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-b753347c42af58d8dc94091e2aed6ba1b61aec9e47ba2aa99ba9fc48f980d143 | Two circles lie outside a regular hexagon \(ABCDEF\). The first circle is tangent to \(\overline{AB}\) and the second circle is tangent to \(\overline{DE}\). Both circles are also tangent to lines \(BC\) and \(FA\). What is the ratio of the area of the second circle to that of the first circle? | [
"81"
] | [
"Two circles lie outside a regular hexagon \\(ABCDEF\\). The first circle is tangent to \\(\\overline{AB}\\) and the second circle is tangent to \\(\\overline{DE}\\). Both circles are also tangent to lines \\(BC\\) and \\(FA\\). What is the ratio of the area of the second circle to that of the first circle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-907666c3e9fb60929b1a4c3c1c86480daaafde908209f0d037e3087d9b2dd1f8 | Two circles lie outside a regular hexagon \(ABCDEF\). The first circle is tangent to \(\overline{AB}\), and the second circle is tangent to \(\overline{DE}\). Both circles are tangent to lines \(BC\) and \(FA\). What is the ratio of the area of the second circle to that of the first circle? | [
"9"
] | [
"Two circles lie outside a regular hexagon \\(ABCDEF\\). The first circle is tangent to \\(\\overline{AB}\\), and the second circle is tangent to \\(\\overline{DE}\\). Both circles are tangent to lines \\(BC\\) and \\(FA\\). What is the ratio of the area of the second circle to that of the first circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-290c5e3f9a47b748daadc8626e0b3e2a6183e9294f7b8146dcd2835febb19c05 | Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. If the tangent to the second circle at $R$ intersects the line $AK$ at... | [
"\\dfrac{3}{4}"
] | [
"Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. If the tangent to the second circle at $R$ intersects the line $A... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-76532f0651e56fe3c0057c27083e78d206467ae7f60d8e36baf43bce89d002e0 | Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. Let $T$ be the intersection of the tangent to the second circle at $R$... | [
"20"
] | [
"Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. Let $T$ be the intersection of the tangent to the second circle a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1232fe8529c75afc18a403e4e384923715648ffbc02c6f114b9f2010ec3d46ea | Two circles of diameter 12 cm overlap, forming a shape with a perimeter of $14\pi$ cm. What is the area of the shape in square cm? | [
"\\(40\\pi + 18\\)"
] | [
"Two circles of diameter 12 cm overlap, forming a shape with a perimeter of $14\\pi$ cm. What is the area of the shape in square cm?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-dcb2b46872fde602aafa493089c5bfb48eb99409708f1e5c4916db7208a3f405 | Two circles of different radii are cut out of cardboard. Each circle is subdivided into 200 equal sectors. On each circle, 100 sectors are painted white and the other 100 are painted black. The smaller circle is then placed on top of the larger circle, so that their centers coincide. Find the minimum number of sectors ... | [
"0"
] | [
"Two circles of different radii are cut out of cardboard. Each circle is subdivided into 200 equal sectors. On each circle, 100 sectors are painted white and the other 100 are painted black. The smaller circle is then placed on top of the larger circle, so that their centers coincide. Find the minimum number of sec... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-a916695f8fd8d7487d4b4204dba73f9dcc151b1dc74f677c9c6a3c96c5d8fb16 | Two circles of different radii are each subdivided into 200 equal sectors. On each circle, 100 sectors are painted white and the other 100 are painted black. The smaller circle is placed on top of the larger circle so that their centers coincide. After optimally rotating the smaller circle, let $N$ be the number of sec... | [
"0"
] | [
"Two circles of different radii are each subdivided into 200 equal sectors. On each circle, 100 sectors are painted white and the other 100 are painted black. The smaller circle is placed on top of the larger circle so that their centers coincide. After optimally rotating the smaller circle, let $N$ be the number o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-c44c0735216e2cd6a40438a9538fe7497a78a2d550572affc6ab0a7c461e33a4 | Two circles of different radii are each subdivided into 200 equal sectors. On each circle, 100 sectors are painted white and the other 100 are painted black. The smaller circle is placed on top of the larger circle so that their centers coincide. Calculate the minimum number of sectors on the small circle that can be m... | [
"0"
] | [
"Two circles of different radii are each subdivided into 200 equal sectors. On each circle, 100 sectors are painted white and the other 100 are painted black. The smaller circle is placed on top of the larger circle so that their centers coincide. Calculate the minimum number of sectors on the small circle that can... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-5f60cdd9361858883aa66059232a92050a85c1c7b2028dcd8f8e7ec5e3408899 | Two circles of different radii intersect at points \( M \) and \( N \). Two common tangents are drawn to these circles, with the first tangent touching the circles at points \( A \) and \( B \), and the second tangent touching the circles at points \( C \) and \( D \). Let \( H_1 \) and \( H_2 \) be the orthocenters of... | [
"\\dfrac{3}{4}"
] | [
"Two circles of different radii intersect at points \\( M \\) and \\( N \\). Two common tangents are drawn to these circles, with the first tangent touching the circles at points \\( A \\) and \\( B \\), and the second tangent touching the circles at points \\( C \\) and \\( D \\). Let \\( H_1 \\) and \\( H_2 \\) b... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-dcdcc72ed183f6d4dc1afdbc3987c7495ef10c8ee9fc0126f51753584c10ed88 | Two circles of equal radii intersect at points $X$ and $Y$ and bisect each other's areas. If the measure of the minor arc $\widehat{XY}$ is $\theta$ degrees, estimate $\left\lfloor1000\theta\right\rfloor$. | [
"132346"
] | [
"Two circles of equal radii intersect at points $X$ and $Y$ and bisect each other's areas. If the measure of the minor arc $\\widehat{XY}$ is $\\theta$ degrees, estimate $\\left\\lfloor1000\\theta\\right\\rfloor$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-55867c8d1cf032ded1cd2d6afcc298ace97202b0a0f446aa30f398d5434c969b | Two circles of equal radius intersect at two distinct points $A$ and $B$. Let their radii be $r$ and their centers be $O_1$ and $O_2$. Find the greatest possible value of the area of the rectangle $O_1AO_2B$. | [
"\\( r^2 \\)"
] | [
"Two circles of equal radius intersect at two distinct points $A$ and $B$. Let their radii be $r$ and their centers be $O_1$ and $O_2$. Find the greatest possible value of the area of the rectangle $O_1AO_2B$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-f227125afb4aa92675f5b9130566a22548b9ec56848723881451366358f82500 | Two circles of equal radius with centers \( O_1 \) and \( O_2 \) intersect at a point \( P \). A third circle of the same radius, also passing through \( P \), intersects the aforementioned circles again at \( P_1 \) and \( P_2 \) respectively. If the distance between \( O_1 \) and \( O_2 \) is \( d \), calculate the l... | [
"d"
] | [
"Two circles of equal radius with centers \\( O_1 \\) and \\( O_2 \\) intersect at a point \\( P \\). A third circle of the same radius, also passing through \\( P \\), intersects the aforementioned circles again at \\( P_1 \\) and \\( P_2 \\) respectively. If the distance between \\( O_1 \\) and \\( O_2 \\) is \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-1f72698eb1a45a7cfcb912835c575a24bd688c2a9121445ea05f62ef0e595414 | Two circles of radii $1$ and $4$ are tangent externally. What is the length of the part of the common tangent line between the points of tangency? | [
