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-c54e6020b3a03ce47ce9db88d41e89cae6324c1de70ffd604d287463f0ea41c2	A point $ P $ is chosen in an arbitrary triangle. Three lines are drawn through $ P $ which are parallel to the sides of the triangle. The lines divide the triangle into three smaller triangles and three parallelograms. Let $ f $ be the ratio between the total area of the three smaller triangles and the area of t...	[ "\\dfrac{1}{3}" ]	[ "A point \$ P \$ is chosen in an arbitrary triangle. Three lines are drawn through \$ P \$ which are parallel to the sides of the triangle. The lines divide the triangle into three smaller triangles and three parallelograms. Let \$ f \$ be the ratio between the total area of the three smaller triangles and th...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 7, "genselect": 2 }	9
omr-ws-9093a56bb401569d41f888dc47ececca32d061b2563fd4ad502294159e1adbe2	A point $ P $ is chosen in the interior of $ \triangle ABC $ so that when lines are drawn through $ P $ parallel to the sides of $ \triangle ABC $, the resulting smaller triangles $ t_1 $, $ t_2 $, and $ t_3 $ have areas 4, 9, and 49, respectively. Find the area of $ \triangle ABC $.	[ "144" ]	[ "A point \$ P \$ is chosen in the interior of \$ \\triangle ABC \$ so that when lines are drawn through \$ P \$ parallel to the sides of \$ \\triangle ABC \$, the resulting smaller triangles \$ t_1 \$, \$ t_2 \$, and \$ t_3 \$ have areas 4, 9, and 49, respectively. Find the area of \\( \\triangle ABC ...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-7176761f73c83fa544cdaa13bc70300edb8fa4fbb6571db2e6bc8db4ef6b87c6	A point $ P $ is chosen in the interior of triangle $ ABC $ such that lines drawn through $ P $ parallel to the sides of triangle $ ABC $ divide $ ABC $ into three triangles and three quadrilaterals. The areas of the three smaller triangles are 4, 9, and 49. Find the area of triangle $ ABC $.	[ "121" ]	[ "A point \$ P \$ is chosen in the interior of triangle \$ ABC \$ such that lines drawn through \$ P \$ parallel to the sides of triangle \$ ABC \$ divide \$ ABC \$ into three triangles and three quadrilaterals. The areas of the three smaller triangles are 4, 9, and 49. Find the area of triangle \\( ABC \\...	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-f23a7d2e83746145bdca3e4fa5a62b4d77db9c9309892387ec55622a4e476786	A point $ P $ is chosen inside the convex quadrilateral $ ABCD $ such that $\angle PAD = \angle ADP = \angle CPD = \angle PCB = \angle PBC$. If $ O $ is the circumcenter of $\triangle CPD$, find the ratio $\frac{OA}{OB}$.	[ "1" ]	[ "A point \$ P \$ is chosen inside the convex quadrilateral \$ ABCD \$ such that \$\\angle PAD = \\angle ADP = \\angle CPD = \\angle PCB = \\angle PBC\$. If \$ O \$ is the circumcenter of \$\\triangle CPD\$, find the ratio \$\\frac{OA}{OB}\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 7, "genselect": 3 }	10
omr-ws-2a1a2a85e468bef9c0317502778487a7f98b87acafb2ff38faa903fae6c0a29b	A point $ P $ is chosen inside the tetrahedron $ ABCD $ with volume $ V $. Denote by $ S_A $ the area of face $ BCD $, and similarly define $ S_B $, $ S_C $, and $ S_D $. If $ AP = 3 $, $ BP = 4 $, $ CP = 5 $, and $ DP = 6 $, and the volume $ V $ is 10, calculate the value of \( S_A + S_B + S_...	[ "60" ]	[ "A point \$ P \$ is chosen inside the tetrahedron \$ ABCD \$ with volume \$ V \$. Denote by \$ S_A \$ the area of face \$ BCD \$, and similarly define \$ S_B \$, \$ S_C \$, and \$ S_D \$. If \$ AP = 3 \$, \$ BP = 4 \$, \$ CP = 5 \$, and \$ DP = 6 \$, and the volume \$ V \$ is 10, calculate...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 3 }	3
omr-ws-9272a6a3384da6d7e72455eda905afae5cd1d322443b770edd39f8b1d8f6c8c5	A point $ P $ is chosen inside triangle $ ABC $ such that $ BC = AP $ and $ \angle APC = 180^\circ - \angle ABC $. A point $ K $ is chosen on side $ AB $ such that $ AK = KB + PC $. If $ AB = 12 $ and $ BC = 10 $, find the length of $ CK $.	[ "8" ]	[ "A point \$ P \$ is chosen inside triangle \$ ABC \$ such that \$ BC = AP \$ and \$ \\angle APC = 180^\\circ - \\angle ABC \$. A point \$ K \$ is chosen on side \$ AB \$ such that \$ AK = KB + PC \$. If \$ AB = 12 \$ and \$ BC = 10 \$, find the length of \$ CK \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-8f8d35cc086cd49827ab58d9e495e69e9212a5ed3da002c3f11d5dd3931d5161	A point $ P $ is given in the square $ ABCD $ such that $ \overline{PA} = 3 $, $ \overline{PB} = 7 $, and $ \overline{PD} = 5 $. Find the area of the square.	[ "58" ]	[ "A point \$ P \$ is given in the square \$ ABCD \$ such that \$ \\overline{PA} = 3 \$, \$ \\overline{PB} = 7 \$, and \$ \\overline{PD} = 5 \$. Find the area of the square." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-bf4bfbf7ab579c94359964c42af79b07f8748155a93bac070585d80ae77834d4	A point $ P $ is in the interior of an equilateral triangle $ \triangle ABC $, where $ PA = 6 $, $ PB = 8 $, and $ PC = 10 $. What is the side length $ s $ of this triangle?	[ "\$ 2\\sqrt{25 + 12\\sqrt{3}} \$" ]	[ "A point \$ P \$ is in the interior of an equilateral triangle \$ \\triangle ABC \$, where \$ PA = 6 \$, \$ PB = 8 \$, and \$ PC = 10 \$. What is the side length \$ s \$ of this triangle?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 28 }	43
omr-ws-92e783c498df0856e0fc87d8d82da8e1ff8e2a13f11eade3220b32371d307c15	A point $ P $ is in the interior of the cyclic quadrilateral $ ABCD $ and has the property $ \angle BPC = \angle BAP + \angle PDC $. The feet of the perpendiculars from $ P $ to $ AB $, $ AD $, and $ DC $ are respectively denoted $ E $, $ F $, and $ G $. If $ \triangle FEG $ and \( \triangle PBC \...	[ "\\dfrac{12}{13}" ]	[ "A point \$ P \$ is in the interior of the cyclic quadrilateral \$ ABCD \$ and has the property \$ \\angle BPC = \\angle BAP + \\angle PDC \$. The feet of the perpendiculars from \$ P \$ to \$ AB \$, \$ AD \$, and \$ DC \$ are respectively denoted \$ E \$, \$ F \$, and \$ G \$. If \\( \\triangle...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-03959c6556d93fd7e5849d84432baeade0490ec955e98d908e40902c9700264f	A point $ P $ is in the interior of triangle $ ABC $ such that $ \angle PAB = 10^\circ $, $ \angle PBA = 20^\circ $, $ \angle PAC = 40^\circ $, and $ \angle PCA = 30^\circ $. Calculate the sum of the measures of $ \angle PBC $ and $ \angle PCB $.	[ "80" ]	[ "A point \$ P \$ is in the interior of triangle \$ ABC \$ such that \$ \\angle PAB = 10^\\circ \$, \$ \\angle PBA = 20^\\circ \$, \$ \\angle PAC = 40^\\circ \$, and \$ \\angle PCA = 30^\\circ \$. Calculate the sum of the measures of \$ \\angle PBC \$ and \$ \\angle PCB \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-b1c1ecb407aad36d207773b9926689c3f3772777fc9c78ed6e0db5da0f052cdd	A point $ P $ is inside $ \triangle ABC $ and the bisectors of $ \angle BPC $, $ \angle CPA $, and $ \angle APB $ meet $ BC $, $ CA $, and $ AB $ at $ D $, $ E $, and $ F $ respectively. If $ AP = 10 $, $ BP = 12 $, $ CP = 14 $, and $ PD + PE + PF = 15 $, calculate the minimum possible val...	[ "6" ]	[ "A point \$ P \$ is inside \$ \\triangle ABC \$ and the bisectors of \$ \\angle BPC \$, \$ \\angle CPA \$, and \$ \\angle APB \$ meet \$ BC \$, \$ CA \$, and \$ AB \$ at \$ D \$, \$ E \$, and \$ F \$ respectively. If \$ AP = 10 \$, \$ BP = 12 \$, \$ CP = 14 \$, and \\( PD + PE + PF = 15 ...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 6 }	6
omr-ws-174927f4a44f3986cf6a7f1471798e4f9fe7e74672d0269181564a7319621f04	A point $ P $ is inside the cyclic quadrilateral $ ABCD $ such that $ \angle BPC = \angle BAP + \angle PDC $. Let $ E $, $ F $, and $ G $ be the projections of $ P $ onto the lines $ AB $, $ AD $, and $ DC $ respectively. If the ratio of the areas of triangles $ FEG $ and $ PBC $ is $ k $, fin...	[ "1" ]	[ "A point \$ P \$ is inside the cyclic quadrilateral \$ ABCD \$ such that \$ \\angle BPC = \\angle BAP + \\angle PDC \$. Let \$ E \$, \$ F \$, and \$ G \$ be the projections of \$ P \$ onto the lines \$ AB \$, \$ AD \$, and \$ DC \$ respectively. If the ratio of the areas of triangles \$ FEG \$...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-710575565a8fc11d87080c7ab39c10270d55495cbc8145c1d4f766c43e839c84	A point $ P $ is inside triangle $ ABC $ such that $ \angle ABP = \angle PCA $. Point $ Q $ is such that $ BPQC $ is a parallelogram. If $ \angle BAC = 60^\circ $, calculate the measure of $ \angle QAB $ in degrees.	[ "90" ]	[ "A point \$ P \$ is inside triangle \$ ABC \$ such that \$ \\angle ABP = \\angle PCA \$. Point \$ Q \$ is such that \$ BPQC \$ is a parallelogram. If \$ \\angle BAC = 60^\\circ \$, calculate the measure of \$ \\angle QAB \$ in degrees." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 2 }	2
