radii of the circles are \\(r_1\\) and \\(r_2\\) respectivel... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-014716044fc7b3948d9099dcff108dde962db5599ead9cfee6cc6c5612591b93 | Two circles \((O_1)\) and \((O_2)\) intersect at points \(A\) and \(B\), with the center \(O_2\) lying on circle \((O_1)\). The chord \(O_2C\) of circle \((O_1)\) intersects circle \((O_2)\) at point \(D\). If \(\angle BAC = 60^\circ\) and \(AB = 6\), find the length of \(AD\). | [
"3(1+\\sqrt{3})"
] | [
"Two circles \\((O_1)\\) and \\((O_2)\\) intersect at points \\(A\\) and \\(B\\), with the center \\(O_2\\) lying on circle \\((O_1)\\). The chord \\(O_2C\\) of circle \\((O_1)\\) intersects circle \\((O_2)\\) at point \\(D\\). If \\(\\angle BAC = 60^\\circ\\) and \\(AB = 6\\), find the length of \\(AD\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-e9fa4475c05cab28cdf5d6177be48fde0f65f1d81c59b25e6122822849af635c | Two circles \((O_1)\) and \((O_2)\) intersect at points \(A\) and \(B\). A line through \(A\) intersects \((O_1)\) at \(C\) and \((O_2)\) at \(D\). Point \(P\) is on \(AB\) such that \(B\) is between \(A\) and \(P\). Line \(PC\) intersects \((O_1)\) again at \(M\), and \(AM\) intersects \(BC\) at \(E\). Line \(PD\) int... | [
"75^\\circ"
] | [
"Two circles \\((O_1)\\) and \\((O_2)\\) intersect at points \\(A\\) and \\(B\\). A line through \\(A\\) intersects \\((O_1)\\) at \\(C\\) and \\((O_2)\\) at \\(D\\). Point \\(P\\) is on \\(AB\\) such that \\(B\\) is between \\(A\\) and \\(P\\). Line \\(PC\\) intersects \\((O_1)\\) again at \\(M\\), and \\(AM\\) in... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-d5f96ec937826c29e38086896cd808b837676ccdfc704d4eb837435edcb2c6c9 | Two circles \((O_1)\) and \((O_2)\) intersect at points \(A\) and \(B\). Point \(M\) lies on the opposite ray of \(AB\). The line through \(B\) intersects \((O_1)\) at \(C\) and \((O_2)\) at \(D\). \(MC\) intersects \((O_1)\) at \(P\) and \(MD\) intersects \((O_2)\) at \(Q\). Points \(E\) and \(F\) are the intersection... | [
"\\dfrac{13}{17}"
] | [
"Two circles \\((O_1)\\) and \\((O_2)\\) intersect at points \\(A\\) and \\(B\\). Point \\(M\\) lies on the opposite ray of \\(AB\\). The line through \\(B\\) intersects \\((O_1)\\) at \\(C\\) and \\((O_2)\\) at \\(D\\). \\(MC\\) intersects \\((O_1)\\) at \\(P\\) and \\(MD\\) intersects \\((O_2)\\) at \\(Q\\). Poin... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-afbb881c8072e6b2972216e90d5836b4c118efec492a566bf06e68eae9af32f9 | Two circles \((O_1)\) and \((O_2)\) intersect at points \(A\) and \(B\). The line \(AB\) is extended to point \(C\). Tangents \(CP\) and \(CQ\) are drawn to circles \((O_1)\) and \((O_2)\) respectively, such that \(P\) is inside \((O_2)\) and \(Q\) is inside \((O_1)\). Let \(D\) and \(E\) be the points where the tangen... | [
"(2, 3)"
] | [
"Two circles \\((O_1)\\) and \\((O_2)\\) intersect at points \\(A\\) and \\(B\\). The line \\(AB\\) is extended to point \\(C\\). Tangents \\(CP\\) and \\(CQ\\) are drawn to circles \\((O_1)\\) and \\((O_2)\\) respectively, such that \\(P\\) is inside \\((O_2)\\) and \\(Q\\) is inside \\((O_1)\\). Let \\(D\\) and \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-cb3defd3f975e96965c405d50d055344f68c92fcebde8e7fdd0140cf16c96b5f | Two circles \((O_1)\) and \((O_2)\) intersect at points \(A\) and \(B\). Their common tangent touches \((O_1)\) at \(P_1\) and \((O_2)\) at \(P_2\). Let \(M_1\) and \(M_2\) be the orthogonal projections of \(P_1\) and \(P_2\) on the line \(O_1O_2\). The line \(AM_1\) cuts \((O_1)\) again at \(N_1\) and \(AM_2\) cuts \(... | [
"14\\sqrt{2}"
] | [
"Two circles \\((O_1)\\) and \\((O_2)\\) intersect at points \\(A\\) and \\(B\\). Their common tangent touches \\((O_1)\\) at \\(P_1\\) and \\((O_2)\\) at \\(P_2\\). Let \\(M_1\\) and \\(M_2\\) be the orthogonal projections of \\(P_1\\) and \\(P_2\\) on the line \\(O_1O_2\\). The line \\(AM_1\\) cuts \\((O_1)\\) ag... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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