Split (1)

train · 500k rows

id stringlengths 71 71	problem stringlengths 8 3.62k	reference_answers listlengths 1 7	original_problem_variants listlengths 1 3	problem_sources listlengths 0 4	source_splits listlengths 1 4	source_split_row_counts unknown	source_row_count int64 1 364
omr-ws-c4f7f15c62725354656a71c7bcfe3d1f8278668d9469ce216d4c111447d3ed94	Three circles $(w_a), (w_b), (w_c)$ intersect at point $P$. The intersections are defined as follows: $w_a \cap w_b = \{D, P\}$, $w_b \cap w_c = \{A, P\}$, and $w_c \cap w_a = \{E, P\}$. Let $Q$ be an arbitrary point on the same plane. Define $B$ as the second intersection point of $AQ$ and $w_b$, \(C...	[ "\\dfrac{3}{5}" ]	[ "Three circles \$(w_a), (w_b), (w_c)\$ intersect at point \$P\$. The intersections are defined as follows: \$w_a \\cap w_b = \\{D, P\\}\$, \$w_b \\cap w_c = \\{A, P\\}\$, and \$w_c \\cap w_a = \\{E, P\\}\$. Let \$Q\$ be an arbitrary point on the same plane. Define \$B\$ as the second intersection poin...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-16b0acae79cd1c7189b5ab1b725c680d2dcb0234495610b2908462b85fc19cb6	Three circles $C_1, C_2, C_3$ with radii 1, 2, and 3 respectively are externally tangent to each other. There is a fourth circle $C_4$ that is externally tangent to all three circles. Find the radius of $C_4$.	[ "\$\\frac{6}{23}\$" ]	[ "Three circles \$C_1, C_2, C_3\$ with radii 1, 2, and 3 respectively are externally tangent to each other. There is a fourth circle \$C_4\$ that is externally tangent to all three circles. Find the radius of \$C_4\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 25 }	41
omr-ws-7c650425d48715c80e2e43597b50a5957106c7b69f8b3638cf9033bfd88e33d9	Three circles $C_1$, $C_2$, and $C_3$ are pairwise tangent to each other: $C_1$ and $C_2$ touch at point $A$, $C_2$ and $C_3$ touch at point $B$, and $C_3$ and $C_1$ touch at point $C$. The lines $AB$ and $AC$ meet circle $C_3$ at points $N$ and $M$ (apart from $B$ and $C$). The li...	[ "\\dfrac{d^2 - R^2}{9}" ]	[ "Three circles \$C_1\$, \$C_2\$, and \$C_3\$ are pairwise tangent to each other: \$C_1\$ and \$C_2\$ touch at point \$A\$, \$C_2\$ and \$C_3\$ touch at point \$B\$, and \$C_3\$ and \$C_1\$ touch at point \$C\$. The lines \$AB\$ and \$AC\$ meet circle \$C_3\$ at points \$N\$ and \\(M\...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-28e24b4e93e199d7f23eedae9181f82621eeef70e7ba011249cdbb968fadc929	Three circles $O_1(r_1)$, $O_2(r_2)$, and $O_3(r_3)$ touch each other externally. The line $l$ is tangent to $O_1(r_1)$ and parallel to the exterior common tangent $m$ to $O_2(r_2)$ and $O_3(r_3)$ which does not intersect $O_1(r_1)$. Find the distance between the lines $l$ and $m$.	[ "\$ r_1 + 2\\sqrt{r_2 r_3} \$" ]	[ "Three circles \$O_1(r_1)\$, \$O_2(r_2)\$, and \$O_3(r_3)\$ touch each other externally. The line \$l\$ is tangent to \$O_1(r_1)\$ and parallel to the exterior common tangent \$m\$ to \$O_2(r_2)\$ and \$O_3(r_3)\$ which does not intersect \$O_1(r_1)\$. Find the distance between the lines \$l\$ a...	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-0cec6c4b2ee8ac9ff11a3d509d7a64fe5b2add926de70e6f90d80a660fbbf499	Three circles $O_1, O_2, O_3$ have a common radical axis. From an arbitrary point $A$ on circle $O_3$, draw two lines tangent to $O_1$ and $O_2$ at points $B$ and $C$ respectively. Line $BC$ intersects circles $O_1$ and $O_2$ again at points $D$ and $E$ respectively. If the radii of $O_1$ and ...	[ "\\dfrac{5}{7}" ]	[ "Three circles \$O_1, O_2, O_3\$ have a common radical axis. From an arbitrary point \$A\$ on circle \$O_3\$, draw two lines tangent to \$O_1\$ and \$O_2\$ at points \$B\$ and \$C\$ respectively. Line \$BC\$ intersects circles \$O_1\$ and \$O_2\$ again at points \$D\$ and \$E\$ respectively....	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 2 }	2
omr-ws-9f1bd0472af37c6ecacc6fe7fb526c8342e3c6b16d91717ea1d499126bd30920	Three circles $O_1$, $O_2$, and $O_3$ touch each other pairwise. Circle $O_1$ and circle $O_2$ both have radii of 1 unit. A common tangent $\overline{AB}$ is drawn to circles $O_1$ and $O_2$, and a semicircle $\widehat{AB}$ touches each of the circles $O_1$, $O_2$, and $O_3$ at different points....	[ "\$\\sqrt{2} - 1\$" ]	[ "Three circles \$O_1\$, \$O_2\$, and \$O_3\$ touch each other pairwise. Circle \$O_1\$ and circle \$O_2\$ both have radii of 1 unit. A common tangent \$\\overline{AB}\$ is drawn to circles \$O_1\$ and \$O_2\$, and a semicircle \$\\widehat{AB}\$ touches each of the circles \$O_1\$, \$O_2\$, and...	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-a040868006972a709e41f09e6e3c445c20899d463e1895a316511b9673450564	Three circles $O_1$, $O_2$, and $O_3$ with radii $r_1 = 3$, $r_2 = 4$, and $r_3 = 5$ respectively are tangent externally in pairs. Let $r$ be the radius of the inscribed circle of triangle $O_1O_2O_3$. Calculate the value of $r$.	[ "\\sqrt{5}" ]	[ "Three circles \$O_1\$, \$O_2\$, and \$O_3\$ with radii \$r_1 = 3\$, \$r_2 = 4\$, and \$r_3 = 5\$ respectively are tangent externally in pairs. Let \$r\$ be the radius of the inscribed circle of triangle \$O_1O_2O_3\$. Calculate the value of \$r\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-f09ca1a7f621bc22d41be9ba60f15c777f60c76d513394c6e3397c96c16fe0bd	Three circles $X$, $Y$, and $Z$ are tangent to each other. Let $A$ and $B$ be the tangency points of circles $X$ and $Y$, and $X$ and $Z$ respectively. Point $P$ is located on circle $X$, and lines $PA$ and $PB$ intersect circles $Y$ and $Z$ at points $M$ and $N$ respectively. Let \(C\...	[ "5r" ]	[ "Three circles \$X\$, \$Y\$, and \$Z\$ are tangent to each other. Let \$A\$ and \$B\$ be the tangency points of circles \$X\$ and \$Y\$, and \$X\$ and \$Z\$ respectively. Point \$P\$ is located on circle \$X\$, and lines \$PA\$ and \$PB\$ intersect circles \$Y\$ and \$Z\$ at points \\(...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-a37c4f98d5fef75b5e4bf44297e7889eb1a1a938375984a1e73c15dfb515bb50	Three circles are arranged in a row so that each is tangent to the circles next to it. The radii of the two circles at the two ends are 5 and 3. What is the length of the line segment AB that passes through the center of each circle? (Endpoint A is on the larger circle and endpoint B is on the smaller circle.)	[ "16 + 2\\sqrt{15}" ]	[ "Three circles are arranged in a row so that each is tangent to the circles next to it. The radii of the two circles at the two ends are 5 and 3. What is the length of the line segment AB that passes through the center of each circle? (Endpoint A is on the larger circle and endpoint B is on the smaller circle.)" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 8, "genselect": 7 }	15
omr-ws-57f414046de4fe1d88a031175e3b159fe9dc587d18f1fa21a3cc0d61213448bc	Three circles are arranged in a row such that each is tangent to the circle(s) next to it and to the straight lines $AB$ and $CD$. If the radii of the two circles at the two ends are 3 and 5 units, respectively, what is the radius of the middle circle?	[ "\\sqrt{15}" ]	[ "Three circles are arranged in a row such that each is tangent to the circle(s) next to it and to the straight lines $AB$ and $CD$. If the radii of the two circles at the two ends are 3 and 5 units, respectively, what is the radius of the middle circle?" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-bedf6c45652c00ff4884a816623a550abd3b1996bc6abb5d7ee77327b6110a3c	Three circles are arranged inside a triangle such that each circle touches the other two circles externally and also touches two sides of the triangle. Let the lengths of the segments of the triangle's sides, located between the points of tangency with the circles, be $a$, $b$, and $c$. If the triangle's sides are of l...	[ "21" ]	[ "Three circles are arranged inside a triangle such that each circle touches the other two circles externally and also touches two sides of the triangle. Let the lengths of the segments of the triangle's sides, located between the points of tangency with the circles, be $a$, $b$, and $c$. If the triangle's sides are...