omr-ws-0fe8a536a2b42ef2a12d2fb5c18e502acc3c169c513dd6fddea488ebecd405d3	A point $ P $ is interior to the triangle $ ABC $ such that $ AP \perp BC $. Let $ E $ and $ F $ be the projections of $ C $ and $ B $ onto $ AP $, respectively. Suppose that the tangents at $ E $ and $ F $ to the circumcircle of triangle $ AEF $ meet at a point $ Q $ on $ BC $. If \( BP = 3 \...	[ "7" ]	[ "A point \$ P \$ is interior to the triangle \$ ABC \$ such that \$ AP \\perp BC \$. Let \$ E \$ and \$ F \$ be the projections of \$ C \$ and \$ B \$ onto \$ AP \$, respectively. Suppose that the tangents at \$ E \$ and \$ F \$ to the circumcircle of triangle \$ AEF \$ meet at a point \\( Q \...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-1dc86113842023c8cb01752205eed67ef44dadba862a3d6ad204d3f06f49871d	A point $ P $ is located in the plane of a convex quadrilateral $ ABCD $. Let $ A_0, B_0, C_0, $ and $ D_0 $ be the midpoints of $ AB, BC, CD, $ and $ DA $ respectively. A point $ A_1 $ is chosen on side $ AB $ such that rays $ PA_0 $ and $ PA_1 $ are symmetric with respect to the angle bisector of ...	[ "\\text{The midpoint of the segment connecting the midpoints of the diagonals } AC \\text{ and } BD" ]	[ "A point \$ P \$ is located in the plane of a convex quadrilateral \$ ABCD \$. Let \$ A_0, B_0, C_0, \$ and \$ D_0 \$ be the midpoints of \$ AB, BC, CD, \$ and \$ DA \$ respectively. A point \$ A_1 \$ is chosen on side \$ AB \$ such that rays \$ PA_0 \$ and \$ PA_1 \$ are symmetric with respect ...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 6, "genselect": 5 }	11
omr-ws-b2e278b5e47439a556e3720928d7923b140fdb20728ba70ea68ffcb9e46b5c38	A point $ P $ is located on the line $ y = 2x $. It is 24 length units from the origin. What are the coordinates of point $ P $?	[ "\$\\left( \\frac{24 \\sqrt{5}}{5}, \\frac{48 \\sqrt{5}}{5} \\right)\$ or \$\\left( -\\frac{24 \\sqrt{5}}{5}, -\\frac{48 \\sqrt{5}}{5} \\right)\$" ]	[ "A point \$ P \$ is located on the line \$ y = 2x \$. It is 24 length units from the origin. What are the coordinates of point \$ P \$?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 31 }	47
omr-ws-9fffdfb178d19ae14d7b48facce8e223996e5735cd0217d5c22b0f9d47e53358	A point $ P $ is marked in a square $ A_1A_2A_3A_4 $ and joined to its vertices. Let $ B_i $ be the foot of the perpendicular from $ A_{i-1} $ onto the line $ PA_i $ for $ i = 1, 2, 3, 4 $ (with $ A_0 $ interpreted as $ A_4 $). If $ P $ is at the center of the square and the side length of the square ...	[ "0" ]	[ "A point \$ P \$ is marked in a square \$ A_1A_2A_3A_4 \$ and joined to its vertices. Let \$ B_i \$ be the foot of the perpendicular from \$ A_{i-1} \$ onto the line \$ PA_i \$ for \$ i = 1, 2, 3, 4 \$ (with \$ A_0 \$ interpreted as \$ A_4 \$). If \$ P \$ is at the center of the square and the sid...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-464d7c5af9b01dc2c4048e696a7be0627b4f2d4473da912edc6a5a4742aef6a7	A point $ P $ is moving along a quarter circle with center $ O $, bounded by points $ A $ and $ B $. Let $ PQ $ be the perpendicular from $ P $ to the radius $ OA $. Choose a point $ M $ on the ray $ OP $ such that the length of $ OM $ equals the sum of the lengths of $ OQ $ and $ QP $. Find the...	[ "The locus of the points \$ M \$ is the semicircle with diameter \$ \\overline{AB} \$ that is outside \$ \\triangle OAB \$." ]	[ "A point \$ P \$ is moving along a quarter circle with center \$ O \$, bounded by points \$ A \$ and \$ B \$. Let \$ PQ \$ be the perpendicular from \$ P \$ to the radius \$ OA \$. Choose a point \$ M \$ on the ray \$ OP \$ such that the length of \$ OM \$ equals the sum of the lengths of \\( OQ...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 11, "genselect": 3, "tir": 6 }	20
omr-ws-b88a866018384d31a0a566b3fb6d792f6c7a9740181ebef695592e033d27e96f	A point $ P $ is performing a random walk on the $ X $-axis. At $ t = 0 $, $ P $ is at $ x_0 $ where $ \|x_0\| \leq N $ and $ x_0 $ and $ N $ are integers with $ N > 0 $. If at time $ t $, $ P $ is at $ x $ with $ \|x\| < N $, then at time $ t+1 $, $ P $ moves to $ x+1 $ or $ x-1 $ each wi...	[ "\$\\frac{1}{2N + 1}\$" ]	[ "A point \$ P \$ is performing a random walk on the \$ X \$-axis. At \$ t = 0 \$, \$ P \$ is at \$ x_0 \$ where \$ \|x_0\| \\leq N \$ and \$ x_0 \$ and \$ N \$ are integers with \$ N > 0 \$. If at time \$ t \$, \$ P \$ is at \$ x \$ with \$ \|x\| < N \$, then at time \$ t+1 \$, \$ P \$ mov...	[ "aops_c7_college_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-4eae55a5452c5cdee614ff8c07a32c885f13d4acc9034b01b0a6c8e5e9d6e2e8	A point $ P $ is randomly chosen in a unit circle such that the probability of a point being chosen as $ P $ is proportional to the distance from the center of the circle to the point. A square $ ABCD $ is inscribed in the circle. What is the probability that $ P $ lies within the square $ ABCD $?	[ "\$\\frac{2}{\\pi}\$" ]	[ "A point \$ P \$ is randomly chosen in a unit circle such that the probability of a point being chosen as \$ P \$ is proportional to the distance from the center of the circle to the point. A square \$ ABCD \$ is inscribed in the circle. What is the probability that \$ P \$ lies within the square \\( ABCD \...	[ "aops_c4_high_school_math" ]	[ "tir" ]	{ "tir": 1 }	1
omr-ws-73588eb215b14924a3ad97406dabc8c7831b69581d2836f3ae1b36c965d884e6	A point $ P $ is randomly selected from the rectangular region with vertices $ (0, 0) $, $ (2, 0) $, $ (2, 1) $, $ (0, 1) $. What is the probability that $ P $ is closer to the origin than it is to the point $ (3, 1) $?	[ "\$\\frac{3}{4}\$" ]	[ "A point \$ P \$ is randomly selected from the rectangular region with vertices \$ (0, 0) \$, \$ (2, 0) \$, \$ (2, 1) \$, \$ (0, 1) \$. What is the probability that \$ P \$ is closer to the origin than it is to the point \$ (3, 1) \$?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 30 }	46
omr-ws-3de39f09fb0064d9625cbd1fc126bfd2d3cd114960d13be03945987e55c1406f	A point $ P $ is selected at random from the interior of the pentagon with vertices $ A = (0,2) $, $ B = (4,0) $, $ C = (2\pi + 1, 0) $, $ D = (2\pi + 1, 4) $, and $ E = (0, 4) $. What is the probability that $ \angle APB $ is obtuse?	[ "\$\\frac{5}{16}\$" ]	[ "A point \$ P \$ is selected at random from the interior of the pentagon with vertices \$ A = (0,2) \$, \$ B = (4,0) \$, \$ C = (2\\pi + 1, 0) \$, \$ D = (2\\pi + 1, 4) \$, and \$ E = (0, 4) \$. What is the probability that \$ \\angle APB \$ is obtuse?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 9, "genselect": 5, "tir": 17 }	31
omr-ws-f9acf6faddeec420379c50f0e252215abfba099c6b8a7c18a6e78153148d35e1	A point $ P $ is selected at random inside a square $ ABCD $ with side length 1. What is the probability that $ AP + BP < DP $?	[ "0.114" ]	[ "A point \$ P \$ is selected at random inside a square \$ ABCD \$ with side length 1. What is the probability that \$ AP + BP < DP \$?" ]	[ "aops_c4_high_school_math" ]	[ "tir" ]	{ "tir": 11 }	11
