id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-450c5760a3a52b78557e0261f7c8fd44251c782fd9b5347fc0f21aa646c2f5b9 | A point $P$ is $a$, $b$, and $a+b$ units from the three vertices of an equilateral triangle. Express the area of the triangle in terms of $a$ and $b$. | [
"\\(\\frac{\\sqrt{3}}{4}(a^2 + ab + b^2)\\)"
] | [
"A point $P$ is $a$, $b$, and $a+b$ units from the three vertices of an equilateral triangle. Express the area of the triangle in terms of $a$ and $b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 11
} | 34 | 0 | 0 |
omr-ws-7f8023cf3a3e5b278e5811f3b8e97627edb21dbfa06a046525cd4100f2e44fa4 | A point $P$ is chosen at random in the coordinate plane. What is the probability that the unit circle with center $P$ contains exactly two lattice points in its interior? | [
"\\pi - \\dfrac{3\\sqrt{3}}{2}"
] | [
"A point $P$ is chosen at random in the coordinate plane. What is the probability that the unit circle with center $P$ contains exactly two lattice points in its interior?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9660b3f61b5f65de46523b8b33bf3515e5012fa71093ae813efd81f5a87b9887 | A point $P$ is chosen at random in the interior of a unit square $S$. Let $d(P)$ denote the distance from $P$ to the closest side of $S$. The probability that $\frac{1}{5}\le d(P)\le\frac{1}{3}$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"281"
] | [
"A point $P$ is chosen at random in the interior of a unit square $S$. Let $d(P)$ denote the distance from $P$ to the closest side of $S$. The probability that $\\frac{1}{5}\\le d(P)\\le\\frac{1}{3}$ is equal to $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-cb9c9a8b6512ad5bf8ae2ab919a929384b7467735cb6909737256d7e15c1c303 | A point $P$ is chosen at random in the interior of equilateral triangle $ABC$. What is the probability that $\triangle ABP$ has a greater area than each of $\triangle ACP$ and $\triangle BCP$? | [
"\\frac{1}{3}"
] | [
"A point $P$ is chosen at random in the interior of equilateral triangle $ABC$. What is the probability that $\\triangle ABP$ has a greater area than each of $\\triangle ACP$ and $\\triangle BCP$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-c0cd8e34044daf7e22ee4fe75bce514f6387b86a9f40771997089cfa3d5d4e59 | A point $P$ is chosen at random inside a square $QRST$. What is the probability that angle $RPQ$ is acute? | [
"\\( 1 - \\frac{\\pi}{8} \\)"
] | [
"A point $P$ is chosen at random inside a square $QRST$. What is the probability that angle $RPQ$ is acute?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-eec39ec4d39b99cb97c80c0c3a2efa554cc039144005fbae26f298267cbc950f | A point $P$ is chosen in the interior of $\triangle ABC$ such 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 $\\triangle ABC$ such 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 | 0 | 0 |
omr-ws-eee3d9d08a65191ed8f714dfca883834c369e3f3d1d1a6244c03fa587b0b4954 | A point $P$ is chosen in the interior of the side $BC$ of triangle $ABC$. The points $D$ and $E$ are symmetric to $P$ with respect to the vertices $B$ and $C$, respectively. The circumcircles of the triangles $ABE$ and $ACD$ intersect at the points $A$ and $X$. The ray $AB$ intersects the segment $XD$ at the point $C_1... | [
"10"
] | [
"A point $P$ is chosen in the interior of the side $BC$ of triangle $ABC$. The points $D$ and $E$ are symmetric to $P$ with respect to the vertices $B$ and $C$, respectively. The circumcircles of the triangles $ABE$ and $ACD$ intersect at the points $A$ and $X$. The ray $AB$ intersects the segment $XD$ at the point... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-a1d12fed6ed515b57ec9fa2d2ebaa732ae3963ebc58947f80cd4ac5305579a71 | A point $P$ is chosen inside a triangle $ABC$ with circumcircle $\Omega$. Let $\Gamma$ be the circle passing through the circumcenters of the triangles $APB$, $BPC$, and $CPA$. Let $\Omega$ and $\Gamma$ intersect at points $X$ and $Y$. Let $Q$ be the reflection of $P$ in the line $XY$. If $\angle BAP = 30^\circ$, find ... | [
"30"
] | [
"A point $P$ is chosen inside a triangle $ABC$ with circumcircle $\\Omega$. Let $\\Gamma$ be the circle passing through the circumcenters of the triangles $APB$, $BPC$, and $CPA$. Let $\\Omega$ and $\\Gamma$ intersect at points $X$ and $Y$. Let $Q$ be the reflection of $P$ in the line $XY$. If $\\angle BAP = 30^\\c... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-4a8a9b6f839f875d97c3507716973ac09590365fbcd7cea9e8023a2aa4212c81 | A point $P$ is chosen on the side $AB$ of an acute-angled triangle $ABC$ so that $AP : BP = 2:1$. It is known that $AC = CP = 1$, $\angle BCP = 15^\circ$. Find the length of side $BC$. | [
"\\( \\sqrt{3.11182} \\approx 1.764 \\)"
] | [
"A point $P$ is chosen on the side $AB$ of an acute-angled triangle $ABC$ so that $AP : BP = 2:1$. It is known that $AC = CP = 1$, $\\angle BCP = 15^\\circ$. Find the length of side $BC$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-966733211b55a573caf69ddf7bbd4a0aa194f81c5436d5feb6f083c933546aa6 | A point $P$ is chosen outside circle $O$ such that $\overrightarrow{PA}$ is tangent to $\odot O$ at point $A$ and $\overrightarrow{PC}$ intersects $\odot O$ at points $B$ and $C$ respectively. Point $D$ is chosen on arc $\widehat{AC}$ such that $\angle ADC=120^\circ$. Given that $\angle APB=30^\circ$, $AD=6$, $PB=4.58$... | [
"\\(\\sqrt{171}\\)"
] | [
"A point $P$ is chosen outside circle $O$ such that $\\overrightarrow{PA}$ is tangent to $\\odot O$ at point $A$ and $\\overrightarrow{PC}$ intersects $\\odot O$ at points $B$ and $C$ respectively. Point $D$ is chosen on arc $\\widehat{AC}$ such that $\\angle ADC=120^\\circ$. Given that $\\angle APB=30^\\circ$, $AD... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 4
} | 16 | 0 | 0 |
omr-ws-78e8c7c2235c949aacebe0b895f538f408b8a5f850508a7982b5f2e2725ffa73 | A point $P$ is chosen randomly in the interior of a square $ABCD$. What is the probability that the angle $\angle APB$ is obtuse? | [
"\\(\\frac{\\pi}{8}\\)"
] | [
"A point $P$ is chosen randomly in the interior of a square $ABCD$. What is the probability that the angle $\\angle APB$ is obtuse?"
