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-f5667877836d8eef90ccf8df0d510d0a565263a144918011fe3d126395b73eba | A quadrilateral $ABCD$ is circumscribed about a circle with center $I$. A point $P \ne I$ is chosen inside $ABCD$ so that the triangles $PAB, PBC, PCD,$ and $PDA$ have equal perimeters. A circle $\Gamma$ centered at $P$ meets the rays $PA, PB, PC$, and $PD$ at $A_1, B_1, C_1$, and $D_1$, respectively. Let $X$ be the in... | [
"3"
] | [
"A quadrilateral $ABCD$ is circumscribed about a circle with center $I$. A point $P \\ne I$ is chosen inside $ABCD$ so that the triangles $PAB, PBC, PCD,$ and $PDA$ have equal perimeters. A circle $\\Gamma$ centered at $P$ meets the rays $PA, PB, PC$, and $PD$ at $A_1, B_1, C_1$, and $D_1$, respectively. Let $X$ be... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-938c8538817d23e3b38aeb1aae6d6a6b84b3989364c4023a81d4237307bcee97 | A quadrilateral $ABCD$ is circumscribed around a circle $\omega$ centered at $I$. Lines $AC$ and $BD$ meet at point $P$, lines $AB$ and $CD$ meet at point $E$, and lines $AD$ and $BC$ meet at point $F$. Point $K$ on the circumcircle of triangle $EFP$ is such that $\angle IKP = 90^\circ$. The ray $PK$ meets $\omega$ at ... | [
"2"
] | [
"A quadrilateral $ABCD$ is circumscribed around a circle $\\omega$ centered at $I$. Lines $AC$ and $BD$ meet at point $P$, lines $AB$ and $CD$ meet at point $E$, and lines $AD$ and $BC$ meet at point $F$. Point $K$ on the circumcircle of triangle $EFP$ is such that $\\angle IKP = 90^\\circ$. The ray $PK$ meets $\\o... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-243559deb6c1272ece1d9a1639d8140c81483ecec27739921d39b66fdc1aabbe | A quadrilateral $ABCD$ is circumscribed around a circle with center $I$. Let $P$ be a point such that the angles $\angle APC$ and $\angle BPD$ share a common angle bisector $\ell$. If $A = (0,0)$, $B = (4,0)$, $C = (5,3)$, and $D = (1,3)$, find the coordinates of the point $I$. | [
"\\left( \\dfrac{5}{2}, \\dfrac{3}{2} \\right)"
] | [
"A quadrilateral $ABCD$ is circumscribed around a circle with center $I$. Let $P$ be a point such that the angles $\\angle APC$ and $\\angle BPD$ share a common angle bisector $\\ell$. If $A = (0,0)$, $B = (4,0)$, $C = (5,3)$, and $D = (1,3)$, find the coordinates of the point $I$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-1bbec6c53178716842165d37e921926437eafd86b10d27443751289a8304416f | A quadrilateral $ABCD$ is circumscribed around a circle, and its diagonals $AC$ and $BD$ are mutually perpendicular. If $AB = 5$, $BC = 12$, and $CD = 13$, find the length of $AD$. | [
"6"
] | [
"A quadrilateral $ABCD$ is circumscribed around a circle, and its diagonals $AC$ and $BD$ are mutually perpendicular. If $AB = 5$, $BC = 12$, and $CD = 13$, find the length of $AD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-2cc87373fe257909922b352d73834a52d7c85c96454440ab37e77fa633ef035e | A quadrilateral $ABCD$ is circumscribed around the circle $\omega$ centered at $I$ and inscribed into the circle $\Gamma$. The lines $AB$ and $CD$ meet at point $P$, and the lines $BC$ and $AD$ meet at point $Q$. If the radius of $\Gamma$ is $R$ and the radius of $\omega$ is $r$, find the value of $R^2 + r^2$ given tha... | [
"4Rr"
] | [
"A quadrilateral $ABCD$ is circumscribed around the circle $\\omega$ centered at $I$ and inscribed into the circle $\\Gamma$. The lines $AB$ and $CD$ meet at point $P$, and the lines $BC$ and $AD$ meet at point $Q$. If the radius of $\\Gamma$ is $R$ and the radius of $\\omega$ is $r$, find the value of $R^2 + r^2$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-a75fa0aaec9e9b7cd59329caaec1d18085d1754f2150c0b8cc25efe4c9d39333 | A quadrilateral $ABCD$ is circumscribed in a circle with center $O$. The distances from $O$ to the vertices are $OA=5$, $OB=6$, $OC=7$, and $OD=8$. Let $M$ and $N$ be the midpoints of the line segments $AC$ and $BD$, respectively. Find the ratio $OM:ON$. | [
"\\( 35:48 \\)"
] | [
"A quadrilateral $ABCD$ is circumscribed in a circle with center $O$. The distances from $O$ to the vertices are $OA=5$, $OB=6$, $OC=7$, and $OD=8$. Let $M$ and $N$ be the midpoints of the line segments $AC$ and $BD$, respectively. Find the ratio $OM:ON$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2821650b136145d1ab90fe5fabec21e335c71b012b1f30aff5de10de6bf5fa8 | A quadrilateral $ABCD$ is inscribed in a circle $(O)$. A circle $(T)$ touches $AB$ at $N$ and $CD$ at $M$ and intersects $(O)$ at points $H$ and $S$. The lines $AC$ and $MN$ intersect at $Q$, and $BD$ and $MN$ intersect at $P$. If the radius of circle $(O)$ is $10$ units and the length of segment $MN$ is $12$ units, ca... | [