omr-ws-9f954be874f408d626c315de6e8b65d5c12f8176546b822e72bf3688f7080fc7	A point $ P $ is selected randomly within the circle $ S = \{(x, y) : x^2 + y^2 \leq R^2\} $. What is the probability that the interval of length $ R $ centered at $ P $ and parallel to the x-axis is entirely contained within $ S $?	[ "\\dfrac{2}{3} - \\dfrac{\\sqrt{3}}{2\\pi}" ]	[ "A point \$ P \$ is selected randomly within the circle \$ S = \\{(x, y) : x^2 + y^2 \\leq R^2\\} \$. What is the probability that the interval of length \$ R \$ centered at \$ P \$ and parallel to the x-axis is entirely contained within \$ S \$?" ]	[ "aops_c7_college_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-1f7150bcbab5b6ea75eb6cb5e08a6ad27318736db88d20567fcb43f7c6b4be99	A point $ P $ is taken in the interior of a right triangle $ ABC $ with $ \angle C = 90^\circ $ such that $ AP = 4 $, $ BP = 2 $, and $ CP = 1 $. Point $ Q $ is symmetric to $ P $ with respect to $ AC $ and lies on the circumcircle of triangle $ ABC $. Find $ \angle ABC $.	[ "60" ]	[ "A point \$ P \$ is taken in the interior of a right triangle \$ ABC \$ with \$ \\angle C = 90^\\circ \$ such that \$ AP = 4 \$, \$ BP = 2 \$, and \$ CP = 1 \$. Point \$ Q \$ is symmetric to \$ P \$ with respect to \$ AC \$ and lies on the circumcircle of triangle \$ ABC \$. Find \\( \\angle ABC...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect", "tir" ]	{ "cot": 46, "genselect": 8, "tir": 19 }	73
omr-ws-4143937bec95939adf69306b701716540af0891ff0c287ccdf8635967afdaa87	A point $ P $ is taken in the interior of a right triangle $ ABC $ with $ \angle C = 90^\circ $ such that $ AP = 4 $, $ BP = 2 $, and $ CP = 1 $. Point $ Q $, which is symmetric to $ P $ with respect to $ AC $, lies on the circumcircle of triangle $ ABC $. Find the angles of triangle $ ABC $.	[ "90^\\circ" ]	[ "A point \$ P \$ is taken in the interior of a right triangle \$ ABC \$ with \$ \\angle C = 90^\\circ \$ such that \$ AP = 4 \$, \$ BP = 2 \$, and \$ CP = 1 \$. Point \$ Q \$, which is symmetric to \$ P \$ with respect to \$ AC \$, lies on the circumcircle of triangle \$ ABC \$. Find the angles ...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 9 }	9
omr-ws-105d1d869ac0659549e326801c38e421f1f0eb25dbe8f94c1732c43305550a58	A point $ P $ is taken inside an acute-angled triangle $ ABC $. The lines $ AP $, $ BP $, $ CP $ intersect the circumcircle of the triangle for the second time at points $ A_1 $, $ B_1 $, $ C_1 $, respectively. Points $ A_2 $, $ B_2 $, $ C_2 $ are the symmetric points of $ P $ with respect to th...	[ "2" ]	[ "A point \$ P \$ is taken inside an acute-angled triangle \$ ABC \$. The lines \$ AP \$, \$ BP \$, \$ CP \$ intersect the circumcircle of the triangle for the second time at points \$ A_1 \$, \$ B_1 \$, \$ C_1 \$, respectively. Points \$ A_2 \$, \$ B_2 \$, \$ C_2 \$ are the symmetric points of...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-c1414d99fc5a24f9b9534401733d2dea60c0e9214b8ca3190d2b1ed3ab671ed9	A point $ P $ lies in $ \triangle ABC $. The lines $ BP $ and $ CP $ meet $ AC $ and $ AB $ at $ Q $ and $ R $ respectively. Given that $ AR = RB = CP $ and $ CQ = PQ $, find $ \angle BRC $.	[ "120^\\circ" ]	[ "A point \$ P \$ lies in \$ \\triangle ABC \$. The lines \$ BP \$ and \$ CP \$ meet \$ AC \$ and \$ AB \$ at \$ Q \$ and \$ R \$ respectively. Given that \$ AR = RB = CP \$ and \$ CQ = PQ \$, find \$ \\angle BRC \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-54670a8b5507849cf0903facf07436fe8e34f3106d0a2d134d5fe49a806ebc9c	A point $ P $ lies in the interior of triangle $ ABC $. The lines $ AP, BP $, and $ CP $ intersect $ BC, CA $, and $ AB $ at points $ D, E $, and $ F $, respectively. Given that $ ABDE $ and $ BCEF $ are concyclic, calculate the value of $ \frac{[ABDE] + [BCEF]}{[ABC]} $.	[ "\\dfrac{1}{2}" ]	[ "A point \$ P \$ lies in the interior of triangle \$ ABC \$. The lines \$ AP, BP \$, and \$ CP \$ intersect \$ BC, CA \$, and \$ AB \$ at points \$ D, E \$, and \$ F \$, respectively. Given that \$ ABDE \$ and \$ BCEF \$ are concyclic, calculate the value of \\( \\frac{[ABDE] + [BCEF]}{[ABC]} \\...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 7, "genselect": 6 }	13
omr-ws-fec648c4de769198ecc7008950238de2799ccbdc9934c0360a2d8a7399867d05	A point $ P $ lies in the same plane as a given square of side 1. Let the vertices of the square, taken counterclockwise, be $ A, B, C, $ and $ D $. Also, let the distances from $ P $ to $ A, B, $ and $ C $, respectively, be $ u, v, $ and $ w $. What is the greatest distance that $ P $ can be from \( ...	[ "2 + \\sqrt{2}" ]	[ "A point \$ P \$ lies in the same plane as a given square of side 1. Let the vertices of the square, taken counterclockwise, be \$ A, B, C, \$ and \$ D \$. Also, let the distances from \$ P \$ to \$ A, B, \$ and \$ C \$, respectively, be \$ u, v, \$ and \$ w \$. What is the greatest distance that \\...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-0986a2e4597e5a779d06e3cca2947be92a38939137fc34aff6e7a6974564eecd	A point $ P $ lies in the same plane as a given square of side $ 1 $. Let the vertices of the square, taken counterclockwise, be $ A $, $ B $, $ C $, and $ D $. Also, let the distances from $ P $ to $ A $, $ B $, and $ C $ be $ u $, $ v $, and $ w $, respectively. What is the greatest distance...	[ "\$2 + \\sqrt{2}\$" ]	[ "A point \$ P \$ lies in the same plane as a given square of side \$ 1 \$. Let the vertices of the square, taken counterclockwise, be \$ A \$, \$ B \$, \$ C \$, and \$ D \$. Also, let the distances from \$ P \$ to \$ A \$, \$ B \$, and \$ C \$ be \$ u \$, \$ v \$, and \$ w \$, respectively...	[ "aops_c5_contests_amp_programs" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 25 }	41
omr-ws-31cf2e0226cb1999ea71b67a6466b574ac88dc884235a047d2973b2ccc2396cb	A point $ P $ lies in the same plane as a given square of side $ 1 $. Let the vertices of the square, taken counterclockwise, be $ A, B, C, D $. Let the distances from $ P $ to $ A, B, C $ be $ u, v, w $ respectively. If $ u^2 + v^2 = w^2 $, what is the greatest possible distance from $ P $ to $ D $?	[ "\$ 2 + \\sqrt{2} \$" ]	[ "A point \$ P \$ lies in the same plane as a given square of side \$ 1 \$. Let the vertices of the square, taken counterclockwise, be \$ A, B, C, D \$. Let the distances from \$ P \$ to \$ A, B, C \$ be \$ u, v, w \$ respectively. If \$ u^2 + v^2 = w^2 \$, what is the greatest possible distance from \...	[ "aops_c5_contests_amp_programs" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 25 }	41
omr-ws-8580a9f659bdd9cca5b249db25ebd1c7c111bbceec9cb115dbef27d955a37d98	A point $ P $ lies in the same plane as a given square with side length 1. Let the vertices of the square, taken counterclockwise, be $ A $, $ B $, $ C $, and $ D $. The distances from $ P $ to $ A $, $ B $, and $ C $ are $ u $, $ v $, and $ w $ respectively, and it is given that \( u^2 + v^2 = ...	[ "\$2 + \\sqrt{2}\$" ]	[ "A point \$ P \$ lies in the same plane as a given square with side length 1. Let the vertices of the square, taken counterclockwise, be \$ A \$, \$ B \$, \$ C \$, and \$ D \$. The distances from \$ P \$ to \$ A \$, \$ B \$, and \$ C \$ are \$ u \$, \$ v \$, and \$ w \$ respectively, and it ...	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 28 }	43
omr-ws-e368c3cd64fe35e069956025c0f2d102b26cc086fc0aba282f50bce78f0cfe49	A point $ P $ lies inside $ \angle ABC (<90^\circ) $. Find the coordinates of a point $ Q $ inside $ \angle ABC $ such that for any two points $ X $ and $ Y $ on the rays $ \overrightarrow{BA} $ and $ \overrightarrow{BC} $ respectively satisfying $ \angle XPY = \angle ABC $, it holds that \( \angle XQ...	[ "no_valid_answer_found" ]	[ "A point \$ P \$ lies inside \$ \\angle ABC (<90^\\circ) \$. Find the coordinates of a point \$ Q \$ inside \$ \\angle ABC \$ such that for any two points \$ X \$ and \$ Y \$ on the rays \$ \\overrightarrow{BA} \$ and \$ \\overrightarrow{BC} \$ respectively satisfying \\( \\angle XPY = \\angle ABC \...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-189afe281cbc1f603c34ee4eec80b8c1a258a6693c78b3fdf8ee40143a3aeda7	A point $ P $ lies inside $ \triangle ABC $ such that the areas of $ \triangle PAB $, $ \triangle PBC $, and $ \triangle PCA $ can form a triangle. Let $ BC = a $, $ CA = b $, $ AB = c $, $ PA = x $, $ PB = y $, and $ PC = z $. Calculate the maximum possible value of the expression \[ \frac{(x + y...	