"4"
] | [
"Two circles of radii $1$ and $4$ are tangent externally. What is the length of the part of the common tangent line between the points of tangency?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e9f53edd1fdb618b8611d6cc60e7280c46bdd2a6a1cabe9c9eeb1fa6da80aff6 | Two circles of radii $20$ and $18$ are an excircle and incircle of a triangle $T$. If the distance between their centers is $47$, compute the area of $T$. | [
"258\\sqrt{10}"
] | [
"Two circles of radii $20$ and $18$ are an excircle and incircle of a triangle $T$. If the distance between their centers is $47$, compute the area of $T$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-a311eaa0d16fd289e73e52327ff2b844e3eab48dce1a4a04b1ef9e9f11691a12 | Two circles of radii $3$ and $12$ are externally tangent at $A$ and internally tangent to a third circle of radius $16$ at $B$ and $C$. What is the radius of the circle that passes through $A$, $B$, and $C$? | [
"24"
] | [
"Two circles of radii $3$ and $12$ are externally tangent at $A$ and internally tangent to a third circle of radius $16$ at $B$ and $C$. What is the radius of the circle that passes through $A$, $B$, and $C$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-32337a5e784ca611dd21d276ce0e9a1777d586f0bfd0bda9681a898a3056778f | Two circles of radii $4$ and $8$ touch at point $A$. A line through point $A$ intersects the larger circle at point $B$ and the smaller circle at point $C$. If $BC = 6\sqrt{2}$, find the length of $AB$. | [
"\\(3\\sqrt{2} + \\sqrt{34}\\)"
] | [
"Two circles of radii $4$ and $8$ touch at point $A$. A line through point $A$ intersects the larger circle at point $B$ and the smaller circle at point $C$. If $BC = 6\\sqrt{2}$, find the length of $AB$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b2088da37e017e4a4964ee39d83441ebd13253b8dd98d72c4947ca9cc674d049 | Two circles of radii $4$ and $9$ touch externally. Find the length of the common tangent to these circles. | [
"12"
] | [
"Two circles of radii $4$ and $9$ touch externally. Find the length of the common tangent to these circles."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7d48f160782b9f95fb7c770b634a23349032b9e8249e401ca653fea463f8f045 | Two circles of radii $6$ and $24$ are tangent externally. Line $\ell$ touches the first circle at point $A$, and the second at point $B$. Find the length of $AB$. | [
"24"
] | [
"Two circles of radii $6$ and $24$ are tangent externally. Line $\\ell$ touches the first circle at point $A$, and the second at point $B$. Find the length of $AB$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-68c09ba45e30d7944ffc4110843975b78fc09820bf0eb7564cd77cc53cebb65a | Two circles of radii $6$ and $4$ touch each other externally. A direct common tangent is drawn to them. Two other circles of radii $p$ and $q$ touch both the given circles and the direct common tangent. Find the values of $p$ and $q$. | [
"\\( p = 12(5 - 2\\sqrt{6}), q = 12(5 + 2\\sqrt{6}) \\)"
] | [
"Two circles of radii $6$ and $4$ touch each other externally. A direct common tangent is drawn to them. Two other circles of radii $p$ and $q$ touch both the given circles and the direct common tangent. Find the values of $p$ and $q$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-55499bc01269bc3d85cca90ed5359be020a223ff67b7f9e33ce61fe944a5847a | Two circles of radii $R$ and $r$ $(R > r)$ are internally tangent. Find the radius of the third circle tangent to the two given circles and to their common diameter. | [
"x=4Rr\\frac{R-r}{(R+r)^2}"
] | [
"Two circles of radii $R$ and $r$ $(R > r)$ are internally tangent. Find the radius of the third circle tangent to the two given circles and to their common diameter."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc9bb032f88ed7b9591d6f9dee8d1fdb6979c34a4cc23d151d3e53fceda7db3b | Two circles of radii $R$ and $r$ ($R > r$) touch internally at the point $M$, and the chord $AB$ of the larger circle touches the smaller circle. What is the greatest value of the perimeter of the triangle $ABM$? | [
"4\\sqrt{r}(\\sqrt{R} + \\sqrt{R - r})"
] | [
"Two circles of radii $R$ and $r$ ($R > r$) touch internally at the point $M$, and the chord $AB$ of the larger circle touches the smaller circle. What is the greatest value of the perimeter of the triangle $ABM$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-249eade47a5f7d0b0e26098b706ec273bf357e67cb46e19005ae3329e8b7a815 | Two circles of radii $R$ and $r$ are externally tangent. A common exterior tangent is drawn to these circles, thus forming a curvilinear triangle. Find the radius of the circle inscribed in this triangle. | [
"The radius of the circle is $\\frac{Rr}{\\left(\\sqrt R+\\sqrt r\\right)^2}$"
] | [
"Two circles of radii $R$ and $r$ are externally tangent. A common exterior tangent is drawn to these circles, thus forming a curvilinear triangle. Find the radius of the circle inscribed in this triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b814fa4347131d3a8cb993e9fb82df02c1c6bd3817720b987a93a0d20a2aeaac | Two circles of radii $R$ and $r$ are tangent to line $L$ at points $A$ and $B$ respectively, and intersect each other at points $C$ and $D$. Let $R_{ABC}$ be the radius of the circumcircle of $\triangle ABC$. If $R=3$ and $r=2$, find the value of $R_{ABC}$. | [
"\\sqrt{6}"
] | [
"Two circles of radii $R$ and $r$ are tangent to line $L$ at points $A$ and $B$ respectively, and intersect each other at points $C$ and $D$. Let $R_{ABC}$ be the radius of the circumcircle of $\\triangle ABC$. If $R=3$ and $r=2$, find the value of $R_{ABC}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe005e38ce51be0c9557b86d771d6d54654f8949d073a944c426497ea91ae34b | Two circles of radii 17 and 23 intersect such that their radii drawn to one of the points of intersection meet at right angles. Calculate the difference of the areas of the non-overlapping portions of the circles. | [
"\\( 240\\pi \\)"
] | [
"Two circles of radii 17 and 23 intersect such that their radii drawn to one of the points of intersection meet at right angles. Calculate the difference of the areas of the non-overlapping portions of the circles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-a3ef90cc4c8547db54035ec029492a06782c570a0f71c460d6a686eb73b4d184 | Two circles of radii 5 cm and 3 cm are internally tangent. A chord of the larger circle touches the smaller circle and is divided by the point of tangency in the ratio $3 :1$. Find the length of the chord. | [
"8"
] | [
"Two circles of radii 5 cm and 3 cm are internally tangent. A chord of the larger circle touches the smaller circle and is divided by the point of tangency in the ratio $3 :1$. Find the length of the chord."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4fd5cc3985e17c6d3e6a47861f49daae819e8f5046776bf1736c8349fd300942 | Two circles of radii 6 and 2 touch externally. Find the distance from the touchpoint to their common tangent that does not pass through the touchpoint. | [
"\\( 4\\sqrt{3} \\)"
] | [
"Two circles of radii 6 and 2 touch externally. Find the distance from the touchpoint to their common tangent that does not pass through the touchpoint."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fee18c034e1758f0a368b8689be9930e5c73ecfa3eedb3813c924774209c3455 | Two circles of radii \( R \) and \( r \) (with \( R > r \)) are tangent to each other externally. The sides adjacent to the base of an isosceles triangle are common tangents to these circles, and the base of the triangle is tangent to the circle of the greater radius. Determine the length of the base of the triangle. | [