"6"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle $(O)$. A circle $(T)$ touches $AB$ at $N$ and $CD$ at $M$ and intersects $(O)$ at points $H$ and $S$. The lines $AC$ and $MN$ intersect at $Q$, and $BD$ and $MN$ intersect at $P$. If the radius of circle $(O)$ is $10$ units and the length of segment $MN$ is $12$ unit... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e917b3df0a8888bd5046d8f96bbca3fb9ac4ada6c65856845e074231a119af8 | A quadrilateral $ABCD$ is inscribed in a circle $(O)$. Another circle $(I)$ is tangent to the diagonals $AC$ and $BD$ at points $M$ and $N$ respectively. Suppose that $MN$ meets $AB$ and $CD$ at points $P$ and $Q$ respectively. The circumcircle of triangle $IMN$ meets the circumcircles of triangles $IAB$ and $ICD$ at p... | [
"1"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle $(O)$. Another circle $(I)$ is tangent to the diagonals $AC$ and $BD$ at points $M$ and $N$ respectively. Suppose that $MN$ meets $AB$ and $CD$ at points $P$ and $Q$ respectively. The circumcircle of triangle $IMN$ meets the circumcircles of triangles $IAB$ and $ICD$... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-2d4e829471d17048abec6c2bdd912b410fad294b4f0f2895b5955d49d864fc3d | A quadrilateral $ABCD$ is inscribed in a circle $(O)$. The angle bisectors of $\angle DAB$ and $\angle ABC$ meet at $P$, and the angle bisectors of $\angle BCD$ and $\angle CDA$ meet at $Q$. Let $M$ be the midpoint of the arc $BC$ of $(O)$ which does not contain $A$ and $D$, and let $N$ be the midpoint of the arc $AD$ ... | [
"90"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle $(O)$. The angle bisectors of $\\angle DAB$ and $\\angle ABC$ meet at $P$, and the angle bisectors of $\\angle BCD$ and $\\angle CDA$ meet at $Q$. Let $M$ be the midpoint of the arc $BC$ of $(O)$ which does not contain $A$ and $D$, and let $N$ be the midpoint of the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-25f0d81e7783e73d30bbc16417fdc3560141309bcac03a08ea03fa14661b2fc4 | A quadrilateral $ABCD$ is inscribed in a circle $\omega$ such that $AB = AD$ and $BC = CD$ (deltoid). It is given that the radius of the inscribed circle of triangle $ABC$ is equal to the radius of the circle tangent to the smaller arc $BC$ of the circle $\omega$ and the side $BC$ at its midpoint. Find the ratio of the... | [
"\\dfrac{5}{2}"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle $\\omega$ such that $AB = AD$ and $BC = CD$ (deltoid). It is given that the radius of the inscribed circle of triangle $ABC$ is equal to the radius of the circle tangent to the smaller arc $BC$ of the circle $\\omega$ and the side $BC$ at its midpoint. Find the ratio... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-50b81c9dac5bd5edf3d5d2d845e60ddaa1086c49b299a24c0af2d26366f131cd | A quadrilateral $ABCD$ is inscribed in a circle $\omega$ with center $O$. Let $M_1$ and $M_2$ be the midpoints of segments $AB$ and $CD$ respectively. Let $\Omega$ be the circumcircle of triangle $OM_1M_2$. Let $X_1$ and $X_2$ be the common points of $\omega$ and $\Omega$, and $Y_1$ and $Y_2$ be the second common point... | [
"\\dfrac{5}{7}"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle $\\omega$ with center $O$. Let $M_1$ and $M_2$ be the midpoints of segments $AB$ and $CD$ respectively. Let $\\Omega$ be the circumcircle of triangle $OM_1M_2$. Let $X_1$ and $X_2$ be the common points of $\\omega$ and $\\Omega$, and $Y_1$ and $Y_2$ be the second com... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-3a1086041239eec38efbec75ad092526cf90446cb1ab40d57a3d2c7c3531c792 | A quadrilateral $ABCD$ is inscribed in a circle $\omega$. The tangent to $\omega$ at $A$ intersects the ray $CB$ at $K$, and the tangent to $\omega$ at $B$ intersects the ray $DA$ at $M$. Given that $AM=AD$ and $BK=BC$, calculate the ratio of the lengths of the parallel sides of $ABCD$ if it is a trapezoid. | [
"2"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle $\\omega$. The tangent to $\\omega$ at $A$ intersects the ray $CB$ at $K$, and the tangent to $\\omega$ at $B$ intersects the ray $DA$ at $M$. Given that $AM=AD$ and $BK=BC$, calculate the ratio of the lengths of the parallel sides of $ABCD$ if it is a trapezoid."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-5dc3f800bb6714a6071036abc21a74731dd887410da48967ac39962a5e25516f | A quadrilateral $ABCD$ is inscribed in a circle $k$. Let $E$ be the intersection point of lines $AB$ and $CD$, and $F$ be the intersection point of lines $AD$ and $BC$. If $P$ is the second intersection point of the circumcircle of triangle $CEF$ and $k$, and $M$ is the intersection point of line $AP$ and segment $EF$,... | [