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-dd928dae15cebbf99a8b6fd5162d298f28f61c282b9e495c1d6fb06421340bf6	Three circles are externally pairwise tangent to each other with radii 1, 2, and 3. The centers of the circles with radii 1 and 2 lie on the X-axis, and the circle with radius 3 touches them from the top. A fourth circle is drawn in the region between these three circles such that it is externally tangent to all three....	[ "\\left( \\dfrac{21}{23}, \\dfrac{20}{23} \\right)" ]	[ "Three circles are externally pairwise tangent to each other with radii 1, 2, and 3. The centers of the circles with radii 1 and 2 lie on the X-axis, and the circle with radius 3 touches them from the top. A fourth circle is drawn in the region between these three circles such that it is externally tangent to all t...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-b3572139eb9b202f7dae236b98ce399bf03671d7e10aa9bffb59d81040f088a0	Three circles are externally tangent to each other and also tangent to a straight line. The radii of two of the circles are $1$ and $\frac{4}{9}$. What is the radius of the remaining circle?	[ "\$ \\frac{4}{25} \$" ]	[ "Three circles are externally tangent to each other and also tangent to a straight line. The radii of two of the circles are $1$ and $\\frac{4}{9}$. What is the radius of the remaining circle?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 21 }	36
omr-ws-07f86ead8c9612c3d91e58945ffd5d5823081167b4e7cce4cefc83b09ae32da7	Three circles are externally tangent to each other with radii 3, 4, and 5. Let P be the point of intersection of the tangents to the circles at the points of contact. Find the distance from P to the points of contact.	[ "\$\\sqrt{5}\$" ]	[ "Three circles are externally tangent to each other with radii 3, 4, and 5. Let P be the point of intersection of the tangents to the circles at the points of contact. Find the distance from P to the points of contact." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 25 }	40
omr-ws-14cdb1be06bd9a9b82fd1c1e384ca55d4ccb206c27ae0feef40025565cb19c6a	Three circles are externally tangent to each other, with each circle having a radius of 1. What is the area of the section bounded by the tangent points (the center area, not in one of the circles)?	[ "\$\\sqrt{3} - \\frac{\\pi}{2}\$" ]	[ "Three circles are externally tangent to each other, with each circle having a radius of 1. What is the area of the section bounded by the tangent points (the center area, not in one of the circles)?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 18 }	33
omr-ws-fd40c8e7d942f778ef1325ce3991d5e97be92896c49c782563564d920516ab87	Three circles are externally tangent to each other. Point $ P $ lies on the first circle. Two lines pass through $ P $ and the tangent points between the first and second circles, and the first and third circles, respectively. These lines intersect the second circle at point $ A $ and the third circle at point \(...	[ "\\dfrac{1}{4}" ]	[ "Three circles are externally tangent to each other. Point \$ P \$ lies on the first circle. Two lines pass through \$ P \$ and the tangent points between the first and second circles, and the first and third circles, respectively. These lines intersect the second circle at point \$ A \$ and the third circle ...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 6, "genselect": 6 }	12
omr-ws-ec2757a147bd203cc10161a84684528197cc6d231e13f2b155e687472f90712a	Three circles are externally tangent to one another with radii $a$, $b$, and $c$. What are the radii of the circles that are tangent to these three original circles, both inside and outside the configuration?	[ "\\dfrac{abc}{2\\sqrt{abc(a + b + c)} - ab - bc - ca}" ]	[ "Three circles are externally tangent to one another with radii \$a\$, \$b\$, and \$c\$. What are the radii of the circles that are tangent to these three original circles, both inside and outside the configuration?" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-18249e7caf2f74ba136316203db44410b35f0ace36cf3bc563d026bd5ee868c6	Three circles are given on the plane with centers $O_1, O_2, O_3$ and radii $r_1, r_2, r_3$, having a common point. Consider three new circles with radii $r_1, r_2, r_3$ and centers $O'_1, O'_2, O'_3$ such that $\|O'_1O'_2\| = \|O_1O_2\| - 1$, $\|O'_2O'_3\| = \|O_2O_3\| - 1$, and $\|O'_3O'_1\| = \|O_3O_1\| - 1$. Calculate the mini...	[ "\\dfrac{1}{2}" ]	[ "Three circles are given on the plane with centers $O_1, O_2, O_3$ and radii $r_1, r_2, r_3$, having a common point. Consider three new circles with radii $r_1, r_2, r_3$ and centers $O'_1, O'_2, O'_3$ such that $\|O'_1O'_2\| = \|O_1O_2\| - 1$, $\|O'_2O'_3\| = \|O_2O_3\| - 1$, and $\|O'_3O'_1\| = \|O_3O_1\| - 1$. Calculate the...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 6 }	11
omr-ws-70ebfc5acaea7eaa3a898c0ed656243b4b9affb3966dddee7680a01381ce21d6	Three circles are inscribed in a semicircle such that each circle is tangent to the semicircle (both the diameter and the arc) and to the other two circles. Calculate the fraction of the area of the semicircle that is covered by the three inscribed circles.	[ "\$ 46 - 32\\sqrt{2} \$ or approximately \$ 0.745166 \$ of the semi-circle's area." ]	[ "Three circles are inscribed in a semicircle such that each circle is tangent to the semicircle (both the diameter and the arc) and to the other two circles. Calculate the fraction of the area of the semicircle that is covered by the three inscribed circles." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-5d9968acefba732e395fb844989da050e51829b61616222ca6d7799f01a14fb6	Three circles are inscribed in a semicircle such that they are all touching each other. The radii of two of the circles are 2 and 3. Find the radius of the third circle.	[ "6" ]	[ "Three circles are inscribed in a semicircle such that they are all touching each other. The radii of two of the circles are 2 and 3. Find the radius of the third circle." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 3 }	3
omr-ws-139f030d94dfbdc7e2fe39121ed1ba97e575f26098f0c67755b823ecda37c32b	Three circles are inscribed in an angle. The radius of the smallest circle is $r$, and the distance from its center to the vertex of the angle is $a$. The medium circle passes through the center of the small circle, and the large circle passes through the center of the medium circle. Determine the radii of the medium a...	[ "\$ s = \\frac{ar}{a-r} \$, \$ t = \\frac{a^2 r}{(a-r)^2} \$" ]	[ "Three circles are inscribed in an angle. The radius of the smallest circle is $r$, and the distance from its center to the vertex of the angle is $a$. The medium circle passes through the center of the small circle, and the large circle passes through the center of the medium circle. Determine the radii of the med...	[ "aops_c4_high_school_math" ]	[ "tir" ]	{ "tir": 12 }	12
omr-ws-b49fa31d5a2dc4f226200bd142490dc6ae8216dd04d8c5d1a4337abe0d5d0868	Three circles are mutually externally tangent. Two of the circles have radii 3 and 7. If the area of the triangle formed by connecting their centers is 84, then the area of the third circle is $k\pi$ for some integer $k$. Determine $k$.	[ "196" ]	[ "Three circles are mutually externally tangent. Two of the circles have radii 3 and 7. If the area of the triangle formed by connecting their centers is 84, then the area of the third circle is \$k\\pi\$ for some integer \$k\$. Determine \$k\$." ]	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-1c2473d4ad1bcaa720239a482e2c569bcf3be8ba4956aedf6b14a4a318318aa9	Three circles are placed inside a square. Two of the circles have radius 2 and are tangent to two sides of the square. These two circles are also externally tangent to the circle with radius 3, which is tangent to the bottom side of the square. Let $x$ be the side length of the square. Then $5x$ can be expressed in the...	[ "59" ]	[ "Three circles are placed inside a square. Two of the circles have radius 2 and are tangent to two sides of the square. These two circles are also externally tangent to the circle with radius 3, which is tangent to the bottom side of the square. Let $x$ be the side length of the square. Then $5x$ can be expressed i...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 15, "genselect": 8 }	23