"\\( 4\\sqrt{Rr} \\)"
] | [
"Two circles of radii \\( R \\) and \\( r \\) (with \\( R > r \\)) are tangent to each other externally. The sides adjacent to the base of an isosceles triangle are common tangents to these circles, and the base of the triangle is tangent to the circle of the greater radius. Determine the length of the base of the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b813184f1c0b2b23daabac111714c25144405dbf866798eb5b810dae642da4c6 | Two circles of radii \( R \) and \( r \) are externally tangent at a point \( A \). Their common external tangent is tangent to the circles at \( B \) and \( C \). Calculate the lengths of the sides of triangle \( ABC \) in terms of \( R \) and \( r \). | [
"BC = 2\\sqrt{Rr}"
] | [
"Two circles of radii \\( R \\) and \\( r \\) are externally tangent at a point \\( A \\). Their common external tangent is tangent to the circles at \\( B \\) and \\( C \\). Calculate the lengths of the sides of triangle \\( ABC \\) in terms of \\( R \\) and \\( r \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-a2abdaf49763ab79928dd3f8866015c13abfae87ce141fe042d025aaf3f70182 | Two circles of radii \( r \) and \( R \) touch each other externally. A straight line \( l \) intersects the circles at points \( A \), \( B \), \( C \), and \( D \) such that \( AB = BC = CD \). Find the length of \( AD \). | [
"\\( AD = 3x \\)"
] | [
"Two circles of radii \\( r \\) and \\( R \\) touch each other externally. A straight line \\( l \\) intersects the circles at points \\( A \\), \\( B \\), \\( C \\), and \\( D \\) such that \\( AB = BC = CD \\). Find the length of \\( AD \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68491f46fa6742cef0c864ecf74db560bbe5f330fe2aae32d4f7d27307315abf | Two circles of radii length 4 units intersect at two points. The distance between the two centers of the circles is 5 units. Find the area of overlap of the two circles. | [
"32 \\cos^{-1}\\left(\\frac{5}{8}\\right) - \\frac{5}{2} \\sqrt{39}"
] | [
"Two circles of radii length 4 units intersect at two points. The distance between the two centers of the circles is 5 units. Find the area of overlap of the two circles."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6b39728a41f81c5df376145de655bd88b5faea7a55ec3f970b767f7a404e5daf | Two circles of radii length 40 units intersect at two points. The distance between the two centers of the circles is \(40\sqrt{2}\) units. Find the area of overlap of the two circles. | [
"\\(800\\pi - 1600\\)"
] | [
"Two circles of radii length 40 units intersect at two points. The distance between the two centers of the circles is \\(40\\sqrt{2}\\) units. Find the area of overlap of the two circles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-52b2021226b531f23575643400d4f02e0478199ec7803559a75e32305570e7b5 | Two circles of radius $1$ are chosen as follows: the center of circle $k_0$ is chosen uniformly at random from the line segment joining $(0,0)$ and $(2,0)$, and the center of circle $k_1$ is chosen uniformly at random from the line segment joining $(0,1)$ to $(2,1)$. What is the probability that $k_0$ and $k_1$ interse... | [
"\\(\\sqrt{3} - \\frac{3}{4}\\)"
] | [
"Two circles of radius $1$ are chosen as follows: the center of circle $k_0$ is chosen uniformly at random from the line segment joining $(0,0)$ and $(2,0)$, and the center of circle $k_1$ is chosen uniformly at random from the line segment joining $(0,1)$ to $(2,1)$. What is the probability that $k_0$ and $k_1$ in... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-3243fec8611eab3d0b546fec8ac612529a5eb69455cbf381a3fd5e03d1979793 | Two circles of radius $1$ intersect at points $X$ and $Y$, with the distance between $X$ and $Y$ also equal to $1$. From a point $C$ on one circle, tangents $CA$ and $CB$ are drawn to the other circle. Line $CB$ intersects the first circle again at point $A'$. Find the distance $AA'$. | [
"\\(\\sqrt{3}\\)"
] | [
"Two circles of radius $1$ intersect at points $X$ and $Y$, with the distance between $X$ and $Y$ also equal to $1$. From a point $C$ on one circle, tangents $CA$ and $CB$ are drawn to the other circle. Line $CB$ intersects the first circle again at point $A'$. Find the distance $AA'$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 18
} | 38 | 0 | 0 |
omr-ws-9e611a1694c077f823b0252e8d37c1afb17d9cc12808689290160f99d7db706c | Two circles of radius $1$ intersect, the center of each being on the circumference of the other. A third circle is drawn touching the line joining the centers of the given circles, and the circumferences of both circles, the righthand one internally and the other externally. Find the radius of the third circle. | [
"\\dfrac{\\sqrt{3}}{4}"
] | [
"Two circles of radius $1$ intersect, the center of each being on the circumference of the other. A third circle is drawn touching the line joining the centers of the given circles, and the circumferences of both circles, the righthand one internally and the other externally. Find the radius of the third circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d348746ca0af1a427f7d2024e9c84a6532d1221a0051d022f5927a713b1db392 | Two circles of radius $5$ are externally tangent to each other and are internally tangent to a circle of radius $13$ at points $A$ and $B$. The distance $AB$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | [
"69",
"\\( m + n = 65 + 4 = 69 \\)"
] | [
"Two circles of radius $5$ are externally tangent to each other and are internally tangent to a circle of radius $13$ at points $A$ and $B$. The distance $AB$ can be written in the form $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 3,
"tir": 3
} | 37 | 0 | 0 |
omr-ws-b87ce15afce2bb6567a30579a0ecd586553db30d80ebf10402e771af6b8f851d | Two circles of radius $5$ are internally tangent to each other and to a circle of radius $13$ at points $A$ and $B$. The distance $AB$ can be written in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime. Find $m+n$. | [
"69"
] | [
"Two circles of radius $5$ are internally tangent to each other and to a circle of radius $13$ at points $A$ and $B$. The distance $AB$ can be written in the form $\\tfrac{m}{n}$ where $m$ and $n$ are relatively prime. Find $m+n$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-70bd710a9b7653bbc75646a127cd1ed86a7685d749c549759984f2044a5138c9 | Two circles of radius $6$ intersect such that they share a common chord of length $6$. The total area covered may be expressed as $a\pi + \sqrt{b}$, where $a$ and $b$ are integers. What is $a + b$? | [
"78"
] | [
"Two circles of radius $6$ intersect such that they share a common chord of length $6$. The total area covered may be expressed as $a\\pi + \\sqrt{b}$, where $a$ and $b$ are integers. What is $a + b$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ec697df6839e97b0cfde2202617daf89dfd40e85f5a80fd5a3eba10fe9e2b6fc | Two circles of radius $R$ and $r$ touch a straight line $\ell$ at points $A$ and $B$ respectively and intersect at points $C$ and $D$ ($D$ lies closer to $I$ than $C$). Let $R_{ABC}$ be the radius of the circle circumscribed around the triangle $ABC$. If $R = 3$ and $r = 2$, find the value of $R_{ABC}$. | [
"\\sqrt{6}"
] | [
"Two circles of radius $R$ and $r$ touch a straight line $\\ell$ at points $A$ and $B$ respectively and intersect at points $C$ and $D$ ($D$ lies closer to $I$ than $C$). Let $R_{ABC}$ be the radius of the circle circumscribed around the triangle $ABC$. If $R = 3$ and $r = 2$, find the value of $R_{ABC}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-144db4e9ceec97de3af485b729a73702d4a33ff35094816049e445512c2e9141 | Two circles of radius $R$ are located such that the center of one lies on the other circle. In one of the circles, a diameter is drawn perpendicular to the line of centers. Find the circumferences of the circles tangent to this diameter and the given circles. | [