] | [
"aops_c4_high_school_math"
] | [
"genselect",
"tir"
] | {
"genselect": 8,
"tir": 27
} | 35 | 0 | 0 |
omr-ws-0bacb23380aeeb98e25f685f52214a09bea533b5edcdadec633713ba1c31adbb | A point $P$ is chosen randomly inside a triangle with sides $13$, $20$, and $21$. Find the probability that the circle centered at $P$ with radius $1$ will intersect at least one of the sides of the triangle. | [
"\\(\\frac{31921}{95256}\\)"
] | [
"A point $P$ is chosen randomly inside a triangle with sides $13$, $20$, and $21$. Find the probability that the circle centered at $P$ with radius $1$ will intersect at least one of the sides of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b3ec78d41c20bda7a84e65081cf55c6f50d3344de71076c80de32a8653293b8e | A point $P$ is considered on the incircle of triangle $ABC$. Tangent segments are drawn from $P$ to the three excircles of $ABC$. If the obtained three tangent segments can form a right triangle, calculate the sum of the squares of the lengths of the segments from $P$ to the midpoints of the sides of $ABC$. | [
"6"
] | [
"A point $P$ is considered on the incircle of triangle $ABC$. Tangent segments are drawn from $P$ to the three excircles of $ABC$. If the obtained three tangent segments can form a right triangle, calculate the sum of the squares of the lengths of the segments from $P$ to the midpoints of the sides of $ABC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6ab874d6e06e34b8c956ace48b1515448a06b13adf7b97876000fb77ef0aa404 | A point $P$ is constructed outside of square $ABCD$ such that $AP = 4$, $BP = 3$, and $\angle APB = 90^\circ$. The angle bisector of $\angle APB$ intersects sides $AB$ and $DC$ at points $E$ and $F$ respectively. The area of quadrilateral $EBCF$ can be expressed as $\frac{m}{n}$. Find $m+n$. | [
"27"
] | [
"A point $P$ is constructed outside of square $ABCD$ such that $AP = 4$, $BP = 3$, and $\\angle APB = 90^\\circ$. The angle bisector of $\\angle APB$ intersects sides $AB$ and $DC$ at points $E$ and $F$ respectively. The area of quadrilateral $EBCF$ can be expressed as $\\frac{m}{n}$. Find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-4ef091c5c359a619cb5566a19262f9d2751a938a6454321e3b476cb7d9dbc68d | A point $P$ is fixed inside a circle with radius $R$. $C$ is an arbitrary point on the circle, and $AB$ is a chord passing through $P$ and perpendicular to the segment $BC$. Points $X$ and $Y$ are the projections of point $B$ onto lines $AC$ and $BC$, respectively. Let $r$ be the radius of the circle to which all line ... | [
"\\dfrac{R^2 - d^2}{2R}"
] | [
"A point $P$ is fixed inside a circle with radius $R$. $C$ is an arbitrary point on the circle, and $AB$ is a chord passing through $P$ and perpendicular to the segment $BC$. Points $X$ and $Y$ are the projections of point $B$ onto lines $AC$ and $BC$, respectively. Let $r$ be the radius of the circle to which all ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-e68a708c5114c298f430b6cff8e0d7dbb52c644dff2fa56cbbf7ac8368454533 | 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",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-97dea6388e26a5799aad5094a642d2fd2bea3912bae40e0c86f56e0b68bd04e7 | A point $P$ is given on the side $AB$ of a convex quadrilateral $ABCD$. Points $I$, $J$, $K$ are the centers of the inscribed circles of the triangles $APD$, $DPC$, $PCB$ respectively. If $AB = 10$, $BC = 15$, $CD = 20$, $DA = 25$, and $AP = 4$, find the perimeter of quadrilateral $PIJK$. | [
"35"
] | [
"A point $P$ is given on the side $AB$ of a convex quadrilateral $ABCD$. Points $I$, $J$, $K$ are the centers of the inscribed circles of the triangles $APD$, $DPC$, $PCB$ respectively. If $AB = 10$, $BC = 15$, $CD = 20$, $DA = 25$, and $AP = 4$, find the perimeter of quadrilateral $PIJK$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-f394cae4437b568f363bd0880180ec129a40064ff8038e6088e5e089076134a8 | A point $P$ is placed in equilateral triangle $\triangle ABC$ such that $\angle APB = 100^\circ$, $\angle BPC = 120^\circ$, and $\angle APC = 140^\circ$. A triangle $\triangle XYZ$ is constructed such that $AP = XY$, $BP = XZ$, and $CP = YZ$. Find the measure of the largest angle of $\triangle XYZ$. | [
"80 degrees"
] | [
"A point $P$ is placed in equilateral triangle $\\triangle ABC$ such that $\\angle APB = 100^\\circ$, $\\angle BPC = 120^\\circ$, and $\\angle APC = 140^\\circ$. A triangle $\\triangle XYZ$ is constructed such that $AP = XY$, $BP = XZ$, and $CP = YZ$. Find the measure of the largest angle of $\\triangle XYZ$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 19
} | 40 | 0 | 0 |
omr-ws-ce779d598636ce9e487789a3c66f221d39d51e90bd0d470e4358036d574a0922 | A point $P$ is randomly chosen on side $AB$ of regular hexagon $ABCDEF$. A point $Q$ is randomly chosen on side $BC$ of the same hexagon. Find the probability that $PQ > AB$. | [
"1 - \\dfrac{\\sqrt{3}}{9} \\pi"
] | [
"A point $P$ is randomly chosen on side $AB$ of regular hexagon $ABCDEF$. A point $Q$ is randomly chosen on side $BC$ of the same hexagon. Find the probability that $PQ > AB$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c7aac1716ce3558d5a80eccfed6abdf05f214bf01d132052f3949d93c3080f85 | 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 $(3,1)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-667897ab1afb8031088480898e8e98346acd3e8a10f153381e89ddcd2645e1ce | 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": 29
} | 45 | 0 | 0 |
omr-ws-74f182861b4bfcba49dcc5d4b0203f4feb0e1cc57648051a5fb00387f022efd1 | A point $P$ is randomly selected from the square region with vertices at $(\pm 2, \pm 2)$. What is the probability that $P$ is within one unit of the origin? Express your answer as a common fraction in terms of $\pi$. | [
"\\frac{\\pi}{16}"
] | [
"A point $P$ is randomly selected from the square region with vertices at $(\\pm 2, \\pm 2)$. What is the probability that $P$ is within one unit of the origin? Express your answer as a common fraction in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e37673852155d52bd1e18062b04373e98ddea54673b04bb75791d1504ff23ba5 | A point $P$ is randomly selected inside an equilateral triangle $ABC$ with side length $2$. Triangle $ABC$ is dilated centered at its orthocenter by a scale factor of $\frac{1}{4}$ to get triangle $A'B'C'$. Let $D, E$, and $F$ be the feet of the perpendiculars from $P$ to lines $A'B', B'C'$, and $C'A'$ respectively. Fi... | [
"\\dfrac{3}{4}"
] | [
"A point $P$ is randomly selected inside an equilateral triangle $ABC$ with side length $2$. Triangle $ABC$ is dilated centered at its orthocenter by a scale factor of $\\frac{1}{4}$ to get triangle $A'B'C'$. Let $D, E$, and $F$ be the feet of the perpendiculars from $P$ to lines $A'B', B'C'$, and $C'A'$ respective... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-254544de3046e2a2fb42edb69431ad638616bc37291b58ae9b751999d6aa8550 | A point $P$ is selected at random from the interior of the pentagon with the 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 the 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",
"tir"
] | {
"cot": 5,
"tir": 8
} | 13 | 0 | 0 |
omr-ws-ae1df27cc2d643ce4a2ca0ddf0c4e0e87ab0ab58949ae0f8e612c278f13c4913 | A point $P$ is selected at random inside square $ABCD$. What is the probability that $AP + BP < DP$? | [
"\\dfrac{1}{8}"
] | [
"A point $P$ is selected at random inside square $ABCD$. What is the probability that $AP + BP < DP$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-cfe0efb2c1f85ba6e5b4d5a92a478827dbbc422c57a5d8b267d8caeffc458e12 | A point $P$ is selected inside the triangle $ABC$. The lines $PA$, $PB$, and $PC$ intersect the sides $BC$, $CA$, and $AB$ of the triangle at the points $A'$, $B'$, and $C'$ respectively. Let $x = \frac{BA'}{BC}$, $y = \frac{CB'}{CA}$, and $z = \frac{AC'}{AB}$. If $x + y + z = \frac{3}{2}$, find the value of $k$ such t... | [
"\\dfrac{1}{3}"
] | [
"A point $P$ is selected inside the triangle $ABC$. The lines $PA$, $PB$, and $PC$ intersect the sides $BC$, $CA$, and $AB$ of the triangle at the points $A'$, $B'$, and $C'$ respectively. Let $x = \\frac{BA'}{BC}$, $y = \\frac{CB'}{CA}$, and $z = \\frac{AC'}{AB}$. If $x + y + z = \\frac{3}{2}$, find the value of $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-d25772e9cfb2b48e936d64d3efc438fc1ae74b191d02d0b3651790708ef5ac8e | 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 the angles of triangle $ABC$. | [