omr-ws-aaf4482791c5bb2dbb0ec5865d5654e176fa6294eb02581b4f48528f4930a999	Three circles are placed to minimize the sum of the distances between each center without one circle being inside another. A rectangle is drawn such that its sides are tangent to the circles. Find the ratio between the shortest and longest side of this rectangle.	[ "\$\\frac{2 + \\sqrt{3}}{4}\$" ]	[ "Three circles are placed to minimize the sum of the distances between each center without one circle being inside another. A rectangle is drawn such that its sides are tangent to the circles. Find the ratio between the shortest and longest side of this rectangle." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 27 }	43
omr-ws-71511c9f5afc1f82ec683cf8bbe6381c1266097ad541a0451f3b2ea0febd33ad	Three circles are positioned such that no two circles overlap each other. Let there be $k$ circles that touch each of the three circles at exactly one point. Determine all possible values of $k$.	[ "\$ k = 2, 3, 4, 5, 6, 7, 8 \$" ]	[ "Three circles are positioned such that no two circles overlap each other. Let there be $k$ circles that touch each of the three circles at exactly one point. Determine all possible values of $k$." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-d9159c3a9c20b8843224312984b28f1ae8276d1f94f6aced776870b4ee36d861	Three circles are tangent to the sides of a 60-degree angle AOB and also to each other. If the radius of the smallest circle is 1, determine the radius of the largest circle.	[ "\$\\frac{6 + \\sqrt{3}}{2}\$" ]	[ "Three circles are tangent to the sides of a 60-degree angle AOB and also to each other. If the radius of the smallest circle is 1, determine the radius of the largest circle." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-a7e84de16cc923ffc78a31453167eee89069946f0c9f0ffff870a85f65644f36	Three circles are touching each other and a line underneath them. The radii of the largest and the smaller circles are 2 and 1, respectively. What is the radius of the smallest circle?	[ "\$6 - 4\\sqrt{2}\$" ]	[ "Three circles are touching each other and a line underneath them. The radii of the largest and the smaller circles are 2 and 1, respectively. What is the radius of the smallest circle?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 24 }	40
omr-ws-6efc9bf534186b96648021f56ee148cf5bd6ba3634302e553bbb2c83f6757ffa	Three circles each having radius $1$ pass through the same point in the plane. Let $A$ be the set of points that lie inside at least two of the circles. Determine the smallest area of $A$.	[ "\$\\pi - \\frac{3\\sqrt{3}}{2}\$" ]	[ "Three circles each having radius $1$ pass through the same point in the plane. Let $A$ be the set of points that lie inside at least two of the circles. Determine the smallest area of $A$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 29 }	45
omr-ws-8bda755df4dfbf9bc8da1d67475bdd492644aaa8205fb7e89114a1bdbd257877	Three circles each having radius $r$ pass through the same point in the plane. Let $A$ be the set of points that lie inside at least two of the circles. Determine the smallest area of $A$.	[ "\$ \\left( \\pi - \\frac{3\\sqrt{3}}{2} \\right) r^2 \$" ]	[ "Three circles each having radius $r$ pass through the same point in the plane. Let $A$ be the set of points that lie inside at least two of the circles. Determine the smallest area of $A$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 23 }	39
omr-ws-4bfce61b7dc0c297f8260da295e8d94f04aa60a16ae5bfd671e4a1dc86a86f28	Three circles each of radius one have centers at $ A $, $ B $, and $ C $. Circles $ A $ and $ B $ touch each other, and circle $ C $ touches $ AB $ at its midpoint. Find the area inside circle $ C $ but outside circles $ A $ and $ B $.	[ "2" ]	[ "Three circles each of radius one have centers at \$ A \$, \$ B \$, and \$ C \$. Circles \$ A \$ and \$ B \$ touch each other, and circle \$ C \$ touches \$ AB \$ at its midpoint. Find the area inside circle \$ C \$ but outside circles \$ A \$ and \$ B \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 47 }	47
omr-ws-6584e390d4a3588a250b149ee521d051b03522438e0d33a4758b735911427f75	Three circles in the plane, whose interiors have no common point, meet each other at three pairs of points: $A_1$ and $A_2$, $B_1$ and $B_2$, and $C_1$ and $C_2$, where points $A_2$, $B_2$, and $C_2$ lie inside the triangle $A_1B_1C_1$. If the lengths $A_1B_2 = 6$, $B_1C_2 = 8$, $C_1A_2 = 10$,...	[ "\\dfrac{20}{7}" ]	[ "Three circles in the plane, whose interiors have no common point, meet each other at three pairs of points: \$A_1\$ and \$A_2\$, \$B_1\$ and \$B_2\$, and \$C_1\$ and \$C_2\$, where points \$A_2\$, \$B_2\$, and \$C_2\$ lie inside the triangle \$A_1B_1C_1\$. If the lengths \$A_1B_2 = 6\$, \\(B_...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 1 }	1
omr-ws-ff1927b9d84f197416176f629326a969f59751804c8ae5cb27b6cc716e77af88	Three circles internally touch the circumcircle of triangle $ABC$ at points $T_a, T_b, T_c$, and the sides $BC$, $CA$, and $AB$ at points $P_a, P_b, P_c$, respectively. The tangents to the circumcircle at points $T_a, T_b, T_c$ intersect the lines $BC, CA, AB$ at points $Q_a, Q_b, Q_c$, respectively. If the lines $AP_a...	[ "2" ]	[ "Three circles internally touch the circumcircle of triangle $ABC$ at points $T_a, T_b, T_c$, and the sides $BC$, $CA$, and $AB$ at points $P_a, P_b, P_c$, respectively. The tangents to the circumcircle at points $T_a, T_b, T_c$ intersect the lines $BC, CA, AB$ at points $Q_a, Q_b, Q_c$, respectively. If the lines ...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-cffd1161304bdbd95adc4e1c5013ea62ba257ae746c8f1dce99ad66fff62f700	Three circles intersect at one point $A$. The centers of the second and third circles lie on the first circle. Let $B$ be the intersection of the second and third circles other than $A$, and let $C$ be the intersection of the first and third circles other than $A$. If the center of the second circle is $O$, calculate t...	[ "\\sqrt{3}" ]	[ "Three circles intersect at one point $A$. The centers of the second and third circles lie on the first circle. Let $B$ be the intersection of the second and third circles other than $A$, and let $C$ be the intersection of the first and third circles other than $A$. If the center of the second circle is $O$, calcul...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 2, "genselect": 4 }	6
omr-ws-6c16af0a8064e68c48017b54fc4060fba2e47b91ad13033c590ba460df53ddc0	Three circles lie inside a square. Each circle touches externally two remaining circles and each circle touches two sides of the square. If the side length of the square is $s$, and the radii of the circles are $r_1$, $r_2$, and $r_3$, find the value of $\frac{s}{r_1 + r_2 + r_3}$.	[ "\\dfrac{5 + 3\\sqrt{2}}{7}" ]	[ "Three circles lie inside a square. Each circle touches externally two remaining circles and each circle touches two sides of the square. If the side length of the square is $s$, and the radii of the circles are $r_1$, $r_2$, and $r_3$, find the value of $\\frac{s}{r_1 + r_2 + r_3}$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-a5a6518a357485d318f0796514e0163d063c7be607768a89e5d918877262ffc2	Three circles lie on a plane so that each of them externally touches the other two. Two of them have radius 3, the third, radius 1. Find the area of the triangle $ABC$, where $A$, $B$, and $C$ are points of tangency of the circles.	[ "$\\frac{9\\sqrt 7}{16}$" ]	[ "Three circles lie on a plane so that each of them externally touches the other two. Two of them have radius 3, the third, radius 1. Find the area of the triangle $ABC$, where $A$, $B$, and $C$ are points of tangency of the circles." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-cf16adeb83c67329a742097e31ac7ba45fd89558559bfd3425b6b11c3705b0fc	Three circles of equal area are situated in the Euclidean plane such that their centers are equidistant. Given that the shared area (the area of the space at the intersection of all three circles) forms a Reuleaux triangle, what is the area of the Reuleaux triangle in terms of the radius of the circles?	[ "\$\\frac{r^2(\\pi-\\sqrt{3})}{2}\$" ]	[ "Three circles of equal area are situated in the Euclidean plane such that their centers are equidistant. Given that the shared area (the area of the space at the intersection of all three circles) forms a Reuleaux triangle, what is the area of the Reuleaux triangle in terms of the radius of the circles?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 17 }	33