"\\(\\pi R\\)"
] | [
"Two circles of radius $R$ are located such that the center of one lies on the other circle. In one of the circles, a diameter is drawn perpendicular to the line of centers. Find the circumferences of the circles tangent to this diameter and the given circles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 1,
"tir": 5
} | 13 | 0 | 0 |
omr-ws-156c6034f4bbd2d54d8e13863dfb82f3b8cc07c7fb360aaacf0494ca358426c7 | Two circles of radius $r$ are spaced so their centers are $2r$ apart. If $A(r)$ is the area of the smallest square containing both circles, what is $\frac{A(r)}{r^2}$? | [
"\\(6 + 4\\sqrt{2}\\)"
] | [
"Two circles of radius $r$ are spaced so their centers are $2r$ apart. If $A(r)$ is the area of the smallest square containing both circles, what is $\\frac{A(r)}{r^2}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-013fdf1a570f2995c0883652a67e8ccf0e21fe3950c3755c24b3fd1f5071349e | Two circles of radius 1 are constructed as follows: The center of circle A is chosen at random from the interval [0,2] on the x-axis, and the center of circle B is chosen from the segment joining (1,0) and (1,2). What is the probability that the circles intersect? | [
"\\(\\frac{4\\sqrt{3} - 3}{4}\\)"
] | [
"Two circles of radius 1 are constructed as follows: The center of circle A is chosen at random from the interval [0,2] on the x-axis, and the center of circle B is chosen from the segment joining (1,0) and (1,2). What is the probability that the circles intersect?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-955dd6372c04dff4a805c16be61e8587c80a341566cdf5e4ba6d9fc91095d6fa | Two circles of radius 1 are to be constructed as follows. The center of circle \( A \) is chosen uniformly and at random from the line segment joining \( (0,0) \) and \( (2,0) \). The center of circle \( B \) is chosen uniformly and at random, and independently of the first choice, from the line segment joining \( (0,1... | [
"\\(\\frac{4\\sqrt{3} - 3}{4}\\)"
] | [
"Two circles of radius 1 are to be constructed as follows. The center of circle \\( A \\) is chosen uniformly and at random from the line segment joining \\( (0,0) \\) and \\( (2,0) \\). The center of circle \\( B \\) is chosen uniformly and at random, and independently of the first choice, from the line segment jo... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-79057f1f369e9dc934672bd31b4f27f03e60e37a608d3b683bd5af8bdc37d7bd | Two circles of radius 1 are to be constructed as follows. The center of circle \( A \) is chosen uniformly and at random from the line segment joining \((0,0)\) and \((2,0)\). The center of circle \( B \) is chosen uniformly and at random, and independently of the first choice, from the line segment joining \((0,1)\) t... | [
"\\(\\frac{4\\sqrt{3} - 3}{4}\\)"
] | [
"Two circles of radius 1 are to be constructed as follows. The center of circle \\( A \\) is chosen uniformly and at random from the line segment joining \\((0,0)\\) and \\((2,0)\\). The center of circle \\( B \\) is chosen uniformly and at random, and independently of the first choice, from the line segment joinin... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-ecf0526b9aadcf4d5ce1466fea921570e0a7f6683e577fd8f07b9bb1730f0297 | Two circles of radius 1 overlap such that the center of each circle lies on the circumference of the other. What is the area of their union? | [
"\\(\\frac{4\\pi}{3} + \\frac{\\sqrt{3}}{2}\\)"
] | [
"Two circles of radius 1 overlap such that the center of each circle lies on the circumference of the other. What is the area of their union?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 28
} | 46 | 0 | 0 |
omr-ws-c6d2159010939ec21300834ad0a2878942c7fd6aaeeaa61741501dc770ca6a5d | Two circles of radius 10 are tangent to each other. A tangent is drawn from the center of one circle to the other circle. Find the area of the shaded region. | [
"\\(50\\sqrt{3} - 25\\pi\\)"
] | [
"Two circles of radius 10 are tangent to each other. A tangent is drawn from the center of one circle to the other circle. Find the area of the shaded region."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 3,
"tir": 3
} | 7 | 0 | 0 |
omr-ws-3269b4a6660976c35c1864bdcc6f6f1ab4746234679a3767347c4093f59db933 | Two circles of radius 12 have their centers on a line such that the point A is the center of the left circle, and AB is a diameter of the right circle. A smaller circle is constructed tangent to AB and the given two circles, internally to the right circle, and externally to the left circle. What is the radius of the sm... | [
"\\(3\\sqrt{3}\\)"
] | [
"Two circles of radius 12 have their centers on a line such that the point A is the center of the left circle, and AB is a diameter of the right circle. A smaller circle is constructed tangent to AB and the given two circles, internally to the right circle, and externally to the left circle. What is the radius of t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-0656cd4a8773933c6603cb8f5a775a2a1fe906a932304529fcea99e5108ce50a | Two circles of radius 12 have their centers on each other. A smaller circle is constructed tangent to the line segment connecting the centers of the two larger circles (denoted as $AB$) and is tangent internally to the lower circle and externally to the upper circle. Find the radius of the smaller circle. | [
"\\( 3\\sqrt{3} \\)"
] | [
"Two circles of radius 12 have their centers on each other. A smaller circle is constructed tangent to the line segment connecting the centers of the two larger circles (denoted as $AB$) and is tangent internally to the lower circle and externally to the upper circle. Find the radius of the smaller circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 24
} | 47 | 0 | 0 |
omr-ws-f200c9e597a943fdcbdea72fa8da3d519e0f3a671f1354166b1640aeccb28bd0 | Two circles of radius 2 are centered at $(2,0)$ and at $(0,2)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$. | [
"2\\pi-4"
] | [
"Two circles of radius 2 are centered at $(2,0)$ and at $(0,2)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1c543480a36052a885e51011647df1a8212d80548e7319706400417fa298cd7e | Two circles of radius 4 and 5 are tangent to each other and to a common external tangent line $\ell$. Between the two circles and the line, there is another circle that is tangent to both circles and the line. Find the radius of this circle. | [
"\\( 180 - 80\\sqrt{5} \\)"
] | [
"Two circles of radius 4 and 5 are tangent to each other and to a common external tangent line $\\ell$. Between the two circles and the line, there is another circle that is tangent to both circles and the line. Find the radius of this circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-095440f7c0a1ccd03f43e32d6ef9d25af5468b778e898f9527fd7f6374167e68 | Two circles of radius 5 and 8 touch each other externally. Calculate the length of the common tangent. | [
"\\( 4\\sqrt{10} \\)"
] | [
"Two circles of radius 5 and 8 touch each other externally. Calculate the length of the common tangent."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 25
} | 48 | 0 | 0 |
omr-ws-cdf02dbd0fd54ed0126233daee6ef92eec20fa9bdb615228cfc8ce0bc13cd4bb | Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points \( A \) and \( B \). The distance \( AB \) can be written in the form \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. What is \( m + n \)? | [
"69"
] | [
"Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points \\( A \\) and \\( B \\). The distance \\( AB \\) can be written in the form \\( \\frac{m}{n} \\), where \\( m \\) and \\( n \\) are relatively prime positive integers. What is \\( m + n \\)?"
... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-be7878aec5d27d2e7a1725815cea8614c847e5c0e00632d262926f6983c37f4b | Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points \( A \) and \( B \). The distance \( AB \) can be written in the form \(\frac{m}{n}\), where \( m \) and \( n \) are relatively prime positive integers. What is \( m+n \)? | [
"69"
] | [
"Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points \\( A \\) and \\( B \\). The distance \\( AB \\) can be written in the form \\(\\frac{m}{n}\\), where \\( m \\) and \\( n \\) are relatively prime positive integers. What is \\( m+n \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-bab94ea129393744684a0bca1ab32e573590416304ab4f57723336e987596b9a | Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points \(A\) and \(B\). The distance \(AB\) can be written in the form \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. What is \(m + n\)? | [
"29"
] | [
"Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points \\(A\\) and \\(B\\). The distance \\(AB\\) can be written in the form \\(\\frac{m}{n}\\), where \\(m\\) and \\(n\\) are relatively prime positive integers. What is \\(m + n\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-786659f153a86ffa1511e48a2282ca19cbb2830bb189aaf236c3e12bfb32e8c7 | Two circles of radius 5 cm are touching each other. Beneath these circles, there is a square whose first two vertices are touching the circles. Find the area of the square. | [
"4"
] | [
"Two circles of radius 5 cm are touching each other. Beneath these circles, there is a square whose first two vertices are touching the circles. Find the area of the square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-2ed4e6bc8953bc629673509f2a069ef0612bee617876ed9fa41be8b724ed1eb4 | Two circles of radius \( r \) and two smaller circles of radius \( t \) are inscribed within a square. This square is itself inscribed in a right triangle \( ABC \). In the resulting smaller triangles, there is again a circle with radius \( r \) and a larger circle with radius \( R \). Determine the ratio \( \frac{R}{t... | [
"2"
] | [
"Two circles of radius \\( r \\) and two smaller circles of radius \\( t \\) are inscribed within a square. This square is itself inscribed in a right triangle \\( ABC \\). In the resulting smaller triangles, there is again a circle with radius \\( r \\) and a larger circle with radius \\( R \\). Determine the rati... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-849d43a737329d530762d3ff7b41cb7d043b910fec21f4835452945e02c25390 | Two circles of radius \( r \) are each tangent to the hypotenuse and a leg of a right triangle with sides 3, 4, and 5, and tangent to each other. Compute \( r \). | [
"\\( \\frac{5}{7} \\)"
] | [
"Two circles of radius \\( r \\) are each tangent to the hypotenuse and a leg of a right triangle with sides 3, 4, and 5, and tangent to each other. Compute \\( r \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 28
} | 76 | 0 | 0 |
omr-ws-ea766807e93fa9f91626d0930da77b9da54bb7308ef28af7de3d7e5f54515bb4 | Two circles of radius one each pass through the center of the other. What is the area of their intersection? | [
"\\(\\frac{4\\pi-3\\sqrt{3}}{6}\\)"
] | [
"Two circles of radius one each pass through the center of the other. What is the area of their intersection?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-fb69052484768d4252fa3bc7e5d0f19a3b1b96f3f8b577062b497507499f23d7 | Two circles pass through a vertex \( A \) of an angle \( \angle BAC \) and an arbitrary point \( P \) on the angle bisector of \( \angle BAC \). Let \( X \) and \( Y \) be the points where the circles intersect the edges \( AB \) and \( AC \) respectively. If \( AB = 10 \) and \( AC = 12 \), find the length of \( AX + ... | [
"22"
] | [
"Two circles pass through a vertex \\( A \\) of an angle \\( \\angle BAC \\) and an arbitrary point \\( P \\) on the angle bisector of \\( \\angle BAC \\). Let \\( X \\) and \\( Y \\) be the points where the circles intersect the edges \\( AB \\) and \\( AC \\) respectively. If \\( AB = 10 \\) and \\( AC = 12 \\), ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-3eba4e474888e5c9d38c20346a57b0f8c17d5fac018ab51e72b42adecabcd8b7 | Two circles pass through points $A$ and $B$. A third circle touches both these circles and meets $AB$ at points $C$ and $D$. If the radii of the two given circles are $r_1$ and $r_2$, and the radius of the third circle is $r_3$, find the value of $\frac{r_1 + r_2}{r_3}$ given that the tangents to the third circle at po... | [
"2"
] | [
"Two circles pass through points $A$ and $B$. A third circle touches both these circles and meets $AB$ at points $C$ and $D$. If the radii of the two given circles are $r_1$ and $r_2$, and the radius of the third circle is $r_3$, find the value of $\\frac{r_1 + r_2}{r_3}$ given that the tangents to the third circle... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-093f2593c7cb8178469644f942137842a56d4983245ee671e33289ee265a1ea0 | Two circles pass through the vertex of angle $A$ and point $B$ on its angle bisector so that the sides of the angle intersect each circle. Let $X$ and $Y$ be the points where one of the circles intersects the sides of the angle, and let $W$ and $Z$ be the points where the other circle intersects the sides of the angle.... | [
"20"
] | [
"Two circles pass through the vertex of angle $A$ and point $B$ on its angle bisector so that the sides of the angle intersect each circle. Let $X$ and $Y$ be the points where one of the circles intersects the sides of the angle, and let $W$ and $Z$ be the points where the other circle intersects the sides of the a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-8348d9f566d139fd17a29e9ecd448e29dee71bd979b5f3e26a114c3bcfa222eb | Two circles that are not equal are tangent externally at point $R$. Suppose point $P$ is the intersection of the external common tangents of the two circles. Let $A$ and $B$ be two points on different circles such that $RA$ is perpendicular to $RB$. If the coordinates of $R$ are $(0,0)$, $A$ is on the circle centered a... | [
"-11"
] | [
"Two circles that are not equal are tangent externally at point $R$. Suppose point $P$ is the intersection of the external common tangents of the two circles. Let $A$ and $B$ be two points on different circles such that $RA$ is perpendicular to $RB$. If the coordinates of $R$ are $(0,0)$, $A$ is on the circle cente... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-0fac9d3ba32781d9fb7f877c2abe91123e61b7ed362d0803bf66d2033b7b7995 | Two circles touch at point \( M \) and lie inside a rectangle \( ABCD \). One circle touches sides \( AB \) and \( AD \), and the other circle touches sides \( AD \), \( BC \), and \( CD \). The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles... | [
"\\( 1:1 \\)"
] | [
"Two circles touch at point \\( M \\) and lie inside a rectangle \\( ABCD \\). One circle touches sides \\( AB \\) and \\( AD \\), and the other circle touches sides \\( AD \\), \\( BC \\), and \\( CD \\). The radius of the second circle is four times that of the first circle. Find the ratio in which the common tan... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 20,
"tir": 17
} | 37 | 0 | 0 |
omr-ws-221c04ab0b884f9fb844d38be34264e183484962364f080d51f377bce0b8f7ad | Two circles touch each other and the sides of triangle \(ABC\). The first circle, with radius \(\frac{1}{18}\), touches sides \(AB\) and \(AC\) at points \(L\) and \(K\), respectively. The second circle, with radius \(\frac{2}{9}\), touches sides \(AC\) and \(BC\) at points \(N\) and \(M\), respectively. Given that \(A... | [
"\\dfrac{3}{11}"
] | [
"Two circles touch each other and the sides of triangle \\(ABC\\). The first circle, with radius \\(\\frac{1}{18}\\), touches sides \\(AB\\) and \\(AC\\) at points \\(L\\) and \\(K\\), respectively. The second circle, with radius \\(\\frac{2}{9}\\), touches sides \\(AC\\) and \\(BC\\) at points \\(N\\) and \\(M\\),... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-6fdc49cfa446435bb7cdda0b9aee1f46ac5d5068de623ca340e1abca7cdaa0ca | Two circles touch each other externally at point A. A straight line through A meets one circle at T and the other at S. Let the tangents to the first circle at T and to the second circle at S intersect at point P. If the distance between the centers of the two circles is 10 units and the radii of the circles are 3 unit... | [
"no_valid_answer_found"
] | [