"1"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle $k$. Let $E$ be the intersection point of lines $AB$ and $CD$, and $F$ be the intersection point of lines $AD$ and $BC$. If $P$ is the second intersection point of the circumcircle of triangle $CEF$ and $k$, and $M$ is the intersection point of line $AP$ and segment ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-6037c0d1da8bb6e7b2dcb185343be788563302811dd20dd8a912063dbb27830a | A quadrilateral $ABCD$ is inscribed in a circle of radius 1 with diameter $AB$. If the quadrilateral $ABCD$ has an incircle, calculate the maximum possible length of $CD$. | [
"2\\sqrt{5} - 4"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle of radius 1 with diameter $AB$. If the quadrilateral $ABCD$ has an incircle, calculate the maximum possible length of $CD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-fbe2d54cb94cebc0de4fd50257b466c39aae9a32a7aec26e1b8858c3424500b4 | A quadrilateral $ABCD$ is inscribed in a circle such that the circle of diameter $CD$ intersects the line segments $AC, AD, BC, BD$ respectively at $A_1, A_2, B_1, B_2$, and the circle of diameter $AB$ meets the line segments $CA, CB, DA, DB$ respectively at $C_1, C_2, D_1, D_2$. Let $P$ be the center of the circle tan... | [
"12"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle such that the circle of diameter $CD$ intersects the line segments $AC, AD, BC, BD$ respectively at $A_1, A_2, B_1, B_2$, and the circle of diameter $AB$ meets the line segments $CA, CB, DA, DB$ respectively at $C_1, C_2, D_1, D_2$. Let $P$ be the center of the circl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-ce0c359111f8931e5cc23a99f3f3a996be2d06b4514387c6922535fada38455f | A quadrilateral $ABCD$ is inscribed in a circle with $AB = BC$ and $AD = 3DC$. Point $R$ is on segment $BD$ such that $DR = 2RB$. Point $Q$ is on segment $AR$ such that $\angle ADQ = \angle BDQ$. Let $P$ be the intersection of $AB$ and $DQ$. Given that $\angle ABQ + \angle CBD = \angle QBD$, calculate the value of $\ta... | [
" -7 -5\\sqrt{2} "
] | [
"A quadrilateral $ABCD$ is inscribed in a circle with $AB = BC$ and $AD = 3DC$. Point $R$ is on segment $BD$ such that $DR = 2RB$. Point $Q$ is on segment $AR$ such that $\\angle ADQ = \\angle BDQ$. Let $P$ be the intersection of $AB$ and $DQ$. Given that $\\angle ABQ + \\angle CBD = \\angle QBD$, calculate the val... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e526f0beaba0af3939e7f310c8e65f99361c0a3ed4f0aebc6528fd172e3b2a02 | A quadrilateral $ABCD$ is inscribed in a circle with $|DA| = |BC| = 2$ and $|AB| = 4$. Let $E$ be the intersection point of lines $BC$ and $DA$. Given that $\angle AEB = 60^\circ$ and $|CD| < |AB|$, calculate the radius of the circle. | [
"2"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle with $|DA| = |BC| = 2$ and $|AB| = 4$. Let $E$ be the intersection point of lines $BC$ and $DA$. Given that $\\angle AEB = 60^\\circ$ and $|CD| < |AB|$, calculate the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-24170d7beb23ec7d2ac37a909dc8520b8c5296b89bf252cf9076dc28cf862724 | A quadrilateral $ABCD$ is inscribed in a circle with center $O$ and radius $r$. The diagonals $AC$ and $BD$ intersect at a fixed point $P$ such that $|OP| = d \in (0, r)$. Find the maximum area of $ABCD$. | [
"2r^2 - d^2"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle with center $O$ and radius $r$. The diagonals $AC$ and $BD$ intersect at a fixed point $P$ such that $|OP| = d \\in (0, r)$. Find the maximum area of $ABCD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-133a22e780adb301e4df7de7cd6b5994bbfa4948c67f00418e2119b7043a0650 | A quadrilateral $ABCD$ is inscribed in a circle with center $O$. A point $P$ is given on the plane. Let $K$, $L$, $M$, $N$ be the circumcenters of triangles $ABP$, $BCP$, $CDP$, $DPA$, respectively. Let $X$, $Y$, and $Z$ be the midpoints of segments $KM$, $LN$, and $OP$, respectively. If $O$ is at the origin and the co... | [
"0"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle with center $O$. A point $P$ is given on the plane. Let $K$, $L$, $M$, $N$ be the circumcenters of triangles $ABP$, $BCP$, $CDP$, $DPA$, respectively. Let $X$, $Y$, and $Z$ be the midpoints of segments $KM$, $LN$, and $OP$, respectively. If $O$ is at the origin and t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-c4bd25c4280bf03eb38c93cd11e62a05561e8685193f8c17039f6099515c670f | A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Its diagonals meet at $M$. The circumcircle of $ABM$ intersects the sides $AD$ and $BC$ at $N$ and $K$ respectively. Calculate the ratio of the area of triangle $NOM$ to the area of triangle $KOM$ if $AB = 12$, $BC = 16$, $CD = 20$, and $DA = 10$. | [