[ "6 - 2\\sqrt{3}" ]	[ "A point \$ P \$ lies inside \$ \\triangle ABC \$ such that the areas of \$ \\triangle PAB \$, \$ \\triangle PBC \$, and \$ \\triangle PCA \$ can form a triangle. Let \$ BC = a \$, \$ CA = b \$, \$ AB = c \$, \$ PA = x \$, \$ PB = y \$, and \$ PC = z \$. Calculate the maximum possible value of...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-db02e19aae447ebe2f10ca50d226cedafd74c5cc02eb2095f360e0e0606a4c98	A point $ P $ lies inside a quadrilateral and is connected to the midpoints of all the sides of the quadrilateral, dividing the quadrilateral into 4 regions. The areas of three of these regions are known. Determine the area of the unknown region (denoted by the question mark).	[ "88" ]	[ "A point \$ P \$ lies inside a quadrilateral and is connected to the midpoints of all the sides of the quadrilateral, dividing the quadrilateral into 4 regions. The areas of three of these regions are known. Determine the area of the unknown region (denoted by the question mark)." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-39d0aa66038e76e66e0a91bd5d5ccdb889aef4e0805c0fce782653598a3dec40	A point $ P $ lies inside an equilateral triangle $ ABC $ such that $ AP = 3 $, $ BP = 4 $, and $ CP = 5 $. Find the area of the equilateral triangle $ ABC $.	[ "\$ 19.625 \\sqrt{3} \$" ]	[ "A point \$ P \$ lies inside an equilateral triangle \$ ABC \$ such that \$ AP = 3 \$, \$ BP = 4 \$, and \$ CP = 5 \$. Find the area of the equilateral triangle \$ ABC \$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 17 }	33
omr-ws-beb2d32a7d938d7cfbedd4bea90ca7714b0fa3c08febf6c0f8b5f4b88fe502d8	A point $ P $ moves in the $ xy $-plane such that $ \max\{PA + PB, PB + PC\} \leq 2 $, where $ A(0,0) $, $ B(1,0) $, and $ C(2,0) $. Determine the area bounded by the possible positions of point $ P $.	[ "\$ 1 \$" ]	[ "A point \$ P \$ moves in the \$ xy \$-plane such that \$ \\max\\{PA + PB, PB + PC\\} \\leq 2 \$, where \$ A(0,0) \$, \$ B(1,0) \$, and \$ C(2,0) \$. Determine the area bounded by the possible positions of point \$ P \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-4f041039bbe6b610efbda1dfb946528b99b47c4dc269931083eea4deacf72baf	A point $ P $ moves inside a unit square in a straight line at unit speed. When it meets a corner, it escapes. When it meets an edge, its line of motion is reflected so that the angle of incidence equals the angle of reflection. Let $ N(t) $ be the number of starting directions from a fixed interior point $ P_0 $...	[ "\$\\pi\$" ]	[ "A point \$ P \$ moves inside a unit square in a straight line at unit speed. When it meets a corner, it escapes. When it meets an edge, its line of motion is reflected so that the angle of incidence equals the angle of reflection. Let \$ N(t) \$ be the number of starting directions from a fixed interior point ...	[ "aops_c7_college_math" ]	[ "cot", "tir" ]	{ "cot": 6, "tir": 9 }	15
omr-ws-93f2df165b89fc94ee63c2b8c23988ea367c93adbc068f5a4770445cc6f925c9	A point $ P $ moves such that the sum of the slopes of the normals drawn from it to the hyperbola $ xy = 16 $ is equal to the sum of the ordinates of the feet of the normals. Find the curve $ C $ on which $ P $ lies.	[ "\$ y = \\frac{x^2}{16} \$" ]	[ "A point \$ P \$ moves such that the sum of the slopes of the normals drawn from it to the hyperbola \$ xy = 16 \$ is equal to the sum of the ordinates of the feet of the normals. Find the curve \$ C \$ on which \$ P \$ lies." ]	[ "aops_c7_college_math", "c7_college_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 14, "genselect": 6, "tir": 22 }	42
omr-ws-8b97faab0f75a094b96f626f568fef929766775360465b49988a2ee92095f3e7	A point $ P(\sqrt{2} \sin A, \cot B) $ lies on the curve $ (x+y)^2 + 4(x-y)^2 = 8 $. If the minimum value of $ (2 \tan C - \sqrt{2} \sin A)^2 + (2 \cot C - \cot B)^2 = a + b \sqrt{2} $ where $ C \in \mathbb{R} - \left\{ \frac{n\pi}{2} \right\} $ and $ n, a, b \in \mathbb{Z} $, then find the value of \( a + b ...	[ "4" ]	[ "A point \$ P(\\sqrt{2} \\sin A, \\cot B) \$ lies on the curve \$ (x+y)^2 + 4(x-y)^2 = 8 \$. If the minimum value of \$ (2 \\tan C - \\sqrt{2} \\sin A)^2 + (2 \\cot C - \\cot B)^2 = a + b \\sqrt{2} \$ where \$ C \\in \\mathbb{R} - \\left\\{ \\frac{n\\pi}{2} \\right\\} \$ and \$ n, a, b \\in \\mathbb{Z} \$...	[ "aops_c7_college_math" ]	[ "cot", "genselect" ]	{ "cot": 13, "genselect": 7 }	20
omr-ws-5d2b26439ed9489d7b413306603f78939f69bfaf7b3a86965e7a4afe99d58d63	A point $ P(a, a^2) $ lies inside the triangle formed by the lines $ 2x + 3y = 1 $, $ x + 2y = 3 $, and $ 6y = 5x - 1 $. Determine the possible values of $ a $.	[ "\\left( -\\dfrac{3}{2}, -1 \\right) \\cup \\left( \\dfrac{1}{2}, 1 \\right)" ]	[ "A point \$ P(a, a^2) \$ lies inside the triangle formed by the lines \$ 2x + 3y = 1 \$, \$ x + 2y = 3 \$, and \$ 6y = 5x - 1 \$. Determine the possible values of \$ a \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 15 }	15
omr-ws-4bd608e19a764e3d0573635aff561fab0ab24da8ed4c9a1c9c2520cc4ecd4b7f	A point $ P(x, y) $ is randomly selected from the region $ \|x\| + \|y\| < 1 $. What is the probability that $ 2x + 3y < 1 $?	[ "\$\\frac{7}{10}\$" ]	[ "A point \$ P(x, y) \$ is randomly selected from the region \$ \|x\| + \|y\| < 1 \$. What is the probability that \$ 2x + 3y < 1 \$?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 13, "genselect": 5, "tir": 17 }	35
omr-ws-c431d49e6340d8716cd8e35a8e6afb29fe4d6d5dc26e4424b48429f7e346cd72	A point $ T $ is chosen inside a triangle $ ABC $. Let $ A_1 $, $ B_1 $, and $ C_1 $ be the reflections of $ T $ in $ BC $, $ CA $, and $ AB $, respectively. Let $ \Omega $ be the circumcircle of the triangle $ A_1B_1C_1 $. The lines $ A_1T $, $ B_1T $, and $ C_1T $ meet $ \Omega $ again a...	[ "\\sqrt{13}" ]	[ "A point \$ T \$ is chosen inside a triangle \$ ABC \$. Let \$ A_1 \$, \$ B_1 \$, and \$ C_1 \$ be the reflections of \$ T \$ in \$ BC \$, \$ CA \$, and \$ AB \$, respectively. Let \$ \\Omega \$ be the circumcircle of the triangle \$ A_1B_1C_1 \$. The lines \$ A_1T \$, \$ B_1T \$, and \\( ...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 5, "genselect": 3 }	8
omr-ws-e4c26b7181e7baa56349a18fab965bc720c6ab8a8d184017c6bbd4d88f47e2a1	A point $ T $ is chosen randomly inside an equilateral triangle with side length $ a $. If $ X $ is the distance from $ T $ to the closest side, find the expected value of $ X $.	[ "\$\\frac{\\sqrt{3}a}{18}\$" ]	[ "A point \$ T \$ is chosen randomly inside an equilateral triangle with side length \$ a \$. If \$ X \$ is the distance from \$ T \$ to the closest side, find the expected value of \$ X \$." ]	[ "aops_c7_college_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 17 }	33
omr-ws-9df25512cdaf4a32761283920c5094de3119e75c0e183d19a689ab13cef639d1	A point $ X $ in the coordinate plane is chosen randomly such that it is exactly $ \sqrt{2} - \frac{\sqrt{6}}{3} $ away from the origin. A point $ Y $ is chosen randomly such that it is less than 4 away from the origin. The probability that a point $ Z $ less than 2 away from the origin forms an equilateral tri...	[ "34" ]	[ "A point \$ X \$ in the coordinate plane is chosen randomly such that it is exactly \$ \\sqrt{2} - \\frac{\\sqrt{6}}{3} \$ away from the origin. A point \$ Y \$ is chosen randomly such that it is less than 4 away from the origin. The probability that a point \$ Z \$ less than 2 away from the origin forms an...	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 13, "tir": 12 }	25