"Two circles touch each other externally at point A. A straight line through A meets one circle at T and the other at S. Let the tangents to the first circle at T and to the second circle at S intersect at point P. If the distance between the centers of the two circles is 10 units and the radii of the circles are 3... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e4691f242558ebd28ea14c0e683941a65244e5b3b5bfa6e76b03772e7712636 | Two circles touch each other externally at point \( A \) and touch a common tangent at points \( B \) and \( C \) (where \( B \neq C \)). Let \( D \) and \( E \) be the points such that \( BD \) and \( CE \) are diameters of the circles. If the radius of the first circle is \( r_1 \) and the radius of the second circle... | [
"\\dfrac{r_1}{r_2}"
] | [
"Two circles touch each other externally at point \\( A \\) and touch a common tangent at points \\( B \\) and \\( C \\) (where \\( B \\neq C \\)). Let \\( D \\) and \\( E \\) be the points such that \\( BD \\) and \\( CE \\) are diameters of the circles. If the radius of the first circle is \\( r_1 \\) and the rad... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72924a15cda5acb2f29c94d96baddcba820d086b3ea142146d41b3e80e193f80 | Two circles touch each other externally. The larger circle has a radius of $5$ and the smaller has a radius of $3$. The common external tangents of the circles intersect at point $O$. What is the distance between $O$ and the center of the smaller circle? | [
"12"
] | [
"Two circles touch each other externally. The larger circle has a radius of $5$ and the smaller has a radius of $3$. The common external tangents of the circles intersect at point $O$. What is the distance between $O$ and the center of the smaller circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-388b66ec0d9a6b79f40f6df7cca21a819f5b59214e64ebc0f2dcd477a3bbfd23 | Two circles touch each other from inside, and an equilateral triangle is inscribed in the larger circle. From the vertices of the triangle, segments are drawn tangent to the smaller circle. If the radius of the larger circle is $R$ and the radius of the smaller circle is $r$, find the value of $\frac{R}{r}$ given that ... | [
"2"
] | [
"Two circles touch each other from inside, and an equilateral triangle is inscribed in the larger circle. From the vertices of the triangle, segments are drawn tangent to the smaller circle. If the radius of the larger circle is $R$ and the radius of the smaller circle is $r$, find the value of $\\frac{R}{r}$ given... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-73abeef4920294f7d4a68aa3b3fa33430fbd04fc1a2cc184e6b1de693c358aa0 | Two circles touch each other internally at point A. A variable chord PQ of the outer circle touches the inner circle. Let I be the incentre of triangle AQP. If the radius of the outer circle is 10 units and the radius of the inner circle is 6 units, find the radius of the circle that represents the locus of point I. | [
"no_valid_answer_found"
] | [
"Two circles touch each other internally at point A. A variable chord PQ of the outer circle touches the inner circle. Let I be the incentre of triangle AQP. If the radius of the outer circle is 10 units and the radius of the inner circle is 6 units, find the radius of the circle that represents the locus of point ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66e2c18a1783374268c1e06167b9aadfdd0cd2efa2ff58113bd03e63402aaa7a | Two circles touch each other internally at point \( A \). A variable chord \( PQ \) of the outer circle touches the inner circle. Let \( I \) be the incenter of \( \triangle AQP \). If the radius of the outer circle is \( R \) and the radius of the inner circle is \( r \), find the radius of the circle that is the locu... | [
"no_valid_answer_found"
] | [
"Two circles touch each other internally at point \\( A \\). A variable chord \\( PQ \\) of the outer circle touches the inner circle. Let \\( I \\) be the incenter of \\( \\triangle AQP \\). If the radius of the outer circle is \\( R \\) and the radius of the inner circle is \\( r \\), find the radius of the circl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2821bc79247d07f956ee8e3ac7500584c0b2fcacacb7086efd291661e30745f1 | Two circles touch externally at a point \( I \). Both circles lie inside a larger circle and are tangent to it. The chord \( BC \) of the larger circle touches both smaller circles (not at \( I \)). The common tangent to the two smaller circles at \( I \) meets the larger circle at point \( A \), where \( A \) and \( I... | [
"4(\\sqrt{6} - \\sqrt{3})"
] | [
"Two circles touch externally at a point \\( I \\). Both circles lie inside a larger circle and are tangent to it. The chord \\( BC \\) of the larger circle touches both smaller circles (not at \\( I \\)). The common tangent to the two smaller circles at \\( I \\) meets the larger circle at point \\( A \\), where \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-1e88d7ac64748c747aa425ea287760d612763db98e1d5668a5eb6e2221b6eed3 | Two circles touch externally at point $P$. The common external tangent $MN$ is drawn to them. A straight line drawn through points $M$ and $P$ intersects the other circle at point $K$. Given that $MN = a$ and $MP = b$. Calculate $PK$. | [
"\\dfrac{a^2 - b^2}{b}"
] | [
"Two circles touch externally at point $P$. The common external tangent $MN$ is drawn to them. A straight line drawn through points $M$ and $P$ intersects the other circle at point $K$. Given that $MN = a$ and $MP = b$. Calculate $PK$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6b5363b9c5598c87b55429bff6861a693b49c1f8356f4dbeb2e9202598df9443 | Two circles touch externally at point \(K\). Line \(AB\) touches the first circle at point \(A\) and the second circle at point \(B\). Line \(BK\) intersects the first circle at point \(B\), and line \(AK\) intersects the second circle at point \(C\). Find the area of triangle \(AKB\) if the radii of the circles are \(... | [
"\\dfrac{16}{5}"
] | [
"Two circles touch externally at point \\(K\\). Line \\(AB\\) touches the first circle at point \\(A\\) and the second circle at point \\(B\\). Line \\(BK\\) intersects the first circle at point \\(B\\), and line \\(AK\\) intersects the second circle at point \\(C\\). Find the area of triangle \\(AKB\\) if the radi... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a74716da0aa5373db68df1423251c862e57e1454f0a1a978ede73e2a57c02030 | Two circles touch externally. One of the circles has a rectangle $ABCD$ inscribed in it. Denote by $\ell_A$, $\ell_B$, $\ell_C$, and $\ell_D$ the lengths of the tangent segments drawn from points $A$, $B$, $C$, and $D$ to the second circle, respectively. If the side lengths of the rectangle $ABCD$ are 6 and 8, calculat... | [
"0"
] | [
"Two circles touch externally. One of the circles has a rectangle $ABCD$ inscribed in it. Denote by $\\ell_A$, $\\ell_B$, $\\ell_C$, and $\\ell_D$ the lengths of the tangent segments drawn from points $A$, $B$, $C$, and $D$ to the second circle, respectively. If the side lengths of the rectangle $ABCD$ are 6 and 8,... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17c0a20521e8fb40c244d32ed896ac4c058da993cd3e3a2ffa0bb4df7b00058c | Two circles touch externally. One of the circles has an inscribed equilateral triangle whose vertex is not at the point of contact with the circles. From each vertex of this triangle, a tangent is drawn to the second circle. If the lengths of the tangents from the vertices are $a$, $b$, and $c$, calculate the value of ... | [
"3 + 2\\sqrt{3}"
] | [
"Two circles touch externally. One of the circles has an inscribed equilateral triangle whose vertex is not at the point of contact with the circles. From each vertex of this triangle, a tangent is drawn to the second circle. If the lengths of the tangents from the vertices are $a$, $b$, and $c$, calculate the valu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-4c7dc61319be544fb2fd3fa665603aeffbf255bf583347ffa4b20620ec99d819 | Two circles touch internally at $M$. A straight line touches the inner circle at $P$ and cuts the outer circle at $Q$ and $R$. If $\angle QMP = 59^{\circ}$, find the value of $\angle RMP$. | [
"\\(59^\\circ\\)"
] | [
"Two circles touch internally at $M$. A straight line touches the inner circle at $P$ and cuts the outer circle at $Q$ and $R$. If $\\angle QMP = 59^{\\circ}$, find the value of $\\angle RMP$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 4
} | 23 | 0 | 0 |
omr-ws-a5db1608a55f7e3e25502dd4db032be10769f4b8bf62321653ec06db7452c119 | Two circles touch internally at point A. The tangent at a point P on the smaller circle intersects the larger circle at points Q and R. If the measure of angle RAQ is 120 degrees, find the measure of angle RAP. | [
"60"
] | [
"Two circles touch internally at point A. The tangent at a point P on the smaller circle intersects the larger circle at points Q and R. If the measure of angle RAQ is 120 degrees, find the measure of angle RAP."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-8386a68f35651dc0f7bccf9f2536bca462f4f5d46f72bc46858c1cad1f17e272 | Two circles touch internally at point M. A straight line touches the inner circle at point P and intersects the outer circle at points Q and R. If the measure of angle QMP is represented by $\theta$, find the value of $\sin^2(\theta) + \cos^2(\theta)$. | [