"\\(\\alpha = 30^\\circ\\), \\(\\beta = 60^\\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 $Q$ is symmetric to $P$ with respect to $AC$ and lies on the circumcircle of triangle $ABC$. Find the angles of triangle $ABC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 18
} | 18 | 0 | 0 |
omr-ws-0d1543a0290fd2e9ce16f577d7ee19aeb06d6cfa2e4233d9d50b9d58fe6d0408 | A point $P$ lies in the interior of a tetrahedron $ABCD$. Given that $AP=1$, $BP=\sqrt{2}$, $CP=\sqrt{3}$, and $DP=2$, and that $\angle APD = \angle BPD = \angle CPD = 120^\circ$, find the length of $AB$ expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square o... | [
"111"
] | [
"A point $P$ lies in the interior of a tetrahedron $ABCD$. Given that $AP=1$, $BP=\\sqrt{2}$, $CP=\\sqrt{3}$, and $DP=2$, and that $\\angle APD = \\angle BPD = \\angle CPD = 120^\\circ$, find the length of $AB$ expressed as $a + b\\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c2705469c4750c3dca825a430fd201c77653a91965f8c1483d33d13421b03d9a | A point $P$ lies in the interior of a tetrahedron $ABCD$. Suppose that $AP=1,BP=\sqrt{2},CP=\sqrt{3}$ and $DP=2$. Furthermore, $\angle APD=\angle BPD=\angle CPD=120^\circ$. If the length of $AB$ can be expressed as $a+b\sqrt{c}$, where $a,b,c$ are positive integers, and $c$ is not divisible by the square of any prime, ... | [
"no_valid_answer_found"
] | [
"A point $P$ lies in the interior of a tetrahedron $ABCD$. Suppose that $AP=1,BP=\\sqrt{2},CP=\\sqrt{3}$ and $DP=2$. Furthermore, $\\angle APD=\\angle BPD=\\angle CPD=120^\\circ$. If the length of $AB$ can be expressed as $a+b\\sqrt{c}$, where $a,b,c$ are positive integers, and $c$ is not divisible by the square of... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7ea707b95c79936c5dd68cf464992770dcfecad907988bced4bcdbd6b5865ca7 | A point $P$ lies inside $\triangle ABC$. The lines $BP$ and $CP$ intersect $AC$ and $AB$ at points $Q$ and $R$, respectively. Given that $AR = RB = CP$ and $CQ = PQ$, find $\angle BRC$. | [
"\\(120^\\circ\\)"
] | [
"A point $P$ lies inside $\\triangle ABC$. The lines $BP$ and $CP$ intersect $AC$ and $AB$ at points $Q$ and $R$, respectively. Given that $AR = RB = CP$ and $CQ = PQ$, find $\\angle BRC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-5776761a674ff33d56079d68d749fe84407251d0edd78d86e15074bb0d3bc046 | A point $P$ lies inside a square. The distances from $P$ to three of the corners are $2$, $3$, and $4$. What is the area of the square? | [
"\\(10 + 3\\sqrt{7}\\)"
] | [
"A point $P$ lies inside a square. The distances from $P$ to three of the corners are $2$, $3$, and $4$. What is the area of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-952d95733ec0d08bc416343987731ca5c16b3b9ccb818361b9fa0d5d7a714383 | A point $P$ lies inside a triangle $ABC$. The points $K$ and $L$ are the projections of $P$ onto $AB$ and $AC$, respectively. The point $M$ lies on the line $BC$ so that $KM = LM$, and the point $P'$ is symmetric to $P$ with respect to $M$. If $\angle BAP = \theta$ and $\angle P'AC = \phi$, find the value of $\theta - ... | [
"0"
] | [
"A point $P$ lies inside a triangle $ABC$. The points $K$ and $L$ are the projections of $P$ onto $AB$ and $AC$, respectively. The point $M$ lies on the line $BC$ so that $KM = LM$, and the point $P'$ is symmetric to $P$ with respect to $M$. If $\\angle BAP = \\theta$ and $\\angle P'AC = \\phi$, find the value of $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eca67333650465df01dfc9d4b0618df35916c2c6726942e3f8a16dba45504672 | A point $P$ lies inside an equilateral triangle $ABC$ such that $AP=15$ and $BP=8$. Find the maximum possible value of the sum of the areas of triangles $ABP$ and $BCP$. | [
"\\( 4(4\\sqrt{3} + \\sqrt{209}) \\)"
] | [
"A point $P$ lies inside an equilateral triangle $ABC$ such that $AP=15$ and $BP=8$. Find the maximum possible value of the sum of the areas of triangles $ABP$ and $BCP$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-225591382bdc01989275e5daefe9fecb570bdd2ac0b9dbf232329d6b9e0fc342 | A point $P$ lies inside equilateral triangle $ABC$. Let $A'$, $B'$, $C'$ be the feet of the perpendiculars from $P$ to $BC$, $AC$, $AB$, respectively. Suppose that $PA = 13$, $PB = 14$, and $PC = 15$. Find the area of $A'B'C'$. | [
"63"
] | [
"A point $P$ lies inside equilateral triangle $ABC$. Let $A'$, $B'$, $C'$ be the feet of the perpendiculars from $P$ to $BC$, $AC$, $AB$, respectively. Suppose that $PA = 13$, $PB = 14$, and $PC = 15$. Find the area of $A'B'C'$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-441a1567a6ecd2f10dcf14050c75f2cf1c10e569983c0a0376261cbe85567605 | A point $P$ lies inside square $ABCD$ such that $PA = 11, PB = 22,$ and $PC = 33.$ The area of $ABCD$ can be expressed in the form $a + b\sqrt{c}$, where $a, b,$ and $c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a + b + c$. | [
"849"
] | [
"A point $P$ lies inside square $ABCD$ such that $PA = 11, PB = 22,$ and $PC = 33.$ The area of $ABCD$ can be expressed in the form $a + b\\sqrt{c}$, where $a, b,$ and $c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a + b + c$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-ed2a407ac93d5c31aad461eb8aa8552de730acaf8b19c5ff6005accbf11600cf | A point $P$ lies on side $AB$ of a convex quadrilateral $ABCD$. Let $\omega$ be the inscribed circle of triangle $CPD$ and $I$ the center of $\omega$. It is known that $\omega$ is tangent to the inscribed circles of triangles $APD$ and $BPC$ at points $K$ and $L$ respectively. Let $E$ be the intersection point of lines... | [
"-1"
] | [
"A point $P$ lies on side $AB$ of a convex quadrilateral $ABCD$. Let $\\omega$ be the inscribed circle of triangle $CPD$ and $I$ the center of $\\omega$. It is known that $\\omega$ is tangent to the inscribed circles of triangles $APD$ and $BPC$ at points $K$ and $L$ respectively. Let $E$ be the intersection point ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-c5092f269b2f90c810c2479adb8354a5735087e3c2d53fbcabd2318f58aa251c | A point $P$ lies outside a circle with center $O$. A secant from $P$ intersects the circle at points $Q$ and $R$ and has a length of $48$. A tangent from $P$ to the circle has a length that is $\frac{2}{3}$ the length of $QR$. If the distance from $O$ to the secant is $24$, find the radius of the circle. | [
"30"
] | [
"A point $P$ lies outside a circle with center $O$. A secant from $P$ intersects the circle at points $Q$ and $R$ and has a length of $48$. A tangent from $P$ to the circle has a length that is $\\frac{2}{3}$ the length of $QR$. If the distance from $O$ to the secant is $24$, find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-a922c0c7e3ea3465823b998f615f8486600e8aba4036ff685d82b1af4a3f1990 | A point $P$ moves along a circle $\Omega$. Let $A$ and $B$ be two fixed points on $\Omega$, and $C$ be an arbitrary point inside $\Omega$. The common external tangents to the circumcircles of triangles $APC$ and $BCP$ meet at point $Q$. If the equations of the two fixed lines on which all points $Q$ lie are $y = m_1x +... | [
"2"
] | [
"A point $P$ moves along a circle $\\Omega$. Let $A$ and $B$ be two fixed points on $\\Omega$, and $C$ be an arbitrary point inside $\\Omega$. The common external tangents to the circumcircles of triangles $APC$ and $BCP$ meet at point $Q$. If the equations of the two fixed lines on which all points $Q$ lie are $y ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-690d2706caef861e7ffb8b1adb8463c73312b241d9e82a8225584d574ce84562 | A point $P$ moves such that the sum of the distances from $P$ to the coordinate axes is equal to the distance from $P$ to the point $(1, 1)$. Determine the equation of the locus of $P$. | [
"|xy| + x + y = 1"
] | [