omr-ws-6fa5b34bf993724bcdfa8032b6abe5dfcb94d68ee473a8516a10c379ea89d1fa	Three circles of equal radii are centered at the vertices $A$, $B$, and $C$ of a triangle $ABC$. The circle centered at $A$ intersects $AB$ and $AC$ at points $A'$ and $A''$, respectively. Let $A_1$ be the midpoint of the arc $\overarc{A'A''}$. Similarly, define points $B'$, $B''$, $C'$, \(C...	[ "10" ]	[ "Three circles of equal radii are centered at the vertices \$A\$, \$B\$, and \$C\$ of a triangle \$ABC\$. The circle centered at \$A\$ intersects \$AB\$ and \$AC\$ at points \$A'\$ and \$A''\$, respectively. Let \$A_1\$ be the midpoint of the arc \$\\overarc{A'A''}\$. Similarly, define points ...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 4 }	9
omr-ws-e83ce06de6b6bc163dd1933671680645df2600e7c5f4a835fe3e26998f38bf47	Three circles of equal radius $ r $ are tangent and their centers are collinear from left to right. Triangle $ ABC $ is constructed where $ A $ is the center of the left circle, $ B $ is the center of the right circle, and $ C $ is the point on the right circle where the tangent line through $ A $ touches t...	[ "\$ r\\sqrt{3} \$" ]	[ "Three circles of equal radius \$ r \$ are tangent and their centers are collinear from left to right. Triangle \$ ABC \$ is constructed where \$ A \$ is the center of the left circle, \$ B \$ is the center of the right circle, and \$ C \$ is the point on the right circle where the tangent line through \\...	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 23 }	39
omr-ws-464989e0626057504be27caac5afc23273e96d976a375191c8ce2ce841def798	Three circles of equal radius have a common point $O$ and lie inside a given triangle $ABC$. Each circle touches a pair of sides of the triangle. If the radius of each circle is $r$, the inradius of the triangle is $r_I$, and the circumradius is $R$, find the value of $\frac{r_I + R}{r}$ when the triangle is equilatera...	[ "\\dfrac{9}{2}" ]	[ "Three circles of equal radius have a common point $O$ and lie inside a given triangle $ABC$. Each circle touches a pair of sides of the triangle. If the radius of each circle is $r$, the inradius of the triangle is $r_I$, and the circumradius is $R$, find the value of $\\frac{r_I + R}{r}$ when the triangle is equi...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-3042c9fd77c2545387a36e72561386f49bd36c34640403ee92d4ff83db1f91fd	Three circles of radii $r$, $r_1$ and $R$ are pairwise externally tangent. Find the length of the chord cut off by the third circle from the internal common tangent of the first two circles.	[ "AB=\\frac{4\\sqrt{rr_1}}{r+r_1}R" ]	[ "Three circles of radii $r$, $r_1$ and $R$ are pairwise externally tangent. Find the length of the chord cut off by the third circle from the internal common tangent of the first two circles." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-9d6b315e678c174cdee042684d5737360d88059ec8bda7ac1b23032b9ce00759	Three circles of radii 1 are mutually tangent. What is the area of the bounded region located between the curves of the three circles?	[ "\$ \\sqrt{3} - \\frac{\\pi}{2} \$" ]	[ "Three circles of radii 1 are mutually tangent. What is the area of the bounded region located between the curves of the three circles?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 13, "genselect": 2, "tir": 28 }	43
omr-ws-55dd524b645684da5d22c772b0bc9b5aa650aa8c9421c0c508c809d9084be343	Three circles of radii $ R_1 $, $ R_2 $, and $ R_3 $ touch each other externally. Let $ r $ and $ R $ be the radii of two circles (with $ r < R $) which touch all three of these circles. If $ R_1 = 3 $, $ R_2 = 4 $, and $ R_3 = 5 $, find the value of $ \frac{1}{r} + \frac{1}{R} $.	[ "\\dfrac{4\\sqrt{5}}{5}" ]	[ "Three circles of radii \$ R_1 \$, \$ R_2 \$, and \$ R_3 \$ touch each other externally. Let \$ r \$ and \$ R \$ be the radii of two circles (with \$ r < R \$) which touch all three of these circles. If \$ R_1 = 3 \$, \$ R_2 = 4 \$, and \$ R_3 = 5 \$, find the value of \\( \\frac{1}{r} + \\frac{1}...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-f031d9d702237c51836cdbf08a468bc4f96395f6633bcd829835123875cad87b	Three circles of radii $ r_1, r_2, r_3 $ touch each other externally and share a common tangent. The tangent touches the circles at points $ A, B, C $ with $ B $ between $ A $ and $ C $. If $ r_1 = 4 $, $ r_2 = 9 $, and $ r_3 = 36 $, calculate the value of $ 16(r_1 + r_2 + r_3) - 9(AB + BC + CA) $.	[ "136" ]	[ "Three circles of radii \$ r_1, r_2, r_3 \$ touch each other externally and share a common tangent. The tangent touches the circles at points \$ A, B, C \$ with \$ B \$ between \$ A \$ and \$ C \$. If \$ r_1 = 4 \$, \$ r_2 = 9 \$, and \$ r_3 = 36 \$, calculate the value of \\( 16(r_1 + r_2 + r_3) - ...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-b2bb7d5b60f4d3761130e53c104a652f900a4832c3d6ee8a63ea0bae7fcd430d	Three circles of radii $ x $, $ 2x $, and $ 3x $ are inscribed inside a circle of radius 12 and are tangent to each other. What is the value of $ x $?	[ "\$ x = 2 \$" ]	[ "Three circles of radii \$ x \$, \$ 2x \$, and \$ 3x \$ are inscribed inside a circle of radius 12 and are tangent to each other. What is the value of \$ x \$?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 15, "genselect": 7 }	22
omr-ws-64bd36df9f02af2159a8f235fa30de2c9da6c6b16c14009b67f61f5825be3fcc	Three circles of radius $1$ are inscribed in a square of side length $s$, such that the circles do not overlap or coincide with each other. What is the minimum $s$ where such a configuration is possible?	[ "4" ]	[ "Three circles of radius $1$ are inscribed in a square of side length $s$, such that the circles do not overlap or coincide with each other. What is the minimum $s$ where such a configuration is possible?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 11, "genselect": 8 }	19
omr-ws-cbb3843c750a48b623ffbe0aa2b1c93f074b3a00d95d14f6bafbb4b8f2853759	Three circles of radius $1$, $2$, and $3$ are placed at the points $(4, 14)$, $(15, 5)$, and $(6, 16)$ respectively. Find the coordinates of the point equidistant from the edges of all three circles.	[ "\\left( \\dfrac{99}{20}, 5 \\right)" ]	[ "Three circles of radius $1$, $2$, and $3$ are placed at the points $(4, 14)$, $(15, 5)$, and $(6, 16)$ respectively. Find the coordinates of the point equidistant from the edges of all three circles." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 13, "genselect": 4 }	17
omr-ws-defb949fb7d0ee04de9aebb7f4ca8890daf56f1db8b562c83eab83b2193962f2	Three circles of radius $R$ intersect in pairs, forming a clover with three common areas. Find the area of the clover in terms of $R$ and the area $S$ of the triangle $IJK$.	[ "3\\pi R^2 - 6S" ]	[ "Three circles of radius $R$ intersect in pairs, forming a clover with three common areas. Find the area of the clover in terms of $R$ and the area $S$ of the triangle $IJK$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 3 }	3
omr-ws-e1ec22364a3b8213a8d91aeb5bfa8e9734005be07bfdc38c31b13d2243dbc914	Three circles of radius $R$ intersect in pairs, forming a clover with three common areas. Find the perimeter of the clover in terms of $R$ and the area $S$ of the triangle $IJK$.	[ "\$2\\pi R\$" ]	[ "Three circles of radius $R$ intersect in pairs, forming a clover with three common areas. Find the perimeter of the clover in terms of $R$ and the area $S$ of the triangle $IJK$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 3, "genselect": 2 }	5