"1"
] | [
"Two circles touch internally at point M. A straight line touches the inner circle at point P and intersects the outer circle at points Q and R. If the measure of angle QMP is represented by $\\theta$, find the value of $\\sin^2(\\theta) + \\cos^2(\\theta)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e7e62077c5216d56d508310f7ae06ffbcdfae977f0d1806a0a51a38390245436 | Two circles touch internally at point \( M \). A line touches the inner circle at point \( P \) and intersects the outer circle at points \( Q \) and \( R \). If the radius of the inner circle is \( r \) and the radius of the outer circle is \( R \), find the value of \( \frac{R}{r} \) given that \( \angle QMP = 30^\ci... | [
"2"
] | [
"Two circles touch internally at point \\( M \\). A line touches the inner circle at point \\( P \\) and intersects the outer circle at points \\( Q \\) and \\( R \\). If the radius of the inner circle is \\( r \\) and the radius of the outer circle is \\( R \\), find the value of \\( \\frac{R}{r} \\) given that \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-7fce9e31b0959a1ff7972fc77c866c33bd4282e2b0e3146d846494607ed11e4b | Two circles touch internally at point \( M \). A straight line touches the inner circle at point \( P \) and intersects the outer circle at points \( Q \) and \( R \). If the measure of angle \( QMP \) is \( \theta \), find the measure of angle \( RMP \) in terms of \( \theta \). | [
"\\theta"
] | [
"Two circles touch internally at point \\( M \\). A straight line touches the inner circle at point \\( P \\) and intersects the outer circle at points \\( Q \\) and \\( R \\). If the measure of angle \\( QMP \\) is \\( \\theta \\), find the measure of angle \\( RMP \\) in terms of \\( \\theta \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-3f65e46fd119fbb4f3b95d690aeb1b3e27342f9738a462840f736bfe402bae75 | Two circles touch internally at point \( P \). A tangent is drawn from a point \( A \) on the inner circle, which meets the outer circle at points \( B \) and \( C \). If \( \angle BPC = 120^\circ \), find the measure of \( \angle BAP \). | [
"60^\\circ"
] | [
"Two circles touch internally at point \\( P \\). A tangent is drawn from a point \\( A \\) on the inner circle, which meets the outer circle at points \\( B \\) and \\( C \\). If \\( \\angle BPC = 120^\\circ \\), find the measure of \\( \\angle BAP \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-3ad10adb404b5c58393fe143a49f0fcb43aeaa2bd85caa64328732cf8f8676b3 | Two circles touch internally at point \( X \). A straight line intersects the circles at points \( A, B, C, \) and \( D \) in that order. If the measure of the angle subtended by segment \( AB \) at point \( X \) is \( \theta \), find the measure of the angle subtended by segment \( CD \) at point \( X \) in terms of \... | [
"\\theta"
] | [
"Two circles touch internally at point \\( X \\). A straight line intersects the circles at points \\( A, B, C, \\) and \\( D \\) in that order. If the measure of the angle subtended by segment \\( AB \\) at point \\( X \\) is \\( \\theta \\), find the measure of the angle subtended by segment \\( CD \\) at point \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-119fef03eab1559e765e2bb4cafb498fec701e17f5e8f0755ab3fd22ce275dc9 | Two circles touch internally at the point $T$. $P$ is a point on the outer circle, and chords $PA$ and $PB$ touch the inner circle at $X$ and $Y$, respectively. If $\angle XTY = \theta$, calculate $\angle ATB$ in terms of $\theta$. | [
"\\theta"
] | [
"Two circles touch internally at the point $T$. $P$ is a point on the outer circle, and chords $PA$ and $PB$ touch the inner circle at $X$ and $Y$, respectively. If $\\angle XTY = \\theta$, calculate $\\angle ATB$ in terms of $\\theta$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-fbb2babf570b6485c9671e12b9fa5f656a4db834c6ebebd60100d075e03b4c6d | Two circles touch one another internally at point $A$. A variable chord $PQ$ of the outer circle touches the inner circle. Let $I$ be the incenter of triangle $AQP$. If the radius of the outer circle is $R$ and the radius of the inner circle is $r$, find the radius of the circle that is the locus of point $I$ in terms ... | [
"no_valid_answer_found"
] | [
"Two circles touch one another internally at point $A$. A variable chord $PQ$ of the outer circle touches the inner circle. Let $I$ be the incenter of triangle $AQP$. If the radius of the outer circle is $R$ and the radius of the inner circle is $r$, find the radius of the circle that is the locus of point $I$ in t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-567fc1fbbfb1208724c78b80585486516781f60302cbb3a2906ab1fce1ff9517 | Two circles touch one another internally at point $A$. A variable chord $PQ$ of the outer circle touches the inner circle. Let $I$ be the incentre of triangle $AQP$. If the radius of the outer circle is $R$ and the radius of the inner circle is $r$, find the radius of the circle that is the locus of point $I$ in terms ... | [
"no_valid_answer_found"
] | [
"Two circles touch one another internally at point $A$. A variable chord $PQ$ of the outer circle touches the inner circle. Let $I$ be the incentre of triangle $AQP$. If the radius of the outer circle is $R$ and the radius of the inner circle is $r$, find the radius of the circle that is the locus of point $I$ in t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7c914d0ab1c58ee5bed4d55c9e3853f9b01e944725bbcabe8c7ddb0f658e7098 | Two circles touch one another internally at point A. A variable chord PQ of the outer circle touches the inner circle. Let I be the incentre of triangle AQP. Find the radius of the circle that is the locus of point I, given that the radii of the outer and inner circles are 10 and 6, respectively. | [
"no_valid_answer_found"
] | [
"Two circles touch one another internally at point A. A variable chord PQ of the outer circle touches the inner circle. Let I be the incentre of triangle AQP. Find the radius of the circle that is the locus of point I, given that the radii of the outer and inner circles are 10 and 6, respectively."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dd510ef0414e24b4cfa1801831b89f662123ad3e1a968af5d5e150e016777962 | Two circles touch one another internally at point \( A \). A variable chord \( PQ \) of the outer circle touches the inner circle. Let \( I \) be the incentre of triangle \( APQ \). Find the radius of the circle that is the locus of point \( I \) if the radii of the outer and inner circles are 10 and 6, respectively. | [
"2"
] | [
"Two circles touch one another internally at point \\( A \\). A variable chord \\( PQ \\) of the outer circle touches the inner circle. Let \\( I \\) be the incentre of triangle \\( APQ \\). Find the radius of the circle that is the locus of point \\( I \\) if the radii of the outer and inner circles are 10 and 6, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-b8dc0585074f3173d02504174abc81e10d428d9c562e56c0e8f0ff7a24262301 | Two circles touch the x-axis and the line \( y = mx \) (where \( m > 0 \)). They intersect at the points \((9, 6)\) and another point, and the product of their radii is 68. Find the value of \( m \). | [
"\\(\\frac{4 \\sqrt{1989}}{49}\\)"
] | [
"Two circles touch the x-axis and the line \\( y = mx \\) (where \\( m > 0 \\)). They intersect at the points \\((9, 6)\\) and another point, and the product of their radii is 68. Find the value of \\( m \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 26
} | 38 | 0 | 0 |
omr-ws-9e745a8747b7eab69de8da8e780fefdceffd8d8f849be248f1afdfad4a6a48e2 | Two circles whose radii are $R_1$ and $R_2$ intersect. The distance between their centers is equal to $d$. Find the radius of the circle which touches the given circles and their common tangent. | [
"$r=\\frac{d^2-(R_1-R_2)^2}{4\\left(\\sqrt {R_1}+\\sqrt {R_2}\\right)^2}$"
] | [
"Two circles whose radii are $R_1$ and $R_2$ intersect. The distance between their centers is equal to $d$. Find the radius of the circle which touches the given circles and their common tangent."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-991613b4bdae85eb585bd70d09bc6ced636c7934bf5360c8f409411babb3a243 | Two circles with centers $A$ and $B$ and radii $5$ and $10$, respectively, lie externally tangent in the coordinate plane. Line $\ell$ lies externally tangent to the two circles, and point $C$ is chosen on $\ell$ such that $\angle ACB=90^\circ$. Suppose that line $\overleftrightarrow{AB}$ and $\ell$ intersect at point ... | [
"17"
] | [