omr-ws-504000fb6a94f012719eb23e5569551ddaae1a92d027cc6fa9d222cbeae47925	A point $ X $ lies inside an isosceles right triangle $ ABC $ with $ \angle B = 90^\circ $ and $ AB = BC = 1 $. Calculate the value of $ AX + BX + \sqrt{2} CX $ when $ X $ is at the point where the equality $ AX + BX + \sqrt{2} CX = \sqrt{5} $ holds.	[ "\\sqrt{5}" ]	[ "A point \$ X \$ lies inside an isosceles right triangle \$ ABC \$ with \$ \\angle B = 90^\\circ \$ and \$ AB = BC = 1 \$. Calculate the value of \$ AX + BX + \\sqrt{2} CX \$ when \$ X \$ is at the point where the equality \$ AX + BX + \\sqrt{2} CX = \\sqrt{5} \$ holds." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-40149553b3c863a5311737c0d761b8639c9bc04a3dda2b66eac16d4036013f80	A point $ X $ lies inside an isosceles right triangle $ ABC $ with the right angle at $ B $. Let $ AB = BC = 1 $. Find the maximum value of the expression $ AX + BX + \sqrt{2}CX $.	[ "3" ]	[ "A point \$ X \$ lies inside an isosceles right triangle \$ ABC \$ with the right angle at \$ B \$. Let \$ AB = BC = 1 \$. Find the maximum value of the expression \$ AX + BX + \\sqrt{2}CX \$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-d46c042476075c33afbfaec92700fd9324f4c711efde3aa186780de8abf115a8	A point $ X $ lies inside or on the boundary of the right triangle $ ABC $ with $ \angle C = 90^\circ $. Points $ P, Q, R $ are the projections of $ X $ onto $ BC, CA, $ and $ AB $ respectively. Let $ AR \cdot RB = BP \cdot PC + AQ \cdot QC $. If $ AB = 13 $, $ AC = 12 $, and $ BC = 5 $, and \( X ...	[ "\\dfrac{13}{2}" ]	[ "A point \$ X \$ lies inside or on the boundary of the right triangle \$ ABC \$ with \$ \\angle C = 90^\\circ \$. Points \$ P, Q, R \$ are the projections of \$ X \$ onto \$ BC, CA, \$ and \$ AB \$ respectively. Let \$ AR \\cdot RB = BP \\cdot PC + AQ \\cdot QC \$. If \$ AB = 13 \$, \\( AC = 12 \\...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-ab78fadaa50837fc82164573c0ded627af255e44db495aff6ffb2b5a8b291aad	A point $ X $ lies inside or on the boundary of the right triangle $ ABC $ with $ \angle C $ being the right angle. The points $ P $, $ Q $, and $ R $ are the perpendicular projections of $ X $ onto the sides $ BC $, $ CA $, and $ AB $ respectively. If $ AR \cdot RB = BP \cdot PC + AQ \cdot QC $, ...	[ "0" ]	[ "A point \$ X \$ lies inside or on the boundary of the right triangle \$ ABC \$ with \$ \\angle C \$ being the right angle. The points \$ P \$, \$ Q \$, and \$ R \$ are the perpendicular projections of \$ X \$ onto the sides \$ BC \$, \$ CA \$, and \$ AB \$ respectively. If \\( AR \\cdot RB = BP...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-65bcc23b030a8dd55076c63d6762a8fb77fbad682b12c299b0379cf57e6be871	A point $ a \in A \subset \mathbb{R}^d $ is called an extreme point of $ A $ provided that if $ \frac{b+c}{2} = a $ for two points $ b, c \in A $, then $ a = b = c $. The set of all extreme points of $ A $ is denoted by $ \text{ex}(A) $.	[ "\\text{ex}(A)" ]	[ "A point \$ a \\in A \\subset \\mathbb{R}^d \$ is called an extreme point of \$ A \$ provided that if \$ \\frac{b+c}{2} = a \$ for two points \$ b, c \\in A \$, then \$ a = b = c \$. The set of all extreme points of \$ A \$ is denoted by \$ \\text{ex}(A) \$." ]	[ "aops_c7_college_math" ]	[ "cot", "genselect" ]	{ "cot": 9, "genselect": 4 }	13
omr-ws-a37a21c9475601a588f7bf5c35f5dc0254bd37e226e367ae8906b1fdd1963583	A point $(X, Y)$ is picked at random, uniformly from the square with corners at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. (a) Compute $Cov(X, Y)$. (b) Compute $P\left(Y > \frac{1}{2} + X\right)$.	[ "\$\\frac{1}{8}\$" ]	[ "A point \$(X, Y)\$ is picked at random, uniformly from the square with corners at \$(0,0)\$, \$(1,0)\$, \$(0,1)\$, and \$(1,1)\$.\n(a) Compute \$Cov(X, Y)\$.\n(b) Compute \$P\\left(Y > \\frac{1}{2} + X\\right)\$." ]	[ "aops_c7_college_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 26 }	42
omr-ws-10b1a9b0546697b23826822341cc82baa67ccd652377fd2a5e1d72d4c123d0f8	A point $(p, q)$ in the coordinate plane is defined as [i]reachable[/i] if there exist $x_1, \dots, x_{17} \in [0, \tfrac{\pi}{2}]$ such that \[ \cos x_1 + \dots + \cos x_{17} = p \quad \text{and} \quad \sin x_1 + \dots + \sin x_{17} = q. \] There exist positive integers $a$, $b$, and $c$ such that \(\gcd(a, ...	[ "205" ]	[ "A point \$(p, q)\$ in the coordinate plane is defined as [i]reachable[/i] if there exist \$x_1, \\dots, x_{17} \\in [0, \\tfrac{\\pi}{2}]\$ such that\n\\[\n\\cos x_1 + \\dots + \\cos x_{17} = p \\quad \\text{and} \\quad \\sin x_1 + \\dots + \\sin x_{17} = q.\n\\]\nThere exist positive integers \$a\$, \$b\$...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 1 }	1
omr-ws-2fa29e71eae8c8b9209229c40d2ca04aa6517fdb44ef02f29c59200ad68b907a	A point $(x(t), y(t))$ moves along $C$ as $t$ increases in all real numbers with a constant horizontal rate of 30 units per second ($x'(t) = 30$ units/sec). What is the vertical rate of change with respect to $t$ when $x(t) = 1$ ($[y']_{x=1}$)?	[ "\$\\sqrt{3} \\pi\$" ]	[ "A point \$(x(t), y(t))\$ moves along \$C\$ as \$t\$ increases in all real numbers with a constant horizontal rate of 30 units per second (\$x'(t) = 30\$ units/sec). What is the vertical rate of change with respect to \$t\$ when \$x(t) = 1\$ (\$[y']_{x=1}\$)?" ]	[ "aops_c7_college_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-0611b531064e01f264eb47f6673f2f6af98ca395b28982a206077d54e403cb4d	A point $(x, y)$ is called integral if both $ x $ and $ y $ are integers. How many integral points are inside the parallelogram whose four sides are on the lines $ x = 100 $, $ x = 300 $, $ y = \frac{x}{3} + 0.1 $, and $ y = \frac{x}{3} + 0.6 $?	[ "67" ]	[ "A point \$(x, y)\$ is called integral if both \$ x \$ and \$ y \$ are integers. How many integral points are inside the parallelogram whose four sides are on the lines \$ x = 100 \$, \$ x = 300 \$, \$ y = \\frac{x}{3} + 0.1 \$, and \$ y = \\frac{x}{3} + 0.6 \$?" ]	[ "aops_c5_contests_amp_programs" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-59dbf596cb99b9f3d2a13fbe563dbc2ff0ae38a4c896e248c8aa6be3160ef689	A point $(x, y, z)$ is randomly chosen in 3-dimensional space such that $0 \leq x \leq 3$, $0 \leq y \leq 4$, and $0 \leq z \leq 5$. What is the probability that none of the three coordinates vary by more than one unit from any of the other two?	[ "\$\\frac{2}{15}\$" ]	[ "A point \$(x, y, z)\$ is randomly chosen in 3-dimensional space such that \$0 \\leq x \\leq 3\$, \$0 \\leq y \\leq 4\$, and \$0 \\leq z \\leq 5\$. What is the probability that none of the three coordinates vary by more than one unit from any of the other two?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 3, "genselect": 4, "tir": 18 }	25
omr-ws-850c39b33e1f5e723f4a586019cf74c26f846347dcbec606411971ef80f5598f	A point $(x,y)$ is chosen at random inside the square with vertices $(0,0)$, $(0,1)$, $(1,1)$, and $(1,0)$. What is the probability that the distance from $(x,y)$ to $(0,1)$ is greater than 1?	[ "\$ 1 - \\frac{\\pi}{4} \$" ]	[ "A point \$(x,y)\$ is chosen at random inside the square with vertices \$(0,0)\$, \$(0,1)\$, \$(1,1)\$, and \$(1,0)\$. What is the probability that the distance from \$(x,y)\$ to \$(0,1)\$ is greater than 1?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 15, "tir": 28 }	43
omr-ws-bde116e9335eff33ed73cd90e99ffced8adb555d0d2ee8eb312092e59a524531	A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(3,0)$, $(3,2)$, and $(0,2)$. What is the probability that $x < y$?	[ "\$\\frac{2}{3}\$" ]	[ "A point \$(x,y)\$ is randomly picked from inside the rectangle with vertices \$(0,0)\$, \$(3,0)\$, \$(3,2)\$, and \$(0,2)\$. What is the probability that \$x < y\$?" ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-2ca041b66b18a28bdf91101815794b64c61b309e7bb667184bc8abe911c74566	A point $(x,y)$ is to be chosen in the coordinate plane so that it is equally distant from the x-axis, y-axis, and the line $x + y = 2$. Determine the value of $x$.	[ "Not uniquely determined." ]	[ "A point \$(x,y)\$ is to be chosen in the coordinate plane so that it is equally distant from the x-axis, y-axis, and the line \$x + y = 2\$. Determine the value of \$x\$." ]	[ "aops_c5_contests_amp_programs" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-62001788637dfdac9e325f3ef76811c1e922e4df4215ae4635aa8ff696d7db4b	A point $A$ is fixed on the circle $\omega$. The chords $BC$ of the circle $\omega$ are chosen so that they pass through a fixed point $P$. Let $O$ be the center of the circle $\omega$ and $R$ its radius. For a particular choice of $BC$, let $O_9$ be the center of the Nine Point Circle of triangle \...	[ "\\dfrac{R - d}{2}" ]	[ "A point \$A\$ is fixed on the circle \$\\omega\$. The chords \$BC\$ of the circle \$\\omega\$ are chosen so that they pass through a fixed point \$P\$. Let \$O\$ be the center of the circle \$\\omega\$ and \$R\$ its radius. For a particular choice of \$BC\$, let \$O_9\$ be the center of the Nin...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 7, "genselect": 7 }	14