"\\dfrac{25}{64}"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Its diagonals meet at $M$. The circumcircle of $ABM$ intersects the sides $AD$ and $BC$ at $N$ and $K$ respectively. Calculate the ratio of the area of triangle $NOM$ to the area of triangle $KOM$ if $AB = 12$, $BC = 16$, $CD = 20$, and $DA = 10$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cbcc5dfa803af73872b99d5cdc0286221d08c11cd3814c9ca52f20ea03ccb97f | A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Let $A_1 B_1 C_1 D_1$ be the image of $ABCD$ after a rotation with center $O$ and angle $\alpha \in (0,90^\circ)$. The points $P, Q, R,$ and $S$ are the intersections of $AB$ and $A_1 B_1$, $BC$ and $B_1 C_1$, $CD$ and $C_1 D_1$, and $DA$ and $D_1 A_1$, r... | [
"1"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Let $A_1 B_1 C_1 D_1$ be the image of $ABCD$ after a rotation with center $O$ and angle $\\alpha \\in (0,90^\\circ)$. The points $P, Q, R,$ and $S$ are the intersections of $AB$ and $A_1 B_1$, $BC$ and $B_1 C_1$, $CD$ and $C_1 D_1$, and $DA$ and $D_1... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-7c9675933d28b54531072ff850d3955194277272b9664aa6fc5febe867bf1065 | A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Points $C'$ and $D'$ are the reflections of the orthocenters of triangles $ABD$ and $ABC$ at point $O$. Given that lines $BD$ and $BD'$ are symmetric with respect to the bisector of angle $ABC$, find the measure of angle $DAB$ if the measure of angle $ABC... | [
"40"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Points $C'$ and $D'$ are the reflections of the orthocenters of triangles $ABD$ and $ABC$ at point $O$. Given that lines $BD$ and $BD'$ are symmetric with respect to the bisector of angle $ABC$, find the measure of angle $DAB$ if the measure of angle... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-edefa54d75ed6a99212017e8d0b1f43c47485df20e5ad0bc743904e1d92d08c3 | A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Points $P$ and $Q$ are the points opposite to $C$ and $D$ respectively. Two tangents drawn to the circle at $P$ and $Q$ meet the line $AB$ at points $E$ and $F$ respectively, with $A$ between $E$ and $B$, and $B$ between $A$ and $F$. The line $EO$ interse... | [
"10"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Points $P$ and $Q$ are the points opposite to $C$ and $D$ respectively. Two tangents drawn to the circle at $P$ and $Q$ meet the line $AB$ at points $E$ and $F$ respectively, with $A$ between $E$ and $B$, and $B$ between $A$ and $F$. The line $EO$ in... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-89aee46ba773a24a484407f7d1ecd4d120fa0d8cc3e3cba3d705f50b79bc2d13 | A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Points $P$ and $Q$ are the points opposite to $C$ and $D$ respectively. Two tangents drawn to the circle at $P$ and $Q$ meet the line $AB$ at points $E$ and $F$ respectively, with $A$ between $E$ and $B$, and $B$ between $A$ and $F$. The line $EO$ interse... | [
"10"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Points $P$ and $Q$ are the points opposite to $C$ and $D$ respectively. Two tangents drawn to the circle at $P$ and $Q$ meet the line $AB$ at points $E$ and $F$ respectively, with $A$ between $E$ and $B$, and $B$ between $A$ and $F$. The line $EO$ in... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-5dee745806bf0b9daeca608a5be67ba2da8458766020b7275a2c163fd69c71ac | A quadrilateral $ABCD$ is inscribed in a circle with diameter $BD$. Point $A_1$ is the reflection of point $A$ over line $BD$, and point $B_1$ is the reflection of point $B$ over line $AC$. Let $P$ be the intersection of lines $CA_1$ and $BD$, and let $Q$ be the intersection of lines $DB_1$ and $AC$. If $AC = 10$ and $... | [
"\\sqrt{119}"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle with diameter $BD$. Point $A_1$ is the reflection of point $A$ over line $BD$, and point $B_1$ is the reflection of point $B$ over line $AC$. Let $P$ be the intersection of lines $CA_1$ and $BD$, and let $Q$ be the intersection of lines $DB_1$ and $AC$. If $AC = 10$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-842a1f70e97f134a1d48cfb0804834abf8cfcd6e5ad3479a7feb71b3341edfac | A quadrilateral $ABCD$ is inscribed in a circle, and its diagonals $AC$ and $BD$ intersect at $G$. Let $M$ be the midpoint of $CD$, and let $E$ and $F$ be points on $BC$ and $AD$, respectively, such that $ME \parallel AC$ and $MF \parallel BD$. Point $H$ is the projection of $G$ onto $CD$. The circumcircle of $\triangl... | [