"A point $P$ moves such that the sum of the distances from $P$ to the coordinate axes is equal to the distance from $P$ to the point $(1, 1)$. Determine the equation of the locus of $P$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-b8c12b63031f85a5a2077759c160144c37c2c2ff316f2153ee6e64174f9fbc40 | A point $P$ on the circumcircle of triangle $ABC$ is projected onto the sides $[BC]$ and $[AC]$ at points $D$ and $E$ respectively. Let $L$ and $M$ be the midpoints of segments $[BD]$ and $[BE]$ respectively. If $LM$ intersects $DE$ at point $N$, find the ratio $\frac{LN}{NM}$ when $AB = 13$, $BC = 14$, and $AC = 15$. | [
"1"
] | [
"A point $P$ on the circumcircle of triangle $ABC$ is projected onto the sides $[BC]$ and $[AC]$ at points $D$ and $E$ respectively. Let $L$ and $M$ be the midpoints of segments $[BD]$ and $[BE]$ respectively. If $LM$ intersects $DE$ at point $N$, find the ratio $\\frac{LN}{NM}$ when $AB = 13$, $BC = 14$, and $AC =... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-26e505c66f51905d419a650fc4d29c6e7e0d39ab7efa0a9559f2c0c63f867f79 | A point $P$ on the ellipse $\frac{x^2}{4} + y^2 = 1$ is a distance of $3.4$ units from one focus and $0.6$ units from the other focus when rounded to the nearest tenth. The smallest number greater than the distance from $P$ to the origin can be written as $\frac{\sqrt{a}}{b}$. Find $a + b \pmod{1000}$. | [
"79"
] | [
"A point $P$ on the ellipse $\\frac{x^2}{4} + y^2 = 1$ is a distance of $3.4$ units from one focus and $0.6$ units from the other focus when rounded to the nearest tenth. The smallest number greater than the distance from $P$ to the origin can be written as $\\frac{\\sqrt{a}}{b}$. Find $a + b \\pmod{1000}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-130c4b09eed88a4ea062a3e65214c08e4a100d798871a993ec00dce024478545 | A point $Q$ (distinct from $P$) is chosen on the major arc $AB$ of $C_{1}$. Lines $QA$ and $QB$ are drawn to intersect $C_{2}$ at $F$ and $G$ respectively. Given that $FG$ is a diameter of $C_{2}$, calculate the coordinates of the intersection point of lines $FB$ and $GA$ if $A = (1,0)$, $B = (-1,0)$, and the center of... | [
"(0,0)"
] | [
"A point $Q$ (distinct from $P$) is chosen on the major arc $AB$ of $C_{1}$. Lines $QA$ and $QB$ are drawn to intersect $C_{2}$ at $F$ and $G$ respectively. Given that $FG$ is a diameter of $C_{2}$, calculate the coordinates of the intersection point of lines $FB$ and $GA$ if $A = (1,0)$, $B = (-1,0)$, and the cent... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-e5e2b9c95bcd1e9d6563b9698d9a0db2c17b558b2d727b11d8f00bf9b182d095 | A point $Q$ is chosen on a circle $\omega$ with diameter $AB$. The point $H$ is the projection of $Q$ onto $AB$. The circle with center $Q$ and radius $QH$ intersects $\omega$ at points $C$ and $D$. Let $M$ be the midpoint of $QH$. If $AB = 10$ and $AQ = 6$, find the length of $CM$. | [
"\\dfrac{12}{5}"
] | [
"A point $Q$ is chosen on a circle $\\omega$ with diameter $AB$. The point $H$ is the projection of $Q$ onto $AB$. The circle with center $Q$ and radius $QH$ intersects $\\omega$ at points $C$ and $D$. Let $M$ be the midpoint of $QH$. If $AB = 10$ and $AQ = 6$, find the length of $CM$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-8c956fc311141225219159a8d2bcbc7a07438460b820e075d69e570f4caa1efa | A point $T$ is randomly chosen inside square $WXYZ$. What is the probability that neither quadrilateral $WXYT$ nor $YZWT$ has an area that is at least twice the area of the other? | [
"\\(\\frac{5}{9}\\)"
] | [
"A point $T$ is randomly chosen inside square $WXYZ$. What is the probability that neither quadrilateral $WXYT$ nor $YZWT$ has an area that is at least twice the area of the other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-26a48754e6dd1eaef9f74b6385f21ab9d0d1bdd7afc3d9a8adfbef16096335a1 | A point $X$ is in the interior of isosceles right triangle $ABC$, where $C$ is the right angle. Given that $CX=3$, $AX=4$, and $BX=5$, find the area of the triangle $ABC$ in the form $a+\sqrt{b}$ where $a$ and $b$ are rational numbers. | [
"\\dfrac{41 + \\sqrt{1071}}{4}"
] | [
"A point $X$ is in the interior of isosceles right triangle $ABC$, where $C$ is the right angle. Given that $CX=3$, $AX=4$, and $BX=5$, find the area of the triangle $ABC$ in the form $a+\\sqrt{b}$ where $a$ and $b$ are rational numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d785805b5abb5df19f6b213fd984d4777b2791905483f5a796cb7fda76c65b63 | A point $X$ lies in the plane of triangle $ABC$. A circle $\Gamma$ passing through $X$ intersects $XA$, $XB$, and $XC$ again at $P$, $Q$, and $R$ respectively. Let $\Gamma$ intersect the circles $BXC$, $AXC$, and $AXB$ again at $T$, $L$, and $M$ respectively. If the coordinates of $P$, $Q$, $R$, $T$, $L$, and $M$ are s... | [
"0"
] | [
"A point $X$ lies in the plane of triangle $ABC$. A circle $\\Gamma$ passing through $X$ intersects $XA$, $XB$, and $XC$ again at $P$, $Q$, and $R$ respectively. Let $\\Gamma$ intersect the circles $BXC$, $AXC$, and $AXB$ again at $T$, $L$, and $M$ respectively. If the coordinates of $P$, $Q$, $R$, $T$, $L$, and $M... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-46a4d4eafc7cb82d521df3381ee63926b6fde12278847f6d581e475141d16d10 | A point 'O' is situated on a circle of radius \( R \). With center 'O', another circle of radius \( \frac{3R}{2} \) is described. Inside the crescent-shaped area intercepted between these circles, a circle of radius \( \frac{R}{8} \) is placed. If the same circle moves in contact with the original circle of radius \( R... | [
"\\dfrac{9R}{4} \\arccos\\left(\\dfrac{1}{6}\\right)"
] | [
"A point 'O' is situated on a circle of radius \\( R \\). With center 'O', another circle of radius \\( \\frac{3R}{2} \\) is described. Inside the crescent-shaped area intercepted between these circles, a circle of radius \\( \\frac{R}{8} \\) is placed. If the same circle moves in contact with the original circle o... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-3e2f3dc013a200ee25513da0be528e6686b4a82dfe6bd70f313b8fa5d66ad9d9 | A point 'P' is chosen randomly inside a regular hexagon. What is the probability that the point will lie closer to the centroid of the hexagon than to the sides of the hexagon? | [
"\\(\\frac{16\\sqrt{3}-27}{3}\\)"
] | [
"A point 'P' is chosen randomly inside a regular hexagon. What is the probability that the point will lie closer to the centroid of the hexagon than to the sides of the hexagon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 3
} | 3 | 0 | 0 |
omr-ws-f2b24083ecd6a28e482f7af0f4ad5db3570b2f114fd7f462450c651140d40d20 | A point A is on a circle of center O and radius 1 with diameter RS such that ∠AOS = 90°. Take a second circle with diameter AO, suppose tangents to this circle from R and S intersect at T and intersect the first circle at R' and S'. Then suppose the triangularish section TR'AS' is inscribed with a circle of radius r. F... | [
"\\dfrac{49 - 21\\sqrt{2}}{155}"
] | [
"A point A is on a circle of center O and radius 1 with diameter RS such that ∠AOS = 90°. Take a second circle with diameter AO, suppose tangents to this circle from R and S intersect at T and intersect the first circle at R' and S'. Then suppose the triangularish section TR'AS' is inscribed with a circle of radius... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36b5abe0eb5fd6d21fac0eb82eca1bdd04d90504bac973d5762790c882c9f467 | A point A is randomly chosen in a square of side length 1. Find the probability that the distance from A to the center of the square does not exceed \( x \). | [
"\\( p = \\begin{cases} \n\\pi x^2, & \\text{for } 0 \\le x \\le \\dfrac{1}{2}; \\\\ \n\\sqrt{4x^2-1} + \\pi x^2 - 4x^2 \\arccos \\dfrac{1}{2x}, & \\text{for } \\dfrac{1}{2} < x \\le \\dfrac{\\sqrt{2}}{2}.\n\\end{cases} \\)"
] | [
"A point A is randomly chosen in a square of side length 1. Find the probability that the distance from A to the center of the square does not exceed \\( x \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-ab8e5850a90c1d9e2cdb2faf9298fe3569655ab332f5f97c7dc2de4b6448c002 | A point A is taken outside a circle of radius R. Two secants are drawn from this point: one passes through the center, and the other is at a distance of R/2 from the center. Find the area of the circular region enclosed between the two secants. | [
"\\(\\frac{1}{6}\\pi R^2 + \\frac{\\sqrt{3}}{4}R^2\\)"
] | [