omr-ws-14318c58c8a590f1d95aff4ab6dc685bba60c5a6853bb6aae7bbdd50e77794f8	A point $B$ is marked on the segment $AC$. Use a compass and a ruler to construct a point $P$ such that $\angle ACD = 90^\circ$ and $\angle ADB = 2\angle CAD$.	[ "P" ]	[ "A point \$B\$ is marked on the segment \$AC\$. Use a compass and a ruler to construct a point \$P\$ such that \$\\angle ACD = 90^\\circ\$ and \$\\angle ADB = 2\\angle CAD\$." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-0b66ca4d332134b1886b95d15f540622e10ca02f5529f27e02bb34ad7db25755	A point $B$ lies on a chord $AC$ of circle $\omega$. Segments $AB$ and $BC$ are diameters of circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ respectively. These circles intersect $\omega$ for the second time in points $D$ and $E$ respectively. The rays $O_1D$ and $O_2E$ meet in...	[ "6\\sqrt{2}" ]	[ "A point \$B\$ lies on a chord \$AC\$ of circle \$\\omega\$. Segments \$AB\$ and \$BC\$ are diameters of circles \$\\omega_1\$ and \$\\omega_2\$ centered at \$O_1\$ and \$O_2\$ respectively. These circles intersect \$\\omega\$ for the second time in points \$D\$ and \$E\$ respectively. The r...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 3 }	3
omr-ws-76199cf660ee519144c56c8921336c875dcd21f166a402e6814f6415d2568ee0	A point $C$ is marked on a chord $AB$ of a circle $\omega$. Let $D$ be the midpoint of $AC$, and $O$ be the center of the circle $\omega$. The circumcircle of the triangle $BOD$ intersects the circle $\omega$ again at point $E$ and the straight line $OC$ again at point $F$. If $AB = 20$ and \(...	[ "4\\sqrt{5}" ]	[ "A point \$C\$ is marked on a chord \$AB\$ of a circle \$\\omega\$. Let \$D\$ be the midpoint of \$AC\$, and \$O\$ be the center of the circle \$\\omega\$. The circumcircle of the triangle \$BOD\$ intersects the circle \$\\omega\$ again at point \$E\$ and the straight line \$OC\$ again at poin...	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-e2ca3bada15825f64a0ba41ce08816c33577c5bd20e49ccb0b66388a548b2b0b	A point $D$ inside an isosceles triangle $ABC$ (with $AB = AC$) is selected such that $AD = DB$ and $\angle BAD = 19.7^\circ$. Suppose $\angle BAC = 99.4^\circ$. What is the degree measure of $\angle BDC$?	[ "149.1" ]	[ "A point \$D\$ inside an isosceles triangle \$ABC\$ (with \$AB = AC\$) is selected such that \$AD = DB\$ and \$\\angle BAD = 19.7^\\circ\$. Suppose \$\\angle BAC = 99.4^\\circ\$. What is the degree measure of \$\\angle BDC\$?" ]	[ "aops_c4_high_school_math", "c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 11, "genselect": 2, "tir": 15 }	28
omr-ws-d454c0bedb5b458d368fb7fb320fccdf64b7735b268c4daf0347e11523d187de	A point $L$ is taken on the median $AM$ of triangle $ABC$ such that $AL:LM=1:2$. In what ratio does the line $BL$ divide the segment $AC$?	[ "\$1:2\$" ]	[ "A point \$L\$ is taken on the median \$AM\$ of triangle \$ABC\$ such that \$AL:LM=1:2\$. In what ratio does the line \$BL\$ divide the segment \$AC\$?" ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-1433de235dab95c83f98ef28710d60c7729c5e14e0d87c1b3efee3267ce2f408	A point $M$ is fixed inside a sphere of radius $R$. Three mutually perpendicular chords are drawn through $M$. If the lengths of these chords are $a$, $b$, and $c$, find the value of $a^2 + b^2 + c^2$ in terms of $R$.	[ "12R^2" ]	[ "A point \$M\$ is fixed inside a sphere of radius \$R\$. Three mutually perpendicular chords are drawn through \$M\$. If the lengths of these chords are \$a\$, \$b\$, and \$c\$, find the value of \$a^2 + b^2 + c^2\$ in terms of \$R\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 4, "genselect": 7 }	11
omr-ws-ba0b41fa020484b58dc09bb86af8a17f990ca80035441c7a3dcafc523f80866b	A point $M$ is taken on the hypotenuse $AB$ of a right-angled triangle $ABC$. Perpendiculars $ME$ and $MF$ are drawn from $M$ to the legs $AC$ and $BC$ respectively, with $E$ and $F$ being the feet of these perpendiculars. Where should point $M$ be chosen so that the length of segment $EF$ is th...	[ "\$M\$ is the foot of the altitude from \$C\$ to \$AB\$." ]	[ "A point \$M\$ is taken on the hypotenuse \$AB\$ of a right-angled triangle \$ABC\$. Perpendiculars \$ME\$ and \$MF\$ are drawn from \$M\$ to the legs \$AC\$ and \$BC\$ respectively, with \$E\$ and \$F\$ being the feet of these perpendiculars. Where should point \$M\$ be chosen so that the len...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 13, "genselect": 8, "tir": 18 }	39
omr-ws-3da2b4894edddd4f6544c42374b8418fa999c727d5167949f3a20368676e99fd	A point $O$ is taken inside a triangular pyramid. Three straight lines are drawn through $O$ parallel to each face of the pyramid. Let the segments of these lines located inside the pyramid be $a$, $b$, $c$, $d$, $e$, and $f$, and let the lengths of the corresponding parallel sides of the pyramid be \(A...	[ "2" ]	[ "A point \$O\$ is taken inside a triangular pyramid. Three straight lines are drawn through \$O\$ parallel to each face of the pyramid. Let the segments of these lines located inside the pyramid be \$a\$, \$b\$, \$c\$, \$d\$, \$e\$, and \$f\$, and let the lengths of the corresponding parallel sides ...	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 6, "genselect": 6 }	12
omr-ws-3e2fd601ca5c6561970d36c40e21fcc51f632de09782b3d2524c206addb1f4d3	A point $P(x, y)$ is randomly selected from the region $\|x\| + \|y\| < 1$. What is the probability that $2x + 3y < 1$?	[ "\$\\frac{7}{10}\$" ]	[ "A point \$P(x, y)\$ is randomly selected from the region \$\|x\| + \|y\| < 1\$. What is the probability that \$2x + 3y < 1\$?" ]	[ "aops_c5_contests_amp_programs" ]	[ "cot", "genselect", "tir" ]	{ "cot": 32, "genselect": 5, "tir": 21 }	58
omr-ws-af1884f3ae585cb71764568929b93ff2f8d79cff298302a4e5224a3e2b6ddcd6	A point $P$ is chosen in the interior of $\Delta ABC$ so that when lines are drawn through $P$ parallel to the sides of $\Delta ABC$, the resulting smaller triangles have areas 4, 9, and 49, respectively. Find the area of $\Delta ABC$.	[ "121" ]	[ "A point \$P\$ is chosen in the interior of \$\\Delta ABC\$ so that when lines are drawn through \$P\$ parallel to the sides of \$\\Delta ABC\$, the resulting smaller triangles have areas 4, 9, and 49, respectively. Find the area of \$\\Delta ABC\$." ]	[ "aops_c5_contests_amp_programs" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-d615c42a379e31339722818d739c7e8920fe664d2d524fd99a37c59e27e1f451	A point $P$ is in the interior of the cyclic quadrilateral $ABCD$ and has the property $\angle BPC = \angle BAP + \angle PDC$. The feet of the perpendiculars from $P$ to $AB$, $AD$, and $DC$ are respectively denoted $E$, $F$, and $G$. If the sides of $\triangle FEG$ are in the ratio $1:2:3$ and ...	[ "14" ]	[ "A point \$P\$ is in the interior of the cyclic quadrilateral \$ABCD\$ and has the property \$\\angle BPC = \\angle BAP + \\angle PDC\$. The feet of the perpendiculars from \$P\$ to \$AB\$, \$AD\$, and \$DC\$ are respectively denoted \$E\$, \$F\$, and \$G\$. If the sides of \$\\triangle FEG\$ ...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 7 }	7
omr-ws-8afca341378af05ed119529c0ce249503476627b00c49d733a41c7255a9b6fab	A point $P$ is located in the interior of a rectangle such that the distance from $P$ to one corner is 5 yards, from the opposite corner is 14 yards, and from a third corner is 10 yards. What is the distance from $P$ to the fourth corner?	[ "11" ]	[ "A point \$P\$ is located in the interior of a rectangle such that the distance from \$P\$ to one corner is 5 yards, from the opposite corner is 14 yards, and from a third corner is 10 yards. What is the distance from \$P\$ to the fourth corner?" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-3eac9daeb393c57323f5a5f92f24f353a7a191adf592302fafee031377919ac9	A point $P$ is randomly chosen inside a regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$. What is the probability that the projections of $P$ onto the lines $\overleftrightarrow{A_i A_{i+1}}$ for $i = 1, 2, \ldots, 8$ lie on the segments $\overline{A_iA_{i+1}}$ for $i = 1, 2, \ldots, 8$ (where indices are taken...	[ "3 - 2\\sqrt{2}" ]	[ "A point \$P\$ is randomly chosen inside a regular octagon \$A_1A_2A_3A_4A_5A_6A_7A_8\$. What is the probability that the projections of \$P\$ onto the lines \$\\overleftrightarrow{A_i A_{i+1}}\$ for \$i = 1, 2, \\ldots, 8\$ lie on the segments \$\\overline{A_iA_{i+1}}\$ for \$i = 1, 2, \\ldots, 8\$ (...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 11, "genselect": 7 }	18