"Two circles with centers $A$ and $B$ and radii $5$ and $10$, respectively, lie externally tangent in the coordinate plane. Line $\\ell$ lies externally tangent to the two circles, and point $C$ is chosen on $\\ell$ such that $\\angle ACB=90^\\circ$. Suppose that line $\\overleftrightarrow{AB}$ and $\\ell$ intersec... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-e5f01e9c4aac68d2d42cee8d438cac1a01ef23b3a4a4280cd76a13a2a03fdc6d | Two circles with centers $O$ and $O'$ and radii $R$ and $R'$ intersect at two points. A variable line $L$ intersects the circles at points $A, C, B, D$ in that order, and it is given that $\frac{AC}{AD} = \frac{CB}{BD}$. The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$. Find the locus of $H$ and $H'$. | [
"\\text{The radical axis of the two circles}"
] | [
"Two circles with centers $O$ and $O'$ and radii $R$ and $R'$ intersect at two points. A variable line $L$ intersects the circles at points $A, C, B, D$ in that order, and it is given that $\\frac{AC}{AD} = \\frac{CB}{BD}$. The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$. Find the locus of $H$ an... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-9edd95ccbc117ac45b90578e5d01d3a6819b2d9a857d50d6ef87c421eda01bbd | Two circles with centers $O_1$ and $O_2$ are externally tangent. A segment $\overline{O_{2}A}$ is drawn from the center of the second circle, tangent to the first circle. The length of this segment is three times greater than the radius of the second circle. What is the ratio of the circumferences of these circles? | [
"$4:1$"
] | [
"Two circles with centers $O_1$ and $O_2$ are externally tangent. A segment $\\overline{O_{2}A}$ is drawn from the center of the second circle, tangent to the first circle. The length of this segment is three times greater than the radius of the second circle. What is the ratio of the circumferences of these circle... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5407070fef95f8c973a5a7548f486c327fa667db76f63d3cba7e760e5c06f867 | Two circles with centers O and P intersect at points A and B. Point M is on the chord AB. Point C is on circle O such that CM is perpendicular to MO. Point D is on circle P such that DM is perpendicular to PM. If the radius of circle O is 5 units and the radius of circle P is 7 units, calculate the sum of the squares o... | [
"no_valid_answer_found"
] | [
"Two circles with centers O and P intersect at points A and B. Point M is on the chord AB. Point C is on circle O such that CM is perpendicular to MO. Point D is on circle P such that DM is perpendicular to PM. If the radius of circle O is 5 units and the radius of circle P is 7 units, calculate the sum of the squa... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-310faed341d4d8bd808e2e52cf146154af3496ef39465b57662cb88510ca033c | Two circles with centers \( A \) and \( B \) have the same radius and intersect at points \( C \) and \( D \). Let \( E \) be the midpoint of \( CD \). Consider arbitrary segments \( FI \) and \( GH \) passing through \( E \), where \( FI \) intersects circle \( (B) \) at \( J \) and \( GH \) intersects circle \( (A) \... | [
"0"
] | [
"Two circles with centers \\( A \\) and \\( B \\) have the same radius and intersect at points \\( C \\) and \\( D \\). Let \\( E \\) be the midpoint of \\( CD \\). Consider arbitrary segments \\( FI \\) and \\( GH \\) passing through \\( E \\), where \\( FI \\) intersects circle \\( (B) \\) at \\( J \\) and \\( GH... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-1128e8a5f608e0dcf7fb5e8cf0d6d8202363e31a05879d6f8e8050a9a2e33a4d | Two circles with centers \( A \) and \( B \) intersect at points \( G \) and \( H \). Lines \( EF \) and \( CD \) pass through \( G \), with \( D \) and \( E \) on one circle and \( C \) and \( F \) on the other. \( I \) is the midpoint of \( EF \) and \( J \) is the midpoint of \( CD \). Given that \( GHIJ \) is a cyc... | [
"5"
] | [
"Two circles with centers \\( A \\) and \\( B \\) intersect at points \\( G \\) and \\( H \\). Lines \\( EF \\) and \\( CD \\) pass through \\( G \\), with \\( D \\) and \\( E \\) on one circle and \\( C \\) and \\( F \\) on the other. \\( I \\) is the midpoint of \\( EF \\) and \\( J \\) is the midpoint of \\( CD ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-265f9b441d4ef8873a71de4878f74cddce9a1cae1a2c006caa2e58531b0a52ca | Two circles with centers \( A \) and \( B \) intersect at points \( M \) and \( N \). Radii \( AP \) and \( BQ \) are parallel (on opposite sides of \( AB \)). If the common external tangents meet \( AB \) at \( D \), and \( PQ \) meets \( AB \) at \( C \), find the value of \( \tan(\angle CND) \). | [
"1"
] | [
"Two circles with centers \\( A \\) and \\( B \\) intersect at points \\( M \\) and \\( N \\). Radii \\( AP \\) and \\( BQ \\) are parallel (on opposite sides of \\( AB \\)). If the common external tangents meet \\( AB \\) at \\( D \\), and \\( PQ \\) meets \\( AB \\) at \\( C \\), find the value of \\( \\tan(\\ang... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-c05c4d66c04a1d4b0fbe8b1d3f45cae0541c2c7d04ceb894b84ef3a78e80fcab | Two circles with centers \( C \) and \( D \) intersect at points \( A \) and \( B \). A line through point \( A \) intersects the circle with center \( C \) at point \( P \) and the circle with center \( D \) at point \( Q \). If \( \angle CAD = 30^\circ \), find the measure of \( \angle PBQ \). | [
"30"
] | [
"Two circles with centers \\( C \\) and \\( D \\) intersect at points \\( A \\) and \\( B \\). A line through point \\( A \\) intersects the circle with center \\( C \\) at point \\( P \\) and the circle with center \\( D \\) at point \\( Q \\). If \\( \\angle CAD = 30^\\circ \\), find the measure of \\( \\angle PB... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-c09169d4a19e15996b67af37ab32e22f9fe0d7c2d4db2e69c6f8254fc6955362 | Two circles with centers \( K \) and \( L \) intersect at points \( A \) and \( B \). A line through \( A \) meets the circles again at points \( C \) and \( D \). Let \( M \) be the midpoint of \( CD \), \( N \) be the midpoint of \( KL \), and \( E \) be the point symmetric to \( A \) with respect to \( N \). If the ... | [
"8"
] | [
"Two circles with centers \\( K \\) and \\( L \\) intersect at points \\( A \\) and \\( B \\). A line through \\( A \\) meets the circles again at points \\( C \\) and \\( D \\). Let \\( M \\) be the midpoint of \\( CD \\), \\( N \\) be the midpoint of \\( KL \\), and \\( E \\) be the point symmetric to \\( A \\) w... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d10f475bfc7a1f1d72df2a1be9f98b07e49c1fa038bd103524b5366f7d776201 | Two circles with centers \( K \) and \( L \) intersect at points \( A \) and \( B \). The tangents to the circles at \( A \) meet the circles again at \( C \) and \( D \). If \( O \) is the center of the circumscribed circle of triangle \( ACD \), calculate the measure of angle \( \angle OBA \) in degrees. | [
"90"
] | [
"Two circles with centers \\( K \\) and \\( L \\) intersect at points \\( A \\) and \\( B \\). The tangents to the circles at \\( A \\) meet the circles again at \\( C \\) and \\( D \\). If \\( O \\) is the center of the circumscribed circle of triangle \\( ACD \\), calculate the measure of angle \\( \\angle OBA \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1eeae34278e9b0392fb3467d2bbc39ba4442cc16f2828c961575ed28925dbed6 | Two circles with centers \( O_1 \) and \( O_2 \) intersect at points \( A \) and \( B \). A point \( C \) is chosen on the ray \( AB \) beyond \( B \). Tangents \( CD \) and \( CE \) are drawn to the circle centered at \( O_2 \). Let \( X \) be the intersection of line \( DE \) and the circle centered at \( O_1 \). The... | [
"5"
] | [
"Two circles with centers \\( O_1 \\) and \\( O_2 \\) intersect at points \\( A \\) and \\( B \\). A point \\( C \\) is chosen on the ray \\( AB \\) beyond \\( B \\). Tangents \\( CD \\) and \\( CE \\) are drawn to the circle centered at \\( O_2 \\). Let \\( X \\) be the intersection of line \\( DE \\) and the circ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-375d31b100fc42ce767cb81f98a837020a4deefbc660e395794d2bb97614e4f0 | Two circles with centers \( O_1 \) and \( O_2 \) intersect at points \( A \) and \( B \). The angle bisector of \( \angle O_1AO_2 \) intersects the circles again at points \( C \) and \( D \). Let \( P \) be the circumcenter of triangle \( CBD \). If \( O_1O_2 = 10 \) and the radius of each circle is \( 13 \), find the... | [
"\\dfrac{65}{12}"
] | [
"Two circles with centers \\( O_1 \\) and \\( O_2 \\) intersect at points \\( A \\) and \\( B \\). The angle bisector of \\( \\angle O_1AO_2 \\) intersects the circles again at points \\( C \\) and \\( D \\). Let \\( P \\) be the circumcenter of triangle \\( CBD \\). If \\( O_1O_2 = 10 \\) and the radius of each ci... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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