omr-ws-07f4dfc8baf9ca2a7143dc27f7cb4b958228675055a606b30ea5c3338fe39b88	Three circles of radius $a$ are drawn on the surface of a sphere of radius $r$. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.	[ "r \\left( \\arccos \\left( \\sqrt{\\frac{4 \\cos^2\\left(\\frac{a}{r}\\right) - 1}{3}} \\right) - \\frac{a}{r} \\right)" ]	[ "Three circles of radius $a$ are drawn on the surface of a sphere of radius $r$. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems", "genselect" ]	{ "additional_problems": 1, "genselect": 1 }	2
omr-ws-a5419f95789f2821f4cc28d6451fed62882acbdc238e253ae863d26b55439de1	Three circles of radius $r$ can be inscribed in a circle of radius $1$ so that they are tangent to each other and tangent to the circle of radius $1$. It turns out that $r$ can be written in the form $r = a\sqrt3+b$, where $a$ and $b$ are integers. Find $a^2 + b^2$.	[ "13" ]	[ "Three circles of radius $r$ can be inscribed in a circle of radius $1$ so that they are tangent to each other and tangent to the circle of radius $1$. It turns out that $r$ can be written in the form $r = a\\sqrt3+b$, where $a$ and $b$ are integers. Find $a^2 + b^2$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-55b4a10917ee622a1e437c90dea37b258e4455a46736339f5f3387cb8c992dfc	Three circles of radius 1 are all tangent to one another. A fourth smaller circle is tangent to the other three circles. Find the radius of the smaller circle.	[ "\$\\frac{2\\sqrt{3} - 3}{3}\$" ]	[ "Three circles of radius 1 are all tangent to one another. A fourth smaller circle is tangent to the other three circles. Find the radius of the smaller circle." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 30 }	46
omr-ws-0650bdd89be0c3b01b7a162f32a3184394dee3a675f3f0a297a1283dd13261cf	Three circles of radius 1 are each tangent to the other two circles. A fourth circle is externally tangent to all three circles. The radius of the fourth circle can be expressed as $\frac{a\sqrt{b}-c}{d}$ for positive integers $a, b, c, d$ where $b$ is not divisible by the square of any prime and $a$ and $d$ are relati...	[ "11" ]	[ "Three circles of radius 1 are each tangent to the other two circles. A fourth circle is externally tangent to all three circles. The radius of the fourth circle can be expressed as $\\frac{a\\sqrt{b}-c}{d}$ for positive integers $a, b, c, d$ where $b$ is not divisible by the square of any prime and $a$ and $d$ are...	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 1 }	17
omr-ws-8f02e7ebf23bb4cfaf3277109ba0c500ac730080d11cc36c641ab40b719a4d70	Three circles of radius 1 are inscribed in a semicircle such that each small circle is tangent to the other two circles and the semicircle. Compute the radius of the large semicircle.	[ "\$ 1 + \\sqrt{5 + 2\\sqrt{3}} \$" ]	[ "Three circles of radius 1 are inscribed in a semicircle such that each small circle is tangent to the other two circles and the semicircle. Compute the radius of the large semicircle." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-1ffe2901f54a549a0519fa66aa409d605e7ea5bfdfa751732faaaf342e9f19cf	Three circles of radius 1 are positioned so that their centers form an equilateral triangle. If the three circles all intersect in exactly one point, find the side length of the triangle formed by their centers.	[ "\$\\sqrt{3}\$" ]	[ "Three circles of radius 1 are positioned so that their centers form an equilateral triangle. If the three circles all intersect in exactly one point, find the side length of the triangle formed by their centers." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 28 }	44
omr-ws-f9597f122ac21f5069592f752f7310d2a7dd9485884cd26bf90a65255d26cd43	Three circles of radius 1 intersect such that each circle passes through the midpoints of the other two circles. Find the area of the region that is the intersection of the three circles. Express your answer in terms of $\pi$.	[ "\$\\frac{\\pi - \\sqrt{3}}{2}\$" ]	[ "Three circles of radius 1 intersect such that each circle passes through the midpoints of the other two circles. Find the area of the region that is the intersection of the three circles. Express your answer in terms of $\\pi$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 13, "genselect": 4, "tir": 9 }	26
omr-ws-99f59e3844a626d33cd05de07d0b87b8c085e798d948a686bdc0d03cfdeaef6a	Three circles of radius 2 are touching each other. Find the area of the enclosed region formed by the circumferences of the circles.	[ "\$ 4\\sqrt{3} - 2\\pi \$", "} 4\\sqrt{3} - 2\\pi \\]" ]	[ "Three circles of radius 2 are touching each other. Find the area of the enclosed region formed by the circumferences of the circles." ]	[ "aops_c4_high_school_math", "aops_c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 8, "genselect": 1, "tir": 20 }	29
omr-ws-062e04ebe3ec78cf74a186d19a8f30b6287a7d6202c005f6327c80bbdebb2962	Three circles of radius $ r $ cover a circle of radius 1. Find the minimum possible value of $ r $.	[ "\$ \\frac{\\sqrt{3}}{2} \$" ]	[ "Three circles of radius \$ r \$ cover a circle of radius 1. Find the minimum possible value of \$ r \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "genselect" ]	{ "genselect": 7 }	7
omr-ws-71168c620333503e6ce4ad904b84dd48c2702de3c86bcdc70d91713ac300b994	Three circles of radius $ s $ are drawn in the first quadrant of the $ xy $-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the $ x $-axis, and the third is tangent to the first circle and the $ y $-axis. A circle of radius $ r > s $ is tangent to both axes and t...	[ "9" ]	[ "Three circles of radius \$ s \$ are drawn in the first quadrant of the \$ xy \$-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the \$ x \$-axis, and the third is tangent to the first circle and the \$ y \$-axis. A circle of radius \$ r > s \$ is tangent to ...	[ "aops_c4_high_school_math", "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 31 }	31
omr-ws-4b67dae8a938620277507ab25870577ee7976bb5b1cd22257cd4f78570e44f7b	Three circles of unit radius pass through a common point $P$ and each circle passes through one of the points $A$, $B$, and $C$. What can be the radius of the circumcircle of triangle $ABC$?	[ "2" ]	[ "Three circles of unit radius pass through a common point $P$ and each circle passes through one of the points $A$, $B$, and $C$. What can be the radius of the circumcircle of triangle $ABC$?" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 13 }	13
omr-ws-2a30fab826915167b30b8fe959237667435b82eb33ea378c2f79481614cf0472	Three circles pass through a point $ P $, and the second points of their intersection $ A, B, C $ lie on a straight line. Let $ A_1, B_1, C_1 $ be the second intersections of lines $ AP, BP, CP $ with the corresponding circles. Let $ C_2 $ be the intersection of lines $ AB_1 $ and $ BA_1 $. Let \( A_2, B_...	[ "60" ]	[ "Three circles pass through a point \$ P \$, and the second points of their intersection \$ A, B, C \$ lie on a straight line. Let \$ A_1, B_1, C_1 \$ be the second intersections of lines \$ AP, BP, CP \$ with the corresponding circles. Let \$ C_2 \$ be the intersection of lines \$ AB_1 \$ and \\( BA_1 ...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-eafc00aeef1f7b9118f816b86eb3f68fe335f6ab4118dd7aab0cb30791c06203	Three circles pass through point X. Their intersection points (other than X) are denoted A, B, C. Let A' be the second point of intersection of line AX and the circle circumscribed around triangle BCX, and define similarly points B', C'. If the circumradius of triangle ABC is R and the circumradius of triangle A'B'C' i...	[ "2" ]	[ "Three circles pass through point X. Their intersection points (other than X) are denoted A, B, C. Let A' be the second point of intersection of line AX and the circle circumscribed around triangle BCX, and define similarly points B', C'. If the circumradius of triangle ABC is R and the circumradius of triangle A'B...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 3, "genselect": 6 }	9
omr-ws-0e9c8a55ae7bed601be93c082d57d136c6d501fe3b6f8f2ba114c948e8620676	Three circles pass through points A and B. A variable line passing through A intersects the three circles at points P, Q, and R. If the radii of the three circles are 5, 7, and 9 respectively, and the distance between A and B is 12, find the value of the constant ratio $\frac{PQ}{QR}$.	[ "\\dfrac{3}{4}" ]	[ "Three circles pass through points A and B. A variable line passing through A intersects the three circles at points P, Q, and R. If the radii of the three circles are 5, 7, and 9 respectively, and the distance between A and B is 12, find the value of the constant ratio $\\frac{PQ}{QR}$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 7 }	12