omr-ws-eda863ab5e1b6cc7978dfe38a0f667dd71806703df0d78cf624acc61641bfdbd	A point $P$ lies inside $\triangle ABC$ such that the areas of $\triangle PAB$, $\triangle PBC$, and $\triangle PCA$ can form a triangle. Let $BC = a$, $CA = b$, $AB = c$, $PA = x$, $PB = y$, and $PC = z$. Calculate the maximum possible value of \(\frac{(x + y)^2 + (y + z)^2 + (z + x)^2}{x + y + z...	[ "56" ]	[ "A point \$P\$ lies inside \$\\triangle ABC\$ such that the areas of \$\\triangle PAB\$, \$\\triangle PBC\$, and \$\\triangle PCA\$ can form a triangle. Let \$BC = a\$, \$CA = b\$, \$AB = c\$, \$PA = x\$, \$PB = y\$, and \$PC = z\$. Calculate the maximum possible value of \\(\\frac{(x + y)^2 +...	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 2 }	2
omr-ws-474f94f9b8b71b3920f54a9ecc3199fb90f863d5e68847697bf04ebc9d635827	A point has rectangular coordinates $(-5,-7,4)$ and spherical coordinates $(\rho, \theta, \phi).$ Find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, -\phi).$	[ "(5,7,4)" ]	[ "A point has rectangular coordinates $(-5,-7,4)$ and spherical coordinates $(\\rho, \\theta, \\phi).$ Find the rectangular coordinates of the point with spherical coordinates $(\\rho, \\theta, -\\phi).$" ]	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-12c822344e69e3ffd9340842f5bc0f19356473574afcea2da4ee617d6d6499e8	A point has rectangular coordinates $(x,y,z)$ and spherical coordinates $\left(2, \frac{8 \pi}{7}, \frac{2 \pi}{9} \right).$ Find the spherical coordinates of the point with rectangular coordinates $(x,y,-z).$ Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2 \pi,$ and $0 \le \phi ...	[ "\\left( 2, \\frac{8 \\pi}{7}, \\frac{7 \\pi}{9} \\right)" ]	[ "A point has rectangular coordinates $(x,y,z)$ and spherical coordinates $\\left(2, \\frac{8 \\pi}{7}, \\frac{2 \\pi}{9} \\right).$ Find the spherical coordinates of the point with rectangular coordinates $(x,y,-z).$ Enter your answer in the form $(\\rho,\\theta,\\phi),$ where $\\rho > 0,$ $0 \\le \\theta < 2 \\p...	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-a838e4e41a4da427c236f5268d16fb6c76e59e3718cfa24f4324a3f899c216df	A point in $3$-dimensional space is called a lattice point if all three of its coordinates $(x, y, z)$ are integers. When making a list of lattice points $a_1, a_2, \ldots, a_n$, what is the minimum $n$ that guarantees the midpoint between some $2$ of the lattice points in the list is a lattice point?	[ "9" ]	[ "A point in $3$-dimensional space is called a lattice point if all three of its coordinates $(x, y, z)$ are integers. When making a list of lattice points $a_1, a_2, \\ldots, a_n$, what is the minimum $n$ that guarantees the midpoint between some $2$ of the lattice points in the list is a lattice point?" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-5b59fad863852eead1d257fea8ec110cb97be659a5f38a44b674640fa5d12bbf	A point in space $(x,y,z)$ is randomly selected so that $-1\le x \le 1$,$-1\le y \le 1$,$-1\le z \le 1$. What is the probability that $x^2+y^2+z^2\le 1$?	[ "\\frac{\\pi}{6}" ]	[ "A point in space $(x,y,z)$ is randomly selected so that $-1\\le x \\le 1$,$-1\\le y \\le 1$,$-1\\le z \\le 1$. What is the probability that $x^2+y^2+z^2\\le 1$?" ]	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-3f712cd31e06d301e6bd03730252883cb951947e3b524e92ef902d78288f34f1	A point in space is 2, 6, 7, 8, and 9 units away from all but one of the vertices of a regular octahedron. Find the distance from the remaining vertex.	[ "\\sqrt{21}" ]	[ "A point in space is 2, 6, 7, 8, and 9 units away from all but one of the vertices of a regular octahedron. Find the distance from the remaining vertex." ]	[ "aops_c4_high_school_math" ]	[ "genselect" ]	{ "genselect": 8 }	8
omr-ws-fe56c7586786aa240164fefc42ac3dd06b1227bdcd42d03558f8344258f6969f	A point in the first quadrant on the circle with the equation $x^2 + y^2 = 1$ has a tangent that makes an angle of 30 degrees with the x-axis. Find its coordinates.	[ "\$\\left(\\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right)\$" ]	[ "A point in the first quadrant on the circle with the equation \$x^2 + y^2 = 1\$ has a tangent that makes an angle of 30 degrees with the x-axis. Find its coordinates." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 28 }	44
omr-ws-43c2cba258ef431fe3e67f268b5f9f687a8b7d59ddb8529d74905538dd1d558a	A point in the plane is called a lattice point if its coordinates are both integers. A triangle whose vertices are all lattice points is called a lattice triangle. Give explicitly the coordinates of the vertices of a lattice triangle $T$ such that the area of $T$ is $1/2$ and the three sides of $T$ each have length gre...	[ "(2012, 4048145)" ]	[ "A point in the plane is called a lattice point if its coordinates are both integers. A triangle whose vertices are all lattice points is called a lattice triangle. Give explicitly the coordinates of the vertices of a lattice triangle $T$ such that the area of $T$ is $1/2$ and the three sides of $T$ each have lengt...	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 3 }	3
omr-ws-ccacd3496a0136082cdf69d928f63d026dbd5d81460ec43375330f300ae992a1	A point in the plane is called a lattice point if its coordinates are both integers. A triangle whose vertices are all lattice points is called a lattice triangle. Give explicitly the coordinates of the vertices of a lattice triangle $T$ such that the area of $T$ is $1/2$ and two sides of $T$ have length greater than $...	[ "(0,0), (2011,1), (2012,1)" ]	[ "A point in the plane is called a lattice point if its coordinates are both integers. A triangle whose vertices are all lattice points is called a lattice triangle. Give explicitly the coordinates of the vertices of a lattice triangle $T$ such that the area of $T$ is $1/2$ and two sides of $T$ have length greater t...	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 7, "genselect": 5 }	12
omr-ws-8185a9d57cda4ae545340d7c57aa9b0205f0d3fcb8648af5ebb19074f202d20c	A point inside a convex quadrilateral is reflected symmetrically with respect to each side. The new points are the vertices of a new quadrilateral, all sides of which pass through the vertices of the original quadrilateral. Find the angles of the original quadrilateral.	[ "90^\\circ" ]	[ "A point inside a convex quadrilateral is reflected symmetrically with respect to each side. The new points are the vertices of a new quadrilateral, all sides of which pass through the vertices of the original quadrilateral. Find the angles of the original quadrilateral." ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-5056136c6ad814fd6b56bf2056aeda0f9d46110069ec827de7b6423ad9441bd7	A point inside a square is positioned so that the distances to the four vertices are $27$, $21$, $6$, and $x$ units. If $x$ is a whole number, what is $x$?	[ "18" ]	[ "A point inside a square is positioned so that the distances to the four vertices are $27$, $21$, $6$, and $x$ units. If $x$ is a whole number, what is $x$?" ]	[ "aops_c4_high_school_math" ]	[ "cot" ]	{ "cot": 16 }	16
omr-ws-86946293d3f822ac8ae37ba45827677170ad1c4cd18b6890bee55cb55f3ac720	A point inside a square is positioned such that the distances to the four vertices are 27, 21, 6, and $ x $ units. If $ x $ is a whole number, what is the value of $ x $?	[ "18" ]	[ "A point inside a square is positioned such that the distances to the four vertices are 27, 21, 6, and \$ x \$ units. If \$ x \$ is a whole number, what is the value of \$ x \$?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 14 }	30