"A point A is taken outside a circle of radius R. Two secants are drawn from this point: one passes through the center, and the other is at a distance of R/2 from the center. Find the area of the circular region enclosed between the two secants."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-7f00fff82a6c6b7f9a11248fb576c84bf8d4699f12f16414ec7325ddddf05764 | A point D is located inside an equilateral triangle ABC such that AD = 7, BD = 9, and CD = 10. Find the area of the equilateral triangle ABC, rounded to the nearest whole number. | [
"118"
] | [
"A point D is located inside an equilateral triangle ABC such that AD = 7, BD = 9, and CD = 10. Find the area of the equilateral triangle ABC, rounded to the nearest whole number."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f296bdf40088761925082f69e36429017fa0dda7a95d9ffac81da96ea0a1cff4 | A point D is on side BC of triangle ABC such that BD = 20 and DC = 14. A circle Γ centered at D is tangent to rays AB and AC. Line l is a tangent of Γ that is parallel to BC, and let l intersect AB at X and AC at Y. How many different integer lengths can XY take on? | [
"27"
] | [
"A point D is on side BC of triangle ABC such that BD = 20 and DC = 14. A circle Γ centered at D is tangent to rays AB and AC. Line l is a tangent of Γ that is parallel to BC, and let l intersect AB at X and AC at Y. How many different integer lengths can XY take on?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-856b11ac01306a8aa9087b09cda5e9a3a982234a46d65f314223710fce7b0a55 | A point D lies inside triangle ABC. Let A1, B1, C1 be the second intersection points of the lines AD, BD, and CD with the circumcircles of BDC, CDA, and ADB, respectively. If AD = 6, BD = 8, CD = 10, AA1 = 18, BB1 = 24, and CC1 = 30, calculate the value of \(\frac{AD}{AA1} + \frac{BD}{BB1} + \frac{CD}{CC1}\). | [
"1"
] | [
"A point D lies inside triangle ABC. Let A1, B1, C1 be the second intersection points of the lines AD, BD, and CD with the circumcircles of BDC, CDA, and ADB, respectively. If AD = 6, BD = 8, CD = 10, AA1 = 18, BB1 = 24, and CC1 = 30, calculate the value of \\(\\frac{AD}{AA1} + \\frac{BD}{BB1} + \\frac{CD}{CC1}\\).... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37cdf221055b67fe308d0d566915197550eef728b55040851e7ce4f78e2d1369 | A point M is chosen inside the square ABCD in such a way that angle MAC = angle MCD = x. Determine angle ABM. | [
"\\(90 - 2x\\)"
] | [
"A point M is chosen inside the square ABCD in such a way that angle MAC = angle MCD = x. Determine angle ABM."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f98939374c02becd5486f20fbf3da3a87f201f156f6e0023b79839f162b8a0a9 | A point M is chosen inside the square ABCD such that angle MAC = angle MCD = x. Express angle ABM in terms of x. | [
"\\( 90^\\circ - 2x \\)"
] | [
"A point M is chosen inside the square ABCD such that angle MAC = angle MCD = x. Express angle ABM in terms of x."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-d00d414d4a0a853e9d7ee935725a5184a82c5df2eaf22918b203f465c6ef23f9 | A point P is chosen in the interior of triangle ABC such 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 of 4, 9, and 49, respectively. Find the area of triangle ABC. | [
"144"
] | [
"A point P is chosen in the interior of triangle ABC such 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 of 4, 9, and 49, respectively. Find the area of triangle ABC."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-dc4520728891617d603f1dacf4e6dd420e849c175ec74826989d7563e5b1bb58 | A point P is chosen randomly inside a triangle with sides 13, 20, and 21. Find the probability that the circle centered at P with radius 1 will intersect at least one of the sides of the triangle. | [
"\\(\\frac{75}{196}\\)"
] | [
"A point P is chosen randomly inside a triangle with sides 13, 20, and 21. Find the probability that the circle centered at P with radius 1 will intersect at least one of the sides of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 22
} | 35 | 0 | 0 |
omr-ws-d7b960856da53a20c9271cc1562dd2a6ebd9990eee136f3c1334196d15c55372 | A point P is given inside a triangle ABC. A line l through P, not containing A, intersects AB and AC at Q and R, respectively. Find the line l such that the perimeter of △AQR is minimized. | [
"\\text{The line joining the reflections of } P \\text{ across } AB \\text{ and } AC"
] | [
"A point P is given inside a triangle ABC. A line l through P, not containing A, intersects AB and AC at Q and R, respectively. Find the line l such that the perimeter of △AQR is minimized."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-d94381d8b9b4ad66f1a4e68be4fd3b134267ffa24d2516c5b1920987e4145911 | A point P is given on the circumference of a circle of radius r. The chord QR is parallel to the tangent line at P. Find the maximum area of triangle PQR. | [
"\\(\\frac{3 \\sqrt{3}}{4} r^2\\)"
] | [
"A point P is given on the circumference of a circle of radius r. The chord QR is parallel to the tangent line at P. Find the maximum area of triangle PQR."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-7dddb2520b8c38e38f6073116644e787f7e4eeb98119c5c2a915c059d96629e2 | A point P is inside an equilateral triangle ABC such that PA = x, PB = y, and PC = z. Determine the area of the triangle in terms of x, y, and z. | [
"\\dfrac{\\sqrt{3}}{8} \\left( x^2 + y^2 + z^2 + \\sqrt{3\\left(4(x^2 y^2 + y^2 z^2 + z^2 x^2) - (x^2 + y^2 + z^2)^2\\right)} \\right)"
] | [
"A point P is inside an equilateral triangle ABC such that PA = x, PB = y, and PC = z. Determine the area of the triangle in terms of x, y, and z."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-9f244f1ad32ccb1f9d72b4c51fc3d66a0825f3bc09d82429671c8f9b1b1d096c | A point P is marked inside a regular hexagon ABCDEF such that the angle BAP is equal to angle DCP = 50 degrees. Find the measure of angle ABP. | [
"80 degrees"
] | [
"A point P is marked inside a regular hexagon ABCDEF such that the angle BAP is equal to angle DCP = 50 degrees. Find the measure of angle ABP."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-4ca3af80a0724a46d1e206c0cd43f02dcef8bbcb7330413aa384bfa802f7a070 | A point P lies inside a square ABCD with side length 120. Let F be a point on CD such that PF is perpendicular to CD. If P is equidistant from A, B, and F, find the length of PA. | [
"75"
] | [
"A point P lies inside a square ABCD with side length 120. Let F be a point on CD such that PF is perpendicular to CD. If P is equidistant from A, B, and F, find the length of PA."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-3aca04a26bbc3cc1d66cb624c615e21037d467f9dd1ecc03651fede5918fb861 | A point T is located inside a rectangle ABCD. Point T is connected to each of the rectangle's vertices A, B, C, and D by segments TA, TB, TC, and TD. Let the lengths of these segments be a, b, c, and d respectively. Calculate the minimum possible value of the sum of the lengths of any three segments that can form a tri... | [
"2"
] | [
"A point T is located inside a rectangle ABCD. Point T is connected to each of the rectangle's vertices A, B, C, and D by segments TA, TB, TC, and TD. Let the lengths of these segments be a, b, c, and d respectively. Calculate the minimum possible value of the sum of the lengths of any three segments that can form ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-285c788bc2664e2ebab59d07cdb3bd17998cc785ff679bc3adcefeb2a328d99d | A point Y moves such that its distance from point B(1,2) is constant. Given that x=k is the tangent of the locus of Y, express the equation of the locus in terms of k. Hence, find the values of k if the locus of Y touches the x-axis. | [
"\\( k = 3 \\)"
] | [
"A point Y moves such that its distance from point B(1,2) is constant. Given that x=k is the tangent of the locus of Y, express the equation of the locus in terms of k. Hence, find the values of k if the locus of Y touches the x-axis."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-fcdf31a046b38ea32e0f6d639e4bf98c558ba3548837b6bab0e503ca93a6cb6c | A point \( A = (a, b) \) in the Cartesian plane is called visible if \( a, b \in \mathbb{Z} \) and there are no other points with integer coordinates on the line segment \( AO \) (where \( O \) is the origin). Find all visible points. | [