omr-ws-9f2f3bee091bc3cf99b1db5db65617a4884c7f90c16d3f93b5080270fefb2d0d	Three circles pass through the points $P$ and $Q$. A line cuts these circles in points $A, B, C, D, E$ and $F$, as shown in the figure. If $AB = 5$, $EF = 4$ and $AF = 20$, determine how many integer values $CD$ can take.	[ "10" ]	[ "Three circles pass through the points \$P\$ and \$Q\$. A line cuts these circles in points \$A, B, C, D, E\$ and \$F\$, as shown in the figure. If \$AB = 5\$, \$EF = 4\$ and \$AF = 20\$, determine how many integer values \$CD\$ can take." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 8 }	13
omr-ws-0b67527b58638f664e0228f103c4c79e53aeaa73fc1801a480ac06eb0eb46c30	Three circles that pass through a point $X$ intersect each other pairwise at points $A$, $B$, and $C$. The line $AX$ intersects one of the circles again at point $D$. Similarly, lines $BX$ and $CY$ intersect their respective circles again at points $E$ and $F$. If the ratio of the lengths of the sides of $\triangle BCD...	[ "2:3:4" ]	[ "Three circles that pass through a point $X$ intersect each other pairwise at points $A$, $B$, and $C$. The line $AX$ intersects one of the circles again at point $D$. Similarly, lines $BX$ and $CY$ intersect their respective circles again at points $E$ and $F$. If the ratio of the lengths of the sides of $\\triang...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-246b558f675b09b4a034583e99108195c9ba1adfda26b0c6ddcac9f2a092e0d5	Three circles that touch each other externally have all their centers on one fourth circle with radius $ R $. If the radii of the three circles are $ r_1 $, $ r_2 $, and $ r_3 $, find the maximum possible value of $ r_1^2 + r_2^2 + r_3^2 $.	[ "2R^2" ]	[ "Three circles that touch each other externally have all their centers on one fourth circle with radius \$ R \$. If the radii of the three circles are \$ r_1 \$, \$ r_2 \$, and \$ r_3 \$, find the maximum possible value of \$ r_1^2 + r_2^2 + r_3^2 \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-b6b79ecb632334c936540c13b5610a5c3f5ec15ea0e9ae7709cfdca4dfe9b42b	Three circles touch each other externally and all three touch a line. If the radius of two circles is equal and the radius of the third circle is 4, find the radius of the two equal circles.	[ "16" ]	[ "Three circles touch each other externally and all three touch a line. If the radius of two circles is equal and the radius of the third circle is 4, find the radius of the two equal circles." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-b064051d01d06ed8dd6a45e8ab92c8bfcf4dbf1579dab3d7df2fce69a8061e42	Three circles touch each other externally and also touch a fixed straight line. Let $ A, B, C $ be the mutual points of contact of these circles. If $ \omega $ denotes the Brocard angle of the triangle $ ABC $, calculate the value of $ \cot^2{\omega} $.	[ "4" ]	[ "Three circles touch each other externally and also touch a fixed straight line. Let \$ A, B, C \$ be the mutual points of contact of these circles. If \$ \\omega \$ denotes the Brocard angle of the triangle \$ ABC \$, calculate the value of \$ \\cot^2{\\omega} \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 6 }	6
omr-ws-2099fe5f59946aa14511c186829aa686a8815f50233ce6e8ed6330c72eaab769	Three circles touch each other externally. The largest circle has center at $ S $ and radius $ c $. The middle-sized circle has center at $ Q $ and radius $ a $. The smallest circle has center at $ R $ and radius $ b $. The centers of the circles are the vertices of a rectangle $ QRST $, where the fourth ...	[ "\$2:1:3\$" ]	[ "Three circles touch each other externally. The largest circle has center at \$ S \$ and radius \$ c \$. The middle-sized circle has center at \$ Q \$ and radius \$ a \$. The smallest circle has center at \$ R \$ and radius \$ b \$. The centers of the circles are the vertices of a rectangle \$ QRST \$...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 42, "genselect": 3, "tir": 1 }	46
omr-ws-c52b7218aa920df3ecda27eac2c7f817c3fc47169c16c42a1a710dbc024e11c0	Three circles touch one another at points $ A $, $ B $, and $ C $ respectively. A point $ D $ is on one of the circles. The line $ DA $ meets one of the other two circles at $ E $, the line $ EB $ meets the other circle at $ F $, and the line $ FC $ meets the first circle at $ G $. If $ O $ is the...	[ "R" ]	[ "Three circles touch one another at points \$ A \$, \$ B \$, and \$ C \$ respectively. A point \$ D \$ is on one of the circles. The line \$ DA \$ meets one of the other two circles at \$ E \$, the line \$ EB \$ meets the other circle at \$ F \$, and the line \$ FC \$ meets the first circle at \\(...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 6, "genselect": 5 }	11
omr-ws-20a22feec18ea61e48462ca1834c6128881b9fb7716a53d42b265bdc2f7ad388	Three circles whose centers are $X$, $Y$, and $Z$ pass through a common point $O$. Let $P$ be the other point of intersection of the circles with centers $Y$ and $Z$. If the coordinates of $X$, $Y$, $Z$, and $O$ are $(0,0)$, $(4,0)$, $(0,3)$, and $(1,1)$ respectively, find the coordinates ...	[ "\\left( \\dfrac{11}{5}, \\dfrac{13}{5} \\right)" ]	[ "Three circles whose centers are \$X\$, \$Y\$, and \$Z\$ pass through a common point \$O\$. Let \$P\$ be the other point of intersection of the circles with centers \$Y\$ and \$Z\$. If the coordinates of \$X\$, \$Y\$, \$Z\$, and \$O\$ are \$(0,0)\$, \$(4,0)\$, \$(0,3)\$, and \$(1,1)\$ ...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-6c9f8473b525652e3279878f9ca5c7f8631ae60f69d762221b981f6ab0021ef7	Three circles with centers A, B, and C touch each other pairwise externally and touch a larger circle $ c $ from inside. If the center of $ c $ coincides with the orthocenter of triangle ABC, find the ratio of the radius of circle $ c $ to the radius of one of the smaller circles.	[ "\\dfrac{3 + 2\\sqrt{3}}{3}" ]	[ "Three circles with centers A, B, and C touch each other pairwise externally and touch a larger circle \$ c \$ from inside. If the center of \$ c \$ coincides with the orthocenter of triangle ABC, find the ratio of the radius of circle \$ c \$ to the radius of one of the smaller circles." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-2507d79a8fd6d02f5a3330a5880f70af8c6db38968a8fddde8549b10df01470c	Three circles with centers $ C_1 $, $ C_2 $, and $ C_3 $ are tangent to each other. $ PC $ is a common tangent to all three circles, touching the circles with centers $ C_1 $ and $ C_2 $ at points $ A $ and $ B $ respectively. Find the relation between $ PA \cdot PB $ and $ PC $.	[ "\$ PA \\cdot PC = PB^2 \$" ]	[ "Three circles with centers \$ C_1 \$, \$ C_2 \$, and \$ C_3 \$ are tangent to each other. \$ PC \$ is a common tangent to all three circles, touching the circles with centers \$ C_1 \$ and \$ C_2 \$ at points \$ A \$ and \$ B \$ respectively. Find the relation between \$ PA \\cdot PB \$ and \\( P...	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-96df1f1f65ea8df2028f04e5f8dacab059825be7e5dd628b6e7c5bdebc5fa33b	Three circles with centers $ V_0 $, $ V_1 $, and $ V_2 $ and radii 33, 30, and 25 respectively are mutually externally tangent. The tangency points between the circles are $ P_0 $, $ P_1 $, and $ P_2 $. For $ i = 0, 1, 2 $, line $ P_{i+1}P_{i+2} $ intersects circle $ V_{i+1} $ at $ P_{i+2} $ and \( ...	[ "208" ]	[ "Three circles with centers \$ V_0 \$, \$ V_1 \$, and \$ V_2 \$ and radii 33, 30, and 25 respectively are mutually externally tangent. The tangency points between the circles are \$ P_0 \$, \$ P_1 \$, and \$ P_2 \$. For \$ i = 0, 1, 2 \$, line \$ P_{i+1}P_{i+2} \$ intersects circle \$ V_{i+1} \$ a...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 7 }	12