omr-ws-d5020ab0be675caa5d24dc7e386be1767343c947db94148b42a62609ffab1193	A point inside a triangle is called "good" if three cevians passing through it are equal. Assume for an isosceles triangle $ABC$ ($AB = BC$) the total number of "good" points is odd. Find all possible values of this number.	[ "1 or 3" ]	[ "A point inside a triangle is called \"good\" if three cevians passing through it are equal. Assume for an isosceles triangle \$ABC\$ (\$AB = BC\$) the total number of \"good\" points is odd. Find all possible values of this number." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot", "genselect" ]	{ "cot": 27, "genselect": 5 }	32
omr-ws-276479102b25b976613db02fb58e4e6b19d2db32a3cf7f82422a9f02933f3fe5	A point inside an equilateral triangle is 6 units away from one vertex, 8 units away from another, and 10 units away from the last. What is the length of one side of the equilateral triangle?	[ "\$2\\sqrt{25 + 12\\sqrt{3}}\$" ]	[ "A point inside an equilateral triangle is 6 units away from one vertex, 8 units away from another, and 10 units away from the last. What is the length of one side of the equilateral triangle?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 27 }	43
omr-ws-824a2edffbc16384e544cf883f77279679878d43ea787c02242a6c124994d31f	A point inside an equilateral triangle is chosen at random. The perpendicular distances from the point to the sides are 1.25, 0.80, and 0.95. What is the area of the triangle?	[ "\$ 3\\sqrt{3} \$" ]	[ "A point inside an equilateral triangle is chosen at random. The perpendicular distances from the point to the sides are 1.25, 0.80, and 0.95. What is the area of the triangle?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 30 }	46
omr-ws-dc1dbfd0ad23e907bb3fafec61335d24b4e44bc2049d1428f2a96afc20eda336	A point inside an equilateral triangle with side length 1 is a distance $a$, $b$, and $c$ from the vertices. Calculate the value of $a + b + c$.	[ "\\sqrt{3}" ]	[ "A point inside an equilateral triangle with side length 1 is a distance \$a\$, \$b\$, and \$c\$ from the vertices. Calculate the value of \$a + b + c\$." ]	[ "aops_c6_high_school_olympiads" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-09186fe6191ec0ec838ac1da257972fddb7cd3186692a9bbb56a20d7f529e9a0	A point inside square $ABCD$ has distances $5\sqrt{2}$, $5\sqrt{10}$, $15\sqrt{2}$, and $5\sqrt{10}$ from the vertices. Find the area of $ABCD$.	[ "400" ]	[ "A point inside square \$ABCD\$ has distances \$5\\sqrt{2}\$, \$5\\sqrt{10}\$, \$15\\sqrt{2}\$, and \$5\\sqrt{10}\$ from the vertices. Find the area of \$ABCD\$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect", "tir" ]	{ "cot": 15, "genselect": 8, "tir": 8 }	31
omr-ws-fbf46a2e15126b9f01508affbf37fbb2d655ca4a2d80f2de33f3813a8cbcae0b	A point is at distances $ \sqrt{2} $, $ 2 $, and $ \sqrt{3}-1 $ from the vertices of a triangle. Find the maximum possible area of the triangle.	[ "\\dfrac{5\\sqrt{2} - \\sqrt{6}}{2}" ]	[ "A point is at distances $ \\sqrt{2} $, $ 2 $, and $ \\sqrt{3}-1 $ from the vertices of a triangle. Find the maximum possible area of the triangle." ]	[ "aops_c6_high_school_olympiads" ]	[ "cot" ]	{ "cot": 9 }	9
omr-ws-ff63034ad1edb37279052433fba9bfdc5d8ab6e760fb4c3d737a52bafeaa5217	A point is chosen at random from the interior of a right triangle. What is the probability that the chosen point is closer to one of the midpoints of a side than it is to a vertex?	[ "\$\\frac{3}{4}\$" ]	[ "A point is chosen at random from the interior of a right triangle. What is the probability that the chosen point is closer to one of the midpoints of a side than it is to a vertex?" ]	[ "aops_c4_high_school_math" ]	[ "tir" ]	{ "tir": 1 }	1
omr-ws-5dc11d6184450ac08c14e6f3cb6bbbff4993854e50b5d3528c1ae37331925a51	A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?	[ "\$\\frac{1}{4}\$" ]	[ "A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?" ]	[ "aops_c5_contests_amp_programs", "c5_contests_amp_programs" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 31 }	47
omr-ws-badeda8c158960eaa2dbdae9d78e2ca928297a94b30fd2a4b56fa1d22c1e464c	A point is chosen at random from within the circumcircle of a triangle with angles $45^\circ$, $75^\circ$, and $60^\circ$. What is the probability that the point is closer to the vertex with an angle of $45^\circ$ than either of the two other vertices?	[ "\\dfrac{3}{8}" ]	[ "A point is chosen at random from within the circumcircle of a triangle with angles $45^\\circ$, $75^\\circ$, and $60^\\circ$. What is the probability that the point is closer to the vertex with an angle of $45^\\circ$ than either of the two other vertices?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 8, "genselect": 7 }	15
omr-ws-2d00a2824a32e5ef640f1b6fa9cc85ee94186bbe825d64dedb5952236a3ccfb3	A point is chosen at random in the interior of a regular octahedron. What is the probability that it lies closer to the center of the octahedron than to any of the vertices?	[ "\$\\frac{5}{8}\$" ]	[ "A point is chosen at random in the interior of a regular octahedron. What is the probability that it lies closer to the center of the octahedron than to any of the vertices?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 13, "tir": 26 }	39
omr-ws-ffcba4393a26ce1c38f7c32fc6a919864b86e97abb38c11fdb1ef4cc41daec6e	A point is chosen at random inside a given square. If segments are drawn joining the point to the four vertices of the square, four angles are formed around the point. What is the probability that at least one of these angles has a measure of $120^\circ$ or more?	[ "\\dfrac{2\\pi + 6 - 3\\sqrt{3}}{9}" ]	[ "A point is chosen at random inside a given square. If segments are drawn joining the point to the four vertices of the square, four angles are formed around the point. What is the probability that at least one of these angles has a measure of $120^\\circ$ or more?" ]	[ "aops_c4_high_school_math" ]	[ "cot", "genselect" ]	{ "cot": 4, "genselect": 2 }	6
omr-ws-ba4738671aee53f7e2b420c3693efb210fb812af919a3b35e7dbf6a63138d2ee	A point is chosen at random inside a triangle with height $ h $ and base of length $ b $. Determine the density of the distance from the point to the base of the triangle.	[ "\$ f(x) = \\begin{cases} \\frac{2}{h^2}(h-x) & \\quad 0 < x < h \\\\ 0 & \\quad \\text{otherwise} \\end{cases} \$" ]	[ "A point is chosen at random inside a triangle with height \$ h \$ and base of length \$ b \$. Determine the density of the distance from the point to the base of the triangle." ]	[ "aops_c7_college_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 14 }	30
omr-ws-7225ce7644faf0f65b7575f305684abd47104bbc767b8e5d4744b79bb8ab7ecf	A point is chosen at random inside the square with vertices $(0,0), (2,0), (2,2),$ and $(0,2)$. What is the probability that this point is closer to $(0,0)$ than to $(3,3)$?	[ "\$\\frac{1}{4}\$" ]	[ "A point is chosen at random inside the square with vertices $(0,0), (2,0), (2,2),$ and $(0,2)$. What is the probability that this point is closer to $(0,0)$ than to $(3,3)$?" ]	[ "aops_c4_high_school_math" ]	[ "additional_problems" ]	{ "additional_problems": 1 }	1
omr-ws-db86024d18171791b64c85234245055e82c897010ce0cfa905f08b5dc0d095fe	A point is chosen at random on a line segment of length $L$. Find the probability that the ratio of the shorter segment to the longer segment is less than $\frac{1}{3}$.	[ "\$\\frac{1}{2}\$" ]	[ "A point is chosen at random on a line segment of length $L$. Find the probability that the ratio of the shorter segment to the longer segment is less than $\\frac{1}{3}$." ]	[ "aops_c4_high_school_math" ]	[ "cot", "tir" ]	{ "cot": 16, "tir": 28 }	44
omr-ws-d5ffacaa7564d6daa9a462edaaf7f1deba0bdeb03776a5184ee0079459d547dc	A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater than the number of the green point, but les...	[ "\\frac{1}{4}" ]	[ "A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater than the number of the green point, bu...	[ "MATH_training_set" ]	[ "cot" ]	{ "cot": 16 }	16

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