"\\(\\gcd(a, b) = 1\\)"
] | [
"A point \\( A = (a, b) \\) in the Cartesian plane is called visible if \\( a, b \\in \\mathbb{Z} \\) and there are no other points with integer coordinates on the line segment \\( AO \\) (where \\( O \\) is the origin). Find all visible points."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 10
} | 28 | 0 | 0 |
omr-ws-791c3b2cd51582a9ced8a31ef0b328b38738aebf17f7f2efb468d3ad935a565e | A point \( A \) is located at \( (3, 3\sqrt{3}) \). If the coordinate axes are rotated \( 150^\circ \) counterclockwise around the origin, what are the new coordinates of point \( A \)? | [
"\\((-3\\sqrt{3}, -3)\\)"
] | [
"A point \\( A \\) is located at \\( (3, 3\\sqrt{3}) \\). If the coordinate axes are rotated \\( 150^\\circ \\) counterclockwise around the origin, what are the new coordinates of point \\( A \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-6ffaab30bddd42113df4ade662eac7fff4c95f0ab40c5a11f7a1a7ffd4ebd6c3 | A point \( A \) lies on a circle \( \Omega \) centered at \( O \). Let \( X \) be the midpoint of \( \overline{AO} \). A point \( P \) distinct from \( A \) and \( O \) is selected on the circle with diameter \( \overline{AO} \), and line \( AP \) meets \( \Omega \) a second time at a point \( Q \). A point \( Y \) is ... | [
"90"
] | [
"A point \\( A \\) lies on a circle \\( \\Omega \\) centered at \\( O \\). Let \\( X \\) be the midpoint of \\( \\overline{AO} \\). A point \\( P \\) distinct from \\( A \\) and \\( O \\) is selected on the circle with diameter \\( \\overline{AO} \\), and line \\( AP \\) meets \\( \\Omega \\) a second time at a poi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-4bfffe46077eb340abba7e573c5fbdf01b36f6f94d5b54803b8745ea9d7cda7d | A point \( C \) is chosen on the line segment \( AB \) such that \( \frac{AC}{BC} = \frac{BC}{5AB} \). Find \( \frac{AC}{BC} \). | [
"\\dfrac{3\\sqrt{5} - 5}{10}"
] | [
"A point \\( C \\) is chosen on the line segment \\( AB \\) such that \\( \\frac{AC}{BC} = \\frac{BC}{5AB} \\). Find \\( \\frac{AC}{BC} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-00c3d43c16c3437f7b2a35ea438cb94b3ccb457ef33cfbc626da1c77cd89b9e2 | A point \( C \) is on a circle with diameter \( AB \), where \( \frac{AB}{CA} = 3 \). The angle bisector \( AI \) intersects \( BC \) and the circle at points \( N \) and \( P \) respectively. \( M \) is the midpoint of \( BC \) and \( AM \) intersects \( BP \) at \( Q \). Find the ratio \( \frac{AM}{MQ} \). | [
"2"
] | [
"A point \\( C \\) is on a circle with diameter \\( AB \\), where \\( \\frac{AB}{CA} = 3 \\). The angle bisector \\( AI \\) intersects \\( BC \\) and the circle at points \\( N \\) and \\( P \\) respectively. \\( M \\) is the midpoint of \\( BC \\) and \\( AM \\) intersects \\( BP \\) at \\( Q \\). Find the ratio \... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b415abff76cec57a197c55cbf062c32680c8b39b8abaac484000b40bae9a680f | A point \( D \) is chosen on side \( BC \) of a triangle \( ABC \) such that the inradii of triangles \( ABD \) and \( ACD \) are equal. Let \( r_1 \) and \( r_2 \) be the radii of the excircles of triangles \( ABD \) and \( ACD \) that touch sides \( BD \) and \( CD \), respectively. If \( AB = 13 \), \( AC = 15 \), a... | [
"12\\sqrt{3} - 12"
] | [
"A point \\( D \\) is chosen on side \\( BC \\) of a triangle \\( ABC \\) such that the inradii of triangles \\( ABD \\) and \\( ACD \\) are equal. Let \\( r_1 \\) and \\( r_2 \\) be the radii of the excircles of triangles \\( ABD \\) and \\( ACD \\) that touch sides \\( BD \\) and \\( CD \\), respectively. If \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6aa8bbb03411df52a970afc09e27ae8d9d6219e87e8d1c176c9948216e144aa4 | A point \( D \) lies inside \( \triangle ABC \). Let \( A_1, B_1, C_1 \) denote the second intersections of \( AD, BD, CD \) with the circumcircles of \( \triangle BDC, \triangle CDA, \triangle ADB \) respectively. If \( AD = 3, BD = 4, CD = 5, AA_1 = 9, BB_1 = 12, CC_1 = 15 \), find the value of \( \dfrac{AD}{AA_1} + ... | [
"1"
] | [
"A point \\( D \\) lies inside \\( \\triangle ABC \\). Let \\( A_1, B_1, C_1 \\) denote the second intersections of \\( AD, BD, CD \\) with the circumcircles of \\( \\triangle BDC, \\triangle CDA, \\triangle ADB \\) respectively. If \\( AD = 3, BD = 4, CD = 5, AA_1 = 9, BB_1 = 12, CC_1 = 15 \\), find the value of \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a882744a7b2d7f1d40e97054f08804664d0019d79caf0a3e0cc9e183b49a700c | A point \( D \) lies inside triangle \( ABC \). Let \( A_1, B_1, C_1 \) be the second intersection points of the lines \( AD, BD, \) and \( CD \) with the circumcircles of \( \triangle BDC, \triangle CDA, \) and \( \triangle ADB \), respectively. Calculate the value of: \[ \frac{AD}{AA_1} + \frac{BD}{BB_1} + \frac{CD}{... | [
"1"
] | [
"A point \\( D \\) lies inside triangle \\( ABC \\). Let \\( A_1, B_1, C_1 \\) be the second intersection points of the lines \\( AD, BD, \\) and \\( CD \\) with the circumcircles of \\( \\triangle BDC, \\triangle CDA, \\) and \\( \\triangle ADB \\), respectively. Calculate the value of:\n\\[\n\\frac{AD}{AA_1} + \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-af41d878e4c53b7c97c11d059e7743477e69f94943ab7edb4905d9595e3c4a4c | A point \( L \) is randomly selected inside circle \( \omega \). Six points \( A_1, A_2, A_3, B_1, B_2, B_3 \) (with \( A \)s and \( B \)s in clockwise order) lie on \( \omega \) such that \( \angle L A_1 A_2 = \angle L A_2 A_3 = \angle L B_1 B_2 = \angle L B_2 B_3 \). Let \( P \) be the intersection point of lines \( ... | [
"2"
] | [
"A point \\( L \\) is randomly selected inside circle \\( \\omega \\). Six points \\( A_1, A_2, A_3, B_1, B_2, B_3 \\) (with \\( A \\)s and \\( B \\)s in clockwise order) lie on \\( \\omega \\) such that \\( \\angle L A_1 A_2 = \\angle L A_2 A_3 = \\angle L B_1 B_2 = \\angle L B_2 B_3 \\). Let \\( P \\) be the inte... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-8d767f6a47b998e1bed9f29efebe7b5b217488ad01b688bcc7c67bf640678953 | A point \( M \) belongs to the small arc \( BC \) of the circumcircle of triangle \( ABC \). The tangents from \( M \) to the incircle of triangle \( ABC \) intersect the circumcircle at points \( N \) and \( P \). Given that \( \measuredangle NMP = \measuredangle BAC \), calculate the ratio of the areas of triangles \... | [
"1"
] | [
"A point \\( M \\) belongs to the small arc \\( BC \\) of the circumcircle of triangle \\( ABC \\). The tangents from \\( M \\) to the incircle of triangle \\( ABC \\) intersect the circumcircle at points \\( N \\) and \\( P \\). Given that \\( \\measuredangle NMP = \\measuredangle BAC \\), calculate the ratio of t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3265247b334c241ac120b55d154119d24b4c650afb807ee5b60e857056c98bc1 | A point \( M \) inside triangle \( ABC \) is such that \( AM = \frac{AB}{2} \) and \( CM = \frac{BC}{2} \). Points \( C_0 \) and \( A_0 \) lie on \( AB \) and \( CB \) respectively, such that \( \frac{BC_0}{AC_0} = \frac{BA_0}{CA_0} = 3 \). If \( AB = 12 \) and \( BC = 16 \), calculate the length of \( MC_0 \). | [
"3\\sqrt{2}"
] | [
"A point \\( M \\) inside triangle \\( ABC \\) is such that \\( AM = \\frac{AB}{2} \\) and \\( CM = \\frac{BC}{2} \\). Points \\( C_0 \\) and \\( A_0 \\) lie on \\( AB \\) and \\( CB \\) respectively, such that \\( \\frac{BC_0}{AC_0} = \\frac{BA_0}{CA_0} = 3 \\). If \\( AB = 12 \\) and \\( BC = 16 \\), calculate th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-5d6ad8e31910b4d5c86bc5adde8f4a5a4915363f2c7b757aa91209617cf9f252 | A point \( M \) is chosen inside square \( ABCD \) such that \( \angle MAC = \angle MCD = x \). Find \( \angle ABM \) in terms of \( x \). | [
"\\( 90^\\circ - 2x \\)"
] | [
"A point \\( M \\) is chosen inside square \\( ABCD \\) such that \\( \\angle MAC = \\angle MCD = x \\). Find \\( \\angle ABM \\) in terms of \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 2
} | 14 | 0 | 0 |