omr-ws-41e4d5518bab7c8e91b08ec8d53be5b3f5b3626ecb44b83973f92cde39aa8f98	Three circles with centers $O_1$, $O_2$, and $O_3$ are externally tangent in pairs at points $A$, $B$, and $C$. Find the area of the triangular region $ABC$ if the radii of the circles are 1, 2, and 3.	[ "\$\\frac{6}{5}\$" ]	[ "Three circles with centers \$O_1\$, \$O_2\$, and \$O_3\$ are externally tangent in pairs at points \$A\$, \$B\$, and \$C\$. Find the area of the triangular region \$ABC\$ if the radii of the circles are 1, 2, and 3." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 30 }	46
omr-ws-6856f599a1f467c78f1d83911dad677d623999cb59e0577279df8698807f77ac	Three circles with integer radii $a$, $b$, $c$ are mutually externally tangent, with $a \le b \le c$ and $a < 10$. The centers of the three circles form a right triangle. Compute the number of possible ordered triples $(a, b, c)$.	[ "30" ]	[ "Three circles with integer radii $a$, $b$, $c$ are mutually externally tangent, with $a \\le b \\le c$ and $a < 10$. The centers of the three circles form a right triangle. Compute the number of possible ordered triples $(a, b, c)$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-9c4ec48ca703be876ea54ab21ef98fbdab30bbe960337e92dd4badd4a5eb70ec	Three circles with radii $1$, $2$, and $3$ all externally touch each other. Find the radius of a circle drawn through these three points of tangency.	[ "1" ]	[ "Three circles with radii $1$, $2$, and $3$ all externally touch each other. Find the radius of a circle drawn through these three points of tangency." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-226223a61000bfb014778174d47d894673dde443977043e6910953dc8ec959f6	Three circles with radii $1$, $2$, and $n$ are pairwise tangent, and their centers form a right triangle. What is the sum of all possible values of $n$?	[ "3" ]	[ "Three circles with radii $1$, $2$, and $n$ are pairwise tangent, and their centers form a right triangle. What is the sum of all possible values of $n$?" ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-ec733902b9a23a285645426c799b8c1094c040be4b165c95fc09aeb6201295e6	Three circles with radii $1, 2,$ and $3$ respectively touch each other in a plane. In the region enclosed by the three circles, another smaller circle $O$ is drawn such that it touches the three circles. Find the radius of the smaller circle $O$.	[ "\$\\frac{6}{23}\$" ]	[ "Three circles with radii $1, 2,$ and $3$ respectively touch each other in a plane. In the region enclosed by the three circles, another smaller circle $O$ is drawn such that it touches the three circles. Find the radius of the smaller circle $O$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 25 }	40
omr-ws-f36058b631227e5d60aaef565b51c181aee74c77d4f00370494fafbeace92369	Three circles with radii $r$ intersect at a common point O and pairwise intersect at points A, B, and C. If the circle passing through points A, B, and C has radius $r$, find the value of $r^2$ given that the distance between any two of the points A, B, and C is $2r$.	[ "no_valid_answer_found" ]	[ "Three circles with radii $r$ intersect at a common point O and pairwise intersect at points A, B, and C. If the circle passing through points A, B, and C has radius $r$, find the value of $r^2$ given that the distance between any two of the points A, B, and C is $2r$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-25d0f8e32ebe64dc13d6a301f33abb7d36cc94e5cdf2dfc57b82d7b6253a5d0a	Three circles with radii $r_1, r_2, r_3$ (where $r_1 < r_2 < r_3$) touch each other externally and have a common tangent. Given that $r_1 = \frac{1}{4}$, find the integer closest to the value of $\sqrt{\frac{50}{r_2}} + \sqrt{\frac{50}{r_3}}$.	[ "14" ]	[ "Three circles with radii $r_1, r_2, r_3$ (where $r_1 < r_2 < r_3$) touch each other externally and have a common tangent. Given that $r_1 = \\frac{1}{4}$, find the integer closest to the value of $\\sqrt{\\frac{50}{r_2}} + \\sqrt{\\frac{50}{r_3}}$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 5 }	10
omr-ws-331259274c8da5837d2fe8663eeaeee809522db7bbeb6bc4039687795252d039	Three circles with radii 1 cm, 2 cm, and 3 cm touch each other externally. A smaller circle lies in the space between them and touches each of the three larger circles. The radius of the smallest circle is given by $\frac{p}{q}$ where $p$ and $q$ are integers with no common factors. What is the value of $p + q$...	[ "13" ]	[ "Three circles with radii 1 cm, 2 cm, and 3 cm touch each other externally. A smaller circle lies in the space between them and touches each of the three larger circles. The radius of the smallest circle is given by \$\\frac{p}{q}\$ where \$p\$ and \$q\$ are integers with no common factors. What is the value ...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-c809e1bddc2e09d32259b1d99fa19dd46b9cd37d4f17655e13da53810b8de6cc	Three circles with radii 1, 2, and 3 are all tangent to each other. A smaller circle is tangent to all three circles. The radius of the smaller circle can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$?	[ "29" ]	[ "Three circles with radii 1, 2, and 3 are all tangent to each other. A smaller circle is tangent to all three circles. The radius of the smaller circle can be expressed in the form $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 8 }	24
omr-ws-6dc2d51e664d189fb562d266311afdb0236f0497b9bcd8b8ffd6d68d3e763ee2	Three circles with radii 1, 2, and 3 are all tangent to each other. What is the length of a "rubber band" around the circles? (A "rubber band" consists of segments of lines tangent to any two circles and arcs between tangency points.)	[ "2\\sqrt{2} + 2\\sqrt{3} + 2\\sqrt{6} + 6\\pi" ]	[ "Three circles with radii 1, 2, and 3 are all tangent to each other. What is the length of a \"rubber band\" around the circles? (A \"rubber band\" consists of segments of lines tangent to any two circles and arcs between tangency points.)" ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-b3ef4c40c76a2aa2efb8afd2cbfcfb13133fad0777c2d6bafb356772e8b4b97e	Three circles with radii 1, 2, and 3 are externally tangent to one another. Find the radius of another circle that is internally tangent to all three circles.	[ "\$\\frac{6}{23}\$" ]	[ "Three circles with radii 1, 2, and 3 are externally tangent to one another. Find the radius of another circle that is internally tangent to all three circles." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 11, "genselect": 8, "tir": 9 }	28
omr-ws-43d022495d23cb42bc161161e442e0e0ca5a90435d1154adc9afd11db92ae529	Three circles with radii 1, 2, and 3 are externally tangent. A triangle is formed by connecting the points of intersection. Find the area of the triangle.	[ "\$\\frac{6}{5}\$" ]	[ "Three circles with radii 1, 2, and 3 are externally tangent. A triangle is formed by connecting the points of intersection. Find the area of the triangle." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 29 }	44
omr-ws-87efc7ffafdf4551798d27e80a6886fd854fbae1442c4c9d04674631a02791e0	Three circles with radii 1, 2, and 3 are tangent to each other externally. Another circle X is tangent to these three circles externally. What is the radius of circle X?	[ "\\dfrac{6}{23}" ]	[ "Three circles with radii 1, 2, and 3 are tangent to each other externally. Another circle X is tangent to these three circles externally. What is the radius of circle X?" ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 12, "genselect": 4 }	16
omr-ws-7648e7c09ec60a69556c03a6036587ce0daf36a457241f4fafc5df125c3a89e1	Three circles with radii 1, 2, and 3 touch each other externally. A larger circle encompasses these three circles such that each of the three smaller circles touches the larger circle internally. Find the radius of the larger circle.	[ "6" ]	[ "Three circles with radii 1, 2, and 3 touch each other externally. A larger circle encompasses these three circles such that each of the three smaller circles touches the larger circle internally. Find the radius of the larger circle." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 1 }	17