"1"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle, and its diagonals $AC$ and $BD$ intersect at $G$. Let $M$ be the midpoint of $CD$, and let $E$ and $F$ be points on $BC$ and $AD$, respectively, such that $ME \\parallel AC$ and $MF \\parallel BD$. Point $H$ is the projection of $G$ onto $CD$. The circumcircle of $\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-d697da1b9606315ef7895c29b8205bc9463d4f1962e5a148eb3a26c121defe75 | A quadrilateral $ABCD$ is inscribed in a circle, and its diagonals intersect at $Q$. The side $DA$, extended from $A$, and side $CB$, extended from $B$, intersect at $P$. Given that $CD = CP = DQ$, find $\angle CAD$. | [
"\\(60^\\circ\\)"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle, and its diagonals intersect at $Q$. The side $DA$, extended from $A$, and side $CB$, extended from $B$, intersect at $P$. Given that $CD = CP = DQ$, find $\\angle CAD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 3,
"tir": 1
} | 20 | 0 | 0 |
omr-ws-fb8f3d51aca7af3fac87d1c4b36b196554c2bbc81d4d7ab10d14fa4a1eac3f5f | A quadrilateral $ABCD$ is inscribed in a circle. Lines $AB$ and $CD$ meet at point $E$, and lines $AD$ and $BC$ meet at point $F$. The bisector of angle $AEC$ intersects side $BC$ at point $M$ and side $AD$ at point $N$. The bisector of angle $BFD$ intersects side $AB$ at point $P$ and side $CD$ at point $Q$. If $AB = ... | [
"24"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle. Lines $AB$ and $CD$ meet at point $E$, and lines $AD$ and $BC$ meet at point $F$. The bisector of angle $AEC$ intersects side $BC$ at point $M$ and side $AD$ at point $N$. The bisector of angle $BFD$ intersects side $AB$ at point $P$ and side $CD$ at point $Q$. If $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-b74421a7620f3c4260ae7e5daf22e6b2d69952ede862839057c904a3e4c6c54a | A quadrilateral $ABCD$ is inscribed in a circle. On each of the sides $AB, BC, CD, DA$, one erects a rectangle towards the interior of the quadrilateral, with the other side of each rectangle being equal to $CD, DA, AB, BC$, respectively. Let the centers of these four rectangles be $P, Q, R, S$. If the coordinates of $... | [
"no_valid_answer_found"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle. On each of the sides $AB, BC, CD, DA$, one erects a rectangle towards the interior of the quadrilateral, with the other side of each rectangle being equal to $CD, DA, AB, BC$, respectively. Let the centers of these four rectangles be $P, Q, R, S$. If the coordinates... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68ec347fb9a97ef3c30183e7a6aa43dddb6bba04c79b293fae8ba01096942a74 | A quadrilateral $ABCD$ is inscribed in a circle. Points $C_1$ and $A_1$ are marked on rays $BA$ and $DC$, respectively, such that $DA = DA_1$ and $BC = BC_1$. Let $M$ be the intersection point of $BD$ and $A_1C_1$. If $BD = 20$ and $A_1C_1 = 24$, calculate the length of $BM$. | [
"12"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle. Points $C_1$ and $A_1$ are marked on rays $BA$ and $DC$, respectively, such that $DA = DA_1$ and $BC = BC_1$. Let $M$ be the intersection point of $BD$ and $A_1C_1$. If $BD = 20$ and $A_1C_1 = 24$, calculate the length of $BM$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-3499cd6165cbe06b8155408dc1180ffee61e597f66a04e477a56acbf945a3614 | A quadrilateral $ABCD$ is inscribed in a circle. Points $K, L, M,$ and $N$ are chosen on segments $AB, BC, CD,$ and $DA$ respectively such that $AK=KB=6$, $BL=3$, $LC=12$, $CM=4$, $MD=9$, $DN=18$, and $NA=2$. Calculate the sum of all possible values of the radius of the circle in which quadrilateral $KLMN$ can be inscr... | [
"no_valid_answer_found"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle. Points $K, L, M,$ and $N$ are chosen on segments $AB, BC, CD,$ and $DA$ respectively such that $AK=KB=6$, $BL=3$, $LC=12$, $CM=4$, $MD=9$, $DN=18$, and $NA=2$. Calculate the sum of all possible values of the radius of the circle in which quadrilateral $KLMN$ can be ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-450b85f03bf454e7284a40b20cd370426fe9b8fa6c718cd1d170b8950274540f | A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| = |BC| = 2$ and $|AB| = 4$. Let $E$ be the point of intersection of lines $BC$ and $DA$. Suppose that $\angle AEB = 60^\circ$ and that $|CD| < |AB|$. Calculate the radius of the circle. | [
"\\dfrac{4\\sqrt{3}}{3}"
] | [
"A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| = |BC| = 2$ and $|AB| = 4$. Let $E$ be the point of intersection of lines $BC$ and $DA$. Suppose that $\\angle AEB = 60^\\circ$ and that $|CD| < |AB|$. Calculate the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-98f46296fd6ce5e8cd9bc7b78b438ab37057e628794d6df8c04e52de5535ad59 | A quadrilateral $ABCD$ is inscribed in a unit circle. Let $P = AB \cap CD$ and $Q = AD \cap BC$. If $p$ and $q$ are the complex numbers representing points $P$ and $Q$ respectively, calculate the value of $p \cdot \overline{q} + q \cdot \overline{p}$. | [