omr-ws-036be98588f11957a1f2480406ef5cefebfa19ffde4932eaff9d37095ff403cb | A point \( M \) is chosen on the side \( AC \) of triangle \( ABC \) such that the radii of the circles inscribed in triangles \( ABM \) and \( BMC \) are equal. If the area of triangle \( ABC \) is \( 50 \) square units, find the value of \( BM^2 \cot \left( \frac{B}{2} \right) \). | [
"100"
] | [
"A point \\( M \\) is chosen on the side \\( AC \\) of triangle \\( ABC \\) such that the radii of the circles inscribed in triangles \\( ABM \\) and \\( BMC \\) are equal. If the area of triangle \\( ABC \\) is \\( 50 \\) square units, find the value of \\( BM^2 \\cot \\left( \\frac{B}{2} \\right) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-bebeb98150644f72edca1f71bb5fa8c95dd99ec177aa01db6b7e4231a34e2f4d | A point \( M \) is found inside a convex quadrilateral \( ABCD \) such that triangles \( AMB \) and \( CMD \) are isosceles (\( AM = MB \), \( CM = MD \)) and \( \angle AMB = \angle CMD = 120^\circ \). Find the measure of \( \angle BNC \) if \( N \) is a point such that triangles \( BNC \) and \( DNA \) are equilateral... | [
"60"
] | [
"A point \\( M \\) is found inside a convex quadrilateral \\( ABCD \\) such that triangles \\( AMB \\) and \\( CMD \\) are isosceles (\\( AM = MB \\), \\( CM = MD \\)) and \\( \\angle AMB = \\angle CMD = 120^\\circ \\). Find the measure of \\( \\angle BNC \\) if \\( N \\) is a point such that triangles \\( BNC \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6bad00fcb198ce462869c1ddd6d977451d390031e526b8c3ec24960a1ebc752f | A point \( M \) is inside a convex quadrilateral \( ABCD \). Perpendiculars from \( M \) are dropped onto the sides of \( ABCD \), such that \( MN \) is perpendicular to \( AB \), \( MI \) is perpendicular to \( BC \), \( MH \) is perpendicular to \( CD \), and \( MK \) is perpendicular to \( DA \). Let \( S \) be the ... | [
"117"
] | [
"A point \\( M \\) is inside a convex quadrilateral \\( ABCD \\). Perpendiculars from \\( M \\) are dropped onto the sides of \\( ABCD \\), such that \\( MN \\) is perpendicular to \\( AB \\), \\( MI \\) is perpendicular to \\( BC \\), \\( MH \\) is perpendicular to \\( CD \\), and \\( MK \\) is perpendicular to \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-2496afe7c63d46b4e0ad718a877134b04c2707afcd52f55125a1b90b3851c0ca | A point \( M \) is uniformly distributed in the interior of the \( n-1 \) simplex \( A_1A_2\ldots A_n \) (where \( n \geq 3 \)). Find the probability that the numbers \( x_i = [A_1 \ldots A_{i-1} M A_{i+1} \ldots A_n] \) for \( i = 1, \ldots, n \) can be the lengths of the sides of a non-degenerate \( n \)-gon. | [
"1 - \\frac{n}{2^{n-1}}"
] | [
"A point \\( M \\) is uniformly distributed in the interior of the \\( n-1 \\) simplex \\( A_1A_2\\ldots A_n \\) (where \\( n \\geq 3 \\)). Find the probability that the numbers \\( x_i = [A_1 \\ldots A_{i-1} M A_{i+1} \\ldots A_n] \\) for \\( i = 1, \\ldots, n \\) can be the lengths of the sides of a non-degenerat... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f66078ab429d8c5f3bfd1f9ce12358f342860a501accba7a7e62e52622a393da | A point \( M \) lies on the interior of \( \triangle ABC \) such that \( \angle MAB = 10^\circ \), \( \angle MBA = 20^\circ \), \( \angle MCA = 30^\circ \), and \( \angle MAC = 40^\circ \). Find the measure of \( \angle MBC \). | [
"\\( 60^\\circ \\)"
] | [
"A point \\( M \\) lies on the interior of \\( \\triangle ABC \\) such that \\( \\angle MAB = 10^\\circ \\), \\( \\angle MBA = 20^\\circ \\), \\( \\angle MCA = 30^\\circ \\), and \\( \\angle MAC = 40^\\circ \\). Find the measure of \\( \\angle MBC \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 22
} | 45 | 0 | 0 |
omr-ws-1fc439f6202d269cff0783082902da6943c0dfd8073e72a75b69bcb2932943bd | A point \( M \) lies on the side \( AC \) of triangle \( ABC \). The circle \( C \) with diameter \( BM \) intersects the lines \( AB \) and \( BC \) at points \( P \) and \( Q \), respectively. Find the locus of the intersection points of the tangents to the circle \( C \) at points \( P \) and \( Q \) as the point \(... | [
"The symmedian of \\( AC \\)."
] | [
"A point \\( M \\) lies on the side \\( AC \\) of triangle \\( ABC \\). The circle \\( C \\) with diameter \\( BM \\) intersects the lines \\( AB \\) and \\( BC \\) at points \\( P \\) and \\( Q \\), respectively. Find the locus of the intersection points of the tangents to the circle \\( C \\) at points \\( P \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e625f05b183981eb8a1fccf61302efac1597edc138266aa1260e888a90d87a7 | A point \( O \) inside an equilateral triangle \( PQR \) has perpendiculars \( OL \), \( OM \), and \( ON \) drawn to the sides \( PQ \), \( QR \), and \( RP \) respectively. The ratios of the lengths of the perpendiculars \( OL:OM:ON \) is \( 1:2:3 \). If \( \frac{\text{area of } LONP}{\text{area of } \Delta PQR} = \f... | [
"47"
] | [
"A point \\( O \\) inside an equilateral triangle \\( PQR \\) has perpendiculars \\( OL \\), \\( OM \\), and \\( ON \\) drawn to the sides \\( PQ \\), \\( QR \\), and \\( RP \\) respectively. The ratios of the lengths of the perpendiculars \\( OL:OM:ON \\) is \\( 1:2:3 \\). If \\( \\frac{\\text{area of } LONP}{\\te... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 2
} | 24 | 0 | 0 |
omr-ws-ea09f5ccffbceb650c21254c22f1b146afc0bd282879355fd1622a109081102e | A point \( O \) is chosen inside a triangle \( ABC \) such that the lengths \( OA = 15 \), \( OB = 12 \), and \( OC = 20 \). The feet of the perpendiculars from \( O \) to the sides of the triangle \( ABC \) form an equilateral triangle. Find the value of the angle \( \angle ABC \). | [
"60^\\circ"
] | [
"A point \\( O \\) is chosen inside a triangle \\( ABC \\) such that the lengths \\( OA = 15 \\), \\( OB = 12 \\), and \\( OC = 20 \\). The feet of the perpendiculars from \\( O \\) to the sides of the triangle \\( ABC \\) form an equilateral triangle. Find the value of the angle \\( \\angle ABC \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ba7ae57fdb274fa4124f2fb7aa2c8f99c99f5b99883cd21a0de47c87cfcc7e2e | A point \( O \) is taken inside a regular triangle \( ABC \) with side length \( a \). Line \( AO \) intersects \( BC \) at \( A_1 \), line \( BO \) intersects \( AC \) at \( B_1 \), and line \( CO \) intersects \( AB \) at \( C_1 \). Calculate the maximum possible value of \( OA_1 + OB_1 + OC_1 \). | [
"a"
] | [
"A point \\( O \\) is taken inside a regular triangle \\( ABC \\) with side length \\( a \\). Line \\( AO \\) intersects \\( BC \\) at \\( A_1 \\), line \\( BO \\) intersects \\( AC \\) at \\( B_1 \\), and line \\( CO \\) intersects \\( AB \\) at \\( C_1 \\). Calculate the maximum possible value of \\( OA_1 + OB_1 ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-2bf16e0c6da02445f324bfaba2dc8eb6dc41aaee01501ba4f59f6229252de75b | A point \( O \) lies inside \( \triangle ABC \) such that \( \angle BOC = 90^\circ - \angle BAC \). Let \( BO \) and \( CO \) meet \( AC \) and \( AB \) at \( K \) and \( L \), respectively. Points \( K_1 \) and \( L_1 \) lie on segments \( CL \) and \( BK \) such that \( K_1B = K_1K \) and \( L_1C = L_1L \). If \( M \... | [
"1"
] | [
"A point \\( O \\) lies inside \\( \\triangle ABC \\) such that \\( \\angle BOC = 90^\\circ - \\angle BAC \\). Let \\( BO \\) and \\( CO \\) meet \\( AC \\) and \\( AB \\) at \\( K \\) and \\( L \\), respectively. Points \\( K_1 \\) and \\( L_1 \\) lie on segments \\( CL \\) and \\( BK \\) such that \\( K_1B = K_1K... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-cd4aa35bb9dee783ee1e3dba10b5433ba7691928073e0fc110175c1452675e89 | A point \( O \) lies inside a scalene triangle \( ABC \). Let \( A', B', C' \) be the reflections of \( O \) across \( BC, CA, AB \) respectively. Let \( O' \) denote the circumcenter of triangle \( A'B'C' \). If \( AO \) bisects \( \angle BAC \), calculate the value of \( \frac{AO}{OO'} \). | [
"2"
] | [
"A point \\( O \\) lies inside a scalene triangle \\( ABC \\). Let \\( A', B', C' \\) be the reflections of \\( O \\) across \\( BC, CA, AB \\) respectively. Let \\( O' \\) denote the circumcenter of triangle \\( A'B'C' \\). If \\( AO \\) bisects \\( \\angle BAC \\), calculate the value of \\( \\frac{AO}{OO'} \\)."... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-b98b9d3383503a71585caf9aaddce051dc3bffb7755a40286ed9e9c5daec20b3 | A point \( P = (X, Y) \) is chosen at random from the unit square \((0,1) \times (0,1)\). Let \(\theta\) be the angle formed between the \(x\)-axis and the segment joining the origin and \(P\). Find the probability density function of \(\theta\). | [