omr-ws-2d572ef0a83e6f2e00d4be523a4da16ec2c2e80bb3d174f73a90f4a17506cc9c	Three circles with radii 3, 4, and 5 touch each other externally. The tangents drawn at the points of tangency meet at point P. Find the distance from P to any of the three points of tangency.	[ "\$\\sqrt{5}\$" ]	[ "Three circles with radii 3, 4, and 5 touch each other externally. The tangents drawn at the points of tangency meet at point P. Find the distance from P to any of the three points of tangency." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 20 }	35
omr-ws-e5a036feb913b51f80c005e372ea0e77bc7b200d6847785886341b0269ef7ea9	Three circles with radii 40 cm, 50 cm, and 60 cm are tangent to each other without overlapping at any point. Determine the angles of the triangle formed when the centers of the circles are joined.	[ "50.5^\\circ" ]	[ "Three circles with radii 40 cm, 50 cm, and 60 cm are tangent to each other without overlapping at any point. Determine the angles of the triangle formed when the centers of the circles are joined." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 8 }	8
omr-ws-d0deb4ef9756fe47bfcb5e8118a7ed9769cb267d3a7af5918ea2ea1d8455d7f2	Three circles with radii $ p $, $ q $, and $ r $ and centers $ A $, $ B $, and $ C $ respectively, touch one another externally at points $ D $, $ E $, and $ F $. Calculate the ratio of the areas of $ \triangle DEF $ and $ \triangle ABC $ when $ p = 3 $, $ q = 4 $, and $ r = 5 $. Express you...	[ "\\dfrac{5}{21}" ]	[ "Three circles with radii \$ p \$, \$ q \$, and \$ r \$ and centers \$ A \$, \$ B \$, and \$ C \$ respectively, touch one another externally at points \$ D \$, \$ E \$, and \$ F \$. Calculate the ratio of the areas of \$ \\triangle DEF \$ and \$ \\triangle ABC \$ when \$ p = 3 \$, \\( q = 4 ...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-313ebb976a401a8a6d499626da3bac2a77f2975245fed93dbf9008d02e2487b4	Three circles with radii $ r < 1 $ unit each are centered at points $(5,1)$, $(3,7)$, and $(0,4)$. What is the center of the circle with the least possible radius that is tangent to all three circles?	[ "\\left( \\dfrac{13}{4}, \\dfrac{15}{4} \\right)" ]	[ "Three circles with radii \$ r < 1 \$ unit each are centered at points \$(5,1)\$, \$(3,7)\$, and \$(0,4)\$. What is the center of the circle with the least possible radius that is tangent to all three circles?" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-9ec4a86d823d42a67bee30a92b764c051e82448850b1c42e6cb271409e2bbced	Three circles with radii $ r $, $ r_1 $, and $ R $ are tangent to each other externally. The internal common tangent of the first two circles intersects the circle with radius $ R $ at points $ A $ and $ B $. Calculate the length of the chord $ \overline{AB} $.	[ "\$\\frac{4R \\sqrt{r r_1}}{r + r_1}\$" ]	[ "Three circles with radii \$ r \$, \$ r_1 \$, and \$ R \$ are tangent to each other externally. The internal common tangent of the first two circles intersects the circle with radius \$ R \$ at points \$ A \$ and \$ B \$. Calculate the length of the chord \$ \\overline{AB} \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "tir" ]	{ "cot": 42, "tir": 6 }	48
omr-ws-3bf564f12a9d842e6908e8679339da99b67bf42afedfb56b01d1bf6f71a2802a	Three circles with radii $ r_1, r_2, r_3 $ are tangent to the line $ y = 0 $ and tangent to each other. The left-most circle $ A $ has its center at $ (0, r_1) $. Find the coordinates of the centers of all three circles in terms of $ r_1 $ and $ r_2 $.	[ "\$ (0, r_1) \$ and \$ (2\\sqrt{r_1 \\cdot r_2}, r_2) \$" ]	[ "Three circles with radii \$ r_1, r_2, r_3 \$ are tangent to the line \$ y = 0 \$ and tangent to each other. The left-most circle \$ A \$ has its center at \$ (0, r_1) \$. Find the coordinates of the centers of all three circles in terms of \$ r_1 \$ and \$ r_2 \$." ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-6ce5e48b6e6ea036606037d183f3281360cfcbe5c69cbc8a0cddfd086a2ce425	Three circles with radii $a$, $b$, and $c$ touch each other externally. A point is chosen at random inside each circle. Find the probability that the triangle formed by joining these three points is acute.	[ "\\dfrac{1}{4}" ]	[ "Three circles with radii \$a\$, \$b\$, and \$c\$ touch each other externally. A point is chosen at random inside each circle. Find the probability that the triangle formed by joining these three points is acute." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-c33416ec87b347e42d36c72e6beab6cdd2675d7deaf58e8cffd7ebebaa79e0c4	Three circles with radius $2$ are mutually tangent. What is the total area of the circles and the region bounded by them?	[ "\$10\\pi + 4\\sqrt{3}\$" ]	[ "Three circles with radius $2$ are mutually tangent. What is the total area of the circles and the region bounded by them?" ]	[ "aops_c5_contests_amp_programs" ]	[ "cot", "genselect", "tir" ]	{ "cot": 5, "genselect": 4, "tir": 13 }	22
omr-ws-f776ab30a4effe916647fb8e221156767c04ba7b71d7e4cdfaf4f59084d3625e	Three circles with radius $5$ are drawn with centers at $A(0,0)$, $B(1,2)$, and a point $C$ such that triangle $ABC$ is equilateral. There is a circle $D$ that circumscribes the whole figure. The area of circle $D$ is $\pi(a+\sqrt{b})$, where $a$ and $b$ are reals with $b \geq 0$ and $a+b=\frac{m}{n}$. Given that $m$ a...	[ "583" ]	[ "Three circles with radius $5$ are drawn with centers at $A(0,0)$, $B(1,2)$, and a point $C$ such that triangle $ABC$ is equilateral. There is a circle $D$ that circumscribes the whole figure. The area of circle $D$ is $\\pi(a+\\sqrt{b})$, where $a$ and $b$ are reals with $b \\geq 0$ and $a+b=\\frac{m}{n}$. Given t...	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 8 }	24
omr-ws-fd663071ba9255bdb237d5924ae2a2de104572cf8869c06d0e467dae88583219	Three circles with radius 1 are all tangent to each other. The area made up by these three circles is denoted as $X$. What is the largest integer that does not exceed $100X$?	[ "942" ]	[ "Three circles with radius 1 are all tangent to each other. The area made up by these three circles is denoted as \$X\$. What is the largest integer that does not exceed \$100X\$?" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-83573839212451804fd7be046c5da9afce6014ef986f64d9429070e92c36682a	Three circles with radius $ r $ all pass through point $ O $. The circles have centers $ O_1 $, $ O_2 $, and $ O_3 $. The intersection of circles $ O_1 $ and $ O_2 $ is point $ A $, the intersection of circles $ O_1 $ and $ O_3 $ is point $ B $, and the intersection of circles $ O_2 $ and \( O_3...	[ "36" ]	[ "Three circles with radius \$ r \$ all pass through point \$ O \$. The circles have centers \$ O_1 \$, \$ O_2 \$, and \$ O_3 \$. The intersection of circles \$ O_1 \$ and \$ O_2 \$ is point \$ A \$, the intersection of circles \$ O_1 \$ and \$ O_3 \$ is point \$ B \$, and the intersection of c...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 4 }	9
omr-ws-ad1ee7d152baf69e5df26d3f0aa325aa6a7a24a0300cc4ca99354357d8530aad	Three circles with the same center $O$ and areas $2\pi, 3\pi$, and $4\pi$ are drawn. From a point $A$ on the largest circle, tangent lines are drawn to points $B$ on the middle circle and $C$ on the smallest circle. If $B, C$ are on the same side of $OA$, find the exact value of $\angle{BAC}$.	[ "\$15^{\\circ}\$" ]	[ "Three circles with the same center $O$ and areas $2\\pi, 3\\pi$, and $4\\pi$ are drawn. From a point $A$ on the largest circle, tangent lines are drawn to points $B$ on the middle circle and $C$ on the smallest circle. If $B, C$ are on the same side of $OA$, find the exact value of $\\angle{BAC}$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 29 }	45
omr-ws-daf14d7c49a017192ff31d9dcd927344ab7c86d11ee0640d7e683579d210c690	Three circles with the same radius $ r $ all pass through a common point $ H $. Let $ A $, $ B $, and $ C $ be the points where pairs of these circles intersect, distinct from $ H $. Calculate the radius of the circle passing through points $ A $, $ B $, and $ C $.	[ "r" ]	[ "Three circles with the same radius \$ r \$ all pass through a common point \$ H \$. Let \$ A \$, \$ B \$, and \$ C \$ be the points where pairs of these circles intersect, distinct from \$ H \$. Calculate the radius of the circle passing through points \$ A \$, \$ B \$, and \$ C \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 6 }	6

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