"2"
] | [
"A quadrilateral $ABCD$ is inscribed in a unit circle. Let $P = AB \\cap CD$ and $Q = AD \\cap BC$. If $p$ and $q$ are the complex numbers representing points $P$ and $Q$ respectively, calculate the value of $p \\cdot \\overline{q} + q \\cdot \\overline{p}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-b22a472871f4cffde609a1377bd06c2ba4f95a21b2993b16b201018d3979d4ce | A quadrilateral $ABCD$ satisfies $BC=1$, $\angle C=90^\circ$, and $AC=AD$. Suppose triangle $ABD$ is similar to triangle $BCD$. Find the area of $ABCD$. | [
"\\(\\frac{5\\sqrt{2}}{4}\\)"
] | [
"A quadrilateral $ABCD$ satisfies $BC=1$, $\\angle C=90^\\circ$, and $AC=AD$. Suppose triangle $ABD$ is similar to triangle $BCD$. Find the area of $ABCD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 8,
"tir": 15
} | 67 | 0 | 0 |
omr-ws-10d10030979887285aecb1687a36fb7bd49993e952695b35fe155300f48e8b8b | A quadrilateral $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle with center $O$. Given that $OH \perp CD$, calculate the ratio $\frac{AB}{OH}$. | [
"2"
] | [
"A quadrilateral $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle with center $O$. Given that $OH \\perp CD$, calculate the ratio $\\frac{AB}{OH}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-57f56e5e0194caa59d45fb29e378b57c3e5298e09510e4a44c351d277b58d1f8 | A quadrilateral $ABCD$ without parallel sides is inscribed in a circle $\omega$. We draw a line $\ell_a \parallel BC$ through the point $A$, a line $\ell_b \parallel CD$ through the point $B$, a line $\ell_c \parallel DA$ through the point $C$, and a line $\ell_d \parallel AB$ through the point $D$. Suppose that the qu... | [
"1"
] | [
"A quadrilateral $ABCD$ without parallel sides is inscribed in a circle $\\omega$. We draw a line $\\ell_a \\parallel BC$ through the point $A$, a line $\\ell_b \\parallel CD$ through the point $B$, a line $\\ell_c \\parallel DA$ through the point $C$, and a line $\\ell_d \\parallel AB$ through the point $D$. Suppo... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-9a31fbcef8a0b395df4271f684346f408956789aad07c659c520a65ded8e1754 | A quadrilateral $ABCE$ is inscribed in circle $O$. Given $AE=8$, $AB=5$, $BC=6$, and $CE=8$, and both $AC$ and $EB$ are integers, the area of $ABCE$ can be expressed in the form $\dfrac{a\sqrt{b}}{c}+d\sqrt{e}$. What is the value of $a+b+c+d+e$? | [
"425"
] | [
"A quadrilateral $ABCE$ is inscribed in circle $O$. Given $AE=8$, $AB=5$, $BC=6$, and $CE=8$, and both $AC$ and $EB$ are integers, the area of $ABCE$ can be expressed in the form $\\dfrac{a\\sqrt{b}}{c}+d\\sqrt{e}$. What is the value of $a+b+c+d+e$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-d3be9b4c0e4f252438293f8ac63352aa7cf68fa55add444e1c40c6528660977c | A quadrilateral ABCD has AB = 10, BC = 7, CD = 7, and AD = 18. Given that diagonal BD is 13 units long, find the area of quadrilateral ABCD. | [
"\\(\\frac{39\\sqrt{3} + 15\\sqrt{287}}{4}\\)"
] | [
"A quadrilateral ABCD has AB = 10, BC = 7, CD = 7, and AD = 18. Given that diagonal BD is 13 units long, find the area of quadrilateral ABCD."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-cb52ec1dbebd1294e26ed5c25000609480dea0869e3c2cb3e8c1ddf9f1d4236d | A quadrilateral \(ABCD\) is circumscribed about a circle \(\omega\). The intersection point of \(\omega\) and the diagonal \(AC\), closest to \(A\), is \(E\). The point \(F\) is diametrically opposite to the point \(E\) on the circle \(\omega\). The tangent to \(\omega\) at the point \(F\) intersects lines \(AB\) and \... | [
"1"
] | [
"A quadrilateral \\(ABCD\\) is circumscribed about a circle \\(\\omega\\). The intersection point of \\(\\omega\\) and the diagonal \\(AC\\), closest to \\(A\\), is \\(E\\). The point \\(F\\) is diametrically opposite to the point \\(E\\) on the circle \\(\\omega\\). The tangent to \\(\\omega\\) at the point \\(F\\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-b76b51b7c78a188a33ccd5e3b6b53768b39f3eb01e1d9752aa3f438f80e1251b | A quadrilateral \(ABCD\) is cyclic. The sides \(AD\) and \(BC\) produced meet at \(P\). The sides \(AB\) and \(DC\) produced meet at \(Q\). Given that \(\angle APQ = 90^\circ\), let \(M\) be the midpoint of the diagonal \(BD\). If the perpendicular from \(P\) to \(AB\) intersects \(AB\) at \(N\), find the ratio \(\frac... | [
"\\dfrac{1}{2}"
] | [