"\\( f(\\theta) = \\begin{cases} \n\\frac{1}{2} \\sec^2 \\theta & \\text{if } 0 < \\theta \\le \\frac{\\pi}{4} \\\\\n\\frac{1}{2} \\csc^2 \\theta & \\text{if } \\frac{\\pi}{4} < \\theta < \\frac{\\pi}{2}\n\\end{cases} \\)"
] | [
"A point \\( P = (X, Y) \\) is chosen at random from the unit square \\((0,1) \\times (0,1)\\). Let \\(\\theta\\) be the angle formed between the \\(x\\)-axis and the segment joining the origin and \\(P\\). Find the probability density function of \\(\\theta\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-9a6b75e6fef938f0527ac59d1bcaef4af7495b09428feed6ae0c290f3d113a37 | A point \( P = (a, b) \) in the plane is called a lattice point if both of its coordinates are integers. Suppose \( P_0 = (0,0), P_1, P_2, \ldots, P_9, P_{10} = (0,0) \) is a sequence of lattice points such that the distance between \( P_i \) and \( P_{i+1} \) is \( \sqrt{2} \) for \( i = 0, 1, 2, \ldots, 9 \). How man... | [
"63504"
] | [
"A point \\( P = (a, b) \\) in the plane is called a lattice point if both of its coordinates are integers. Suppose \\( P_0 = (0,0), P_1, P_2, \\ldots, P_9, P_{10} = (0,0) \\) is a sequence of lattice points such that the distance between \\( P_i \\) and \\( P_{i+1} \\) is \\( \\sqrt{2} \\) for \\( i = 0, 1, 2, \\l... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-b5a5ea7326921f2efbf75c9b1c030dfac5855fb9492afc42245d65322c0bbcbb | A point \( P \) in the interior of a cyclic quadrilateral \( ABCD \) satisfies \( \angle BPC = \angle BAP + \angle PDC \). Denote by \( E \), \( F \), and \( G \) the feet of the perpendiculars from \( P \) to the lines \( AB \), \( AD \), and \( DC \), respectively. If the ratio of the area of triangle \( FEG \) to th... | [
"1"
] | [
"A point \\( P \\) in the interior of a cyclic quadrilateral \\( ABCD \\) satisfies \\( \\angle BPC = \\angle BAP + \\angle PDC \\). Denote by \\( E \\), \\( F \\), and \\( G \\) the feet of the perpendiculars from \\( P \\) to the lines \\( AB \\), \\( AD \\), and \\( DC \\), respectively. If the ratio of the area... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-66ad606e1b9c4e0971ef38fef04424c178e685f8971dc92e11985936acec669f | A point \( P \) in the interior of a cyclic quadrilateral \( ABCD \) satisfies \( \angle BPC = \angle BAP + \angle PDC \). Denote by \( E, F \), and \( G \) the feet of the perpendiculars from \( P \) to the lines \( AB \), \( AD \), and \( DC \), respectively. If the lengths of the sides of \( \triangle FEG \) are in ... | [
"\\dfrac{6}{25}"
] | [
"A point \\( P \\) in the interior of a cyclic quadrilateral \\( ABCD \\) satisfies \\( \\angle BPC = \\angle BAP + \\angle PDC \\). Denote by \\( E, F \\), and \\( G \\) the feet of the perpendiculars from \\( P \\) to the lines \\( AB \\), \\( AD \\), and \\( DC \\), respectively. If the lengths of the sides of \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-d6c5035d00935c70ee56440f83ad6654f3888cc663bf04e7a838eb9bc247ff1d | A point \( P \) in the interior of triangle \( ABC \) satisfies the following angle conditions: \[ \angle BPC - \angle BAC = \angle CPA - \angle CBA = \angle APB - \angle ACB. \] Let \( k \) be the common value of \( \bar{PA} \cdot \bar{BC} \), \( \bar{PB} \cdot \bar{AC} \), and \( \bar{PC} \cdot \bar{AB} \) for a spec... | [
"112"
] | [
"A point \\( P \\) in the interior of triangle \\( ABC \\) satisfies the following angle conditions:\n\\[ \\angle BPC - \\angle BAC = \\angle CPA - \\angle CBA = \\angle APB - \\angle ACB. \\]\nLet \\( k \\) be the common value of \\( \\bar{PA} \\cdot \\bar{BC} \\), \\( \\bar{PB} \\cdot \\bar{AC} \\), and \\( \\bar... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-6bba71144cbaacc33e7e958d5b4104cc3b28fd8aa50a28396864cd2c8f5c5ffe | A point \( P \) inside a regular tetrahedron \( ABCD \) is such that \( PA = PB = \sqrt{11} \) and \( PC = PD = \sqrt{17} \). What is the side length of \( ABCD \)? | [
"6"
] | [
"A point \\( P \\) inside a regular tetrahedron \\( ABCD \\) is such that \\( PA = PB = \\sqrt{11} \\) and \\( PC = PD = \\sqrt{17} \\). What is the side length of \\( ABCD \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7150cb50796614c803e7d44d17009ffb38f2b6d3c7958095169693dfbeae31c1 | A point \( P \) inside an acute triangle \( ABC \) is such that \( \angle BAP = \angle CAP \). Point \( M \) is the midpoint of side \( BC \). The line \( MP \) intersects the circumscribed circles of triangles \( ABP \) and \( ACP \) at points \( D \) and \( E \), respectively (point \( P \) lies between the points \(... | [
"2"
] | [
"A point \\( P \\) inside an acute triangle \\( ABC \\) is such that \\( \\angle BAP = \\angle CAP \\). Point \\( M \\) is the midpoint of side \\( BC \\). The line \\( MP \\) intersects the circumscribed circles of triangles \\( ABP \\) and \\( ACP \\) at points \\( D \\) and \\( E \\), respectively (point \\( P \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-65279c76deb0e78c16f143bf384f765d77c6a6ce90016cd1fb03d719576fc949 | A point \( P \) inside an equilateral triangle with side length \( s \) has distances of 3, 5, and 4 from the vertices \( A \), \( B \), and \( C \) respectively. Find the side length \( s \) of the equilateral triangle. | [
"\\(\\sqrt{25 + 12\\sqrt{3}}\\)"
] | [
"A point \\( P \\) inside an equilateral triangle with side length \\( s \\) has distances of 3, 5, and 4 from the vertices \\( A \\), \\( B \\), and \\( C \\) respectively. Find the side length \\( s \\) of the equilateral triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 26
} | 74 | 0 | 0 |
omr-ws-f42277b066d514e39f7d99b6c707331a75d36d295ab8999aa58d4e0985c31668 | A point \( P \) is chosen at random in the interior of a unit square \( S \). Let \( d(P) \) denote the distance from \( P \) to the closest side of \( S \). The probability that \( \frac{1}{5} \leq d(P) \leq \frac{1}{3} \) is equal to \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers.... | [
"281"
] | [
"A point \\( P \\) is chosen at random in the interior of a unit square \\( S \\). Let \\( d(P) \\) denote the distance from \\( P \\) to the closest side of \\( S \\). The probability that \\( \\frac{1}{5} \\leq d(P) \\leq \\frac{1}{3} \\) is equal to \\( \\frac{m}{n} \\), where \\( m \\) and \\( n \\) are relativ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ddae15531ff331a408804e06042a289f0b92aa280788a77e513cfd27e13aee29 | A point \( P \) is chosen at random on the circle \( x^2 + y^2 = 1 \). Let \( D \) be the distance of \( P \) from \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \). Find the mean of \( D \). | [
"\\(\\frac{4}{\\pi}\\)"
] | [
"A point \\( P \\) is chosen at random on the circle \\( x^2 + y^2 = 1 \\). Let \\( D \\) be the distance of \\( P \\) from \\( \\left( \\frac{1}{2}, \\frac{\\sqrt{3}}{2} \\right) \\). Find the mean of \\( D \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-9c49af60f43ac56b55971b79e94ba3df5a2a07fce7e35a3e75b74dbd25d68252 | A point \( P \) is chosen at random on the coordinate plane. What is the probability that the unit circle with center \( P \) contains exactly 2 lattice points? | [
"\\pi - 3"
] | [
"A point \\( P \\) is chosen at random on the coordinate plane. What is the probability that the unit circle with center \\( P \\) contains exactly 2 lattice points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-a8dba82307b8b9bf857030b5029b336128d8b13a0e73ba691fb4692619935dbd | A point \( P \) is chosen in an arbitrary triangle \( ABC \). 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 \( E_1, E_2 \), and \( E_3 \) be the areas of the three smaller triangles. Calcula... | [
"3"
] | [
"A point \\( P \\) is chosen in an arbitrary triangle \\( ABC \\). 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 \\( E_1, E_2 \\), and \\( E_3 \\) be the areas of the three smaller tri... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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