"A quadrilateral \\(ABCD\\) is cyclic. The sides \\(AD\\) and \\(BC\\) produced meet at \\(P\\). The sides \\(AB\\) and \\(DC\\) produced meet at \\(Q\\). Given that \\(\\angle APQ = 90^\\circ\\), let \\(M\\) be the midpoint of the diagonal \\(BD\\). If the perpendicular from \\(P\\) to \\(AB\\) intersects \\(AB\\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-3d804544296502ed2462577dc212e3f797aa0f906b4a3b5ab2dfce51be6ca1eb | A quadrilateral \(ABCD\) is given. A circle \(k\) touches \(AD\) at point \(D\) and \(BC\) at point \(C\). The intersections of the side \(AB\) and \(k\) are denoted by \(K\) and \(L\), with \(DL = CL\). Let \(M\) be the midpoint of the side \(CD\). If the intersection point of \(AC\) and \(BD\) is \(P\), calculate the... | [
"\\dfrac{3}{5}"
] | [
"A quadrilateral \\(ABCD\\) is given. A circle \\(k\\) touches \\(AD\\) at point \\(D\\) and \\(BC\\) at point \\(C\\). The intersections of the side \\(AB\\) and \\(k\\) are denoted by \\(K\\) and \\(L\\), with \\(DL = CL\\). Let \\(M\\) be the midpoint of the side \\(CD\\). If the intersection point of \\(AC\\) a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-95984c4343e850d902faa85535f3e1a183d78490e1c9b5f3e70351056ab67a56 | A quadrilateral \(ABCD\) is inscribed in a circle \(\omega\) with center \(O\). Let \(M\) and \(N\) be the midpoints of segments \(AB\) and \(CD\) respectively. Let \(\Omega\) be the circumcircle of triangle \(OMN\). Let \(P\) and \(Q\) be the common points of \(\omega\) and \(\Omega\), and \(X\) and \(Y\) the second c... | [
"10"
] | [
"A quadrilateral \\(ABCD\\) is inscribed in a circle \\(\\omega\\) with center \\(O\\). Let \\(M\\) and \\(N\\) be the midpoints of segments \\(AB\\) and \\(CD\\) respectively. Let \\(\\Omega\\) be the circumcircle of triangle \\(OMN\\). Let \\(P\\) and \\(Q\\) be the common points of \\(\\omega\\) and \\(\\Omega\\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95c52ee9521aeabf0913527886e285f479f486d32047f77dfc4602e2ebe522b5 | A quadrilateral \(ABCD\) is inscribed in a circle \(\omega\) with center \(O\). The circumcircle of triangle \(AOC\) intersects the lines \(AB\), \(BC\), \(CD\), and \(DA\) at points \(M\), \(N\), \(K\), and \(L\) respectively (other than \(A\) and \(C\)). Let \(T_1\) and \(T_2\) be the tangents to \(\omega\) at points... | [
"\\dfrac{1}{2}"
] | [
"A quadrilateral \\(ABCD\\) is inscribed in a circle \\(\\omega\\) with center \\(O\\). The circumcircle of triangle \\(AOC\\) intersects the lines \\(AB\\), \\(BC\\), \\(CD\\), and \\(DA\\) at points \\(M\\), \\(N\\), \\(K\\), and \\(L\\) respectively (other than \\(A\\) and \\(C\\)). Let \\(T_1\\) and \\(T_2\\) b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9352a0fbc279a14b7af7f65040f90b622b511ae8d91de48bcfed294678da08d2 | A quadrilateral \(ABCD\) is inscribed in a circle \(k\) where \(AB > CD\) and \(AB\) is not parallel to \(CD\). Point \(M\) is the intersection of diagonals \(AC\) and \(BD\), and the perpendicular from \(M\) to \(AB\) intersects the segment \(AB\) at a point \(E\). If \(EM\) bisects the angle \(CED\), find the length ... | [
"10"
] | [
"A quadrilateral \\(ABCD\\) is inscribed in a circle \\(k\\) where \\(AB > CD\\) and \\(AB\\) is not parallel to \\(CD\\). Point \\(M\\) is the intersection of diagonals \\(AC\\) and \\(BD\\), and the perpendicular from \\(M\\) to \\(AB\\) intersects the segment \\(AB\\) at a point \\(E\\). If \\(EM\\) bisects the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-669bebfaa4c4ca99d64ff65b3034fa5131f4ef2dd9c85311165cbf07c0c80607 | A quadrilateral \(ABCD\) is inscribed in a circle \(k\), where \(AB > CD\) and \(AB\) is not parallel to \(CD\). Point \(M\) is the intersection of the diagonals \(AC\) and \(BD\), and the perpendicular from \(M\) to \(AB\) intersects the segment \(AB\) at the point \(E\). If \(EM\) bisects the angle \(CED\), find the ... | [
"1"
] | [
"A quadrilateral \\(ABCD\\) is inscribed in a circle \\(k\\), where \\(AB > CD\\) and \\(AB\\) is not parallel to \\(CD\\). Point \\(M\\) is the intersection of the diagonals \\(AC\\) and \\(BD\\), and the perpendicular from \\(M\\) to \\(AB\\) intersects the segment \\(AB\\) at the point \\(E\\). If \\(EM\\) bisec... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-fa072bff3e325280ca3275000094080898c7988eb2c5e269932896465ba6e78a | A quadrilateral \(ABCD\) is inscribed in a circle centered at \(I\) and circumscribed by another circle with center at \(O\). The diagonals \(AC\) and \(BD\) intersect at \(E\). If the coordinates of \(I\) and \(O\) are \((0, 0)\) and \((5, 0)\) respectively, and the coordinates of \(E\) are \((x, y)\), find the value ... | [
"25"
] | [
"A quadrilateral \\(ABCD\\) is inscribed in a circle centered at \\(I\\) and circumscribed by another circle with center at \\(O\\). The diagonals \\(AC\\) and \\(BD\\) intersect at \\(E\\). If the coordinates of \\(I\\) and \\(O\\) are \\((0, 0)\\) and \\((5, 0)\\) respectively, and the coordinates of \\(E\\) are ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-c567103dfd60393526d34b2535286fa017f3a105a8b5233470639c7704f41ecf | A quadrilateral \(ABCD\) is inscribed in a circle with center \(O\). A circle \(O'\) touches side \(BD\) at \(P\) and side \(AC\) at \(Q\), and also touches the circle \(O\). Let \(I_1\) and \(I_2\) be the incenters of \(\triangle ABC\) and \(\triangle DBC\) respectively. If the coordinates of \(P\) and \(Q\) are \((p_... | [
"-1"
] | [
"A quadrilateral \\(ABCD\\) is inscribed in a circle with center \\(O\\). A circle \\(O'\\) touches side \\(BD\\) at \\(P\\) and side \\(AC\\) at \\(Q\\), and also touches the circle \\(O\\). Let \\(I_1\\) and \\(I_2\\) be the incenters of \\(\\triangle ABC\\) and \\(\\triangle DBC\\) respectively. If the coordinat... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-5a406771dbd7e71c43a10ee44db0318fba89ce70e7c12ea845ba4d9d6b22ecc2 | A quadrilateral \(ABCD\) is inscribed in a circle with center \(O\). Its diagonals meet at \(M\). The circumcircle of \(ABM\) intersects the sides \(AD\) and \(BC\) at \(N\) and \(K\) respectively. Let \(S_1\) be the area of \( \triangle NOMD\) and \(S_2\) be the area of \( \triangle KOMC\). Calculate the value of \( \... | [
"1"
] | [
"A quadrilateral \\(ABCD\\) is inscribed in a circle with center \\(O\\). Its diagonals meet at \\(M\\). The circumcircle of \\(ABM\\) intersects the sides \\(AD\\) and \\(BC\\) at \\(N\\) and \\(K\\) respectively. Let \\(S_1\\) be the area of \\( \\triangle NOMD\\) and \\(S_2\\) be the area of \\( \\triangle KOMC\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-82b99aa9fdb007c676924b3fc308b2f5d08c9a52f7a7fe0e26230e540539e4af | A quadrilateral \(ABCD\) is inscribed in a circle with center \(O\). The circumcircle of triangle \(AOC\) passes through the midpoint \(M\) of \(BD\). Let \(N\) be the midpoint of \(AC\). If the length of \(BD\) is \(20\) units and the length of \(AC\) is \(16\) units, find the length of \(ON\). | [
"6"
] | [
"A quadrilateral \\(ABCD\\) is inscribed in a circle with center \\(O\\). The circumcircle of triangle \\(AOC\\) passes through the midpoint \\(M\\) of \\(BD\\). Let \\(N\\) be the midpoint of \\(AC\\). If the length of \\(BD\\) is \\(20\\) units and the length of \\(AC\\) is \\(16\\) units, find the length of \\(O... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-7a50d9b1b0a32e7272f40576c3098b686ca72702148b59aa16e7dcef35a99e38 | A quadrilateral \(ABCD\) is inscribed in a circle with no parallel sides. The sides \(DA\) and \(CB\) meet at \(E\), and the sides \(BA\) and \(CD\) meet at \(Z\). The bisectors of \(\angle DEC\) and \(\angle CZB\) intersect the sides of the quadrilateral at points \(K, L, M, N\). If the area of the rhombus \(KLMN\) is... | [
"8"
] | [
"A quadrilateral \\(ABCD\\) is inscribed in a circle with no parallel sides. The sides \\(DA\\) and \\(CB\\) meet at \\(E\\), and the sides \\(BA\\) and \\(CD\\) meet at \\(Z\\). The bisectors of \\(\\angle DEC\\) and \\(\\angle CZB\\) intersect the sides of the quadrilateral at points \\(K, L, M, N\\). If the area... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f412941176aa1e258ad12b9d2e0c5ff405c9c1310ddcea7cb486f39c86565c8c | A quadrilateral \(ABCD\) is inscribed in a circle. Its diagonals intersect at point \(K\). Let \(M_1, M_2, M_3, M_4\) be the midpoints of the arcs \(AB, BC, CD, DA\) (not containing the other vertices) respectively. Let \(I_1, I_2, I_3, I_4\) be the incenters of triangles \(ABK, BCK, CDK, DAK\) respectively. Calculate ... | [
"1"
] | [
"A quadrilateral \\(ABCD\\) is inscribed in a circle. Its diagonals intersect at point \\(K\\). Let \\(M_1, M_2, M_3, M_4\\) be the midpoints of the arcs \\(AB, BC, CD, DA\\) (not containing the other vertices) respectively. Let \\(I_1, I_2, I_3, I_4\\) be the incenters of triangles \\(ABK, BCK, CDK, DAK\\) respect... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-f0583f13001a7594e4fe2a98e6ff82b0985e16499af9627022e9b1ef9d8aae86 | A quadrilateral \(ABCD\) is inscribed in a square of area one unit. Calculate the sum of the maximum and minimum possible values of \(|AB|^2 + |BC|^2 + |CD|^2 + |DA|^2\). | [
"8"
] | [
"A quadrilateral \\(ABCD\\) is inscribed in a square of area one unit. Calculate the sum of the maximum and minimum possible values of \\(|AB|^2 + |BC|^2 + |CD|^2 + |DA|^2\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-466ef64c67cf331a76f343f4a5df34f4a5ab0411a20292b752404069ca6fa7bb | A quadrilateral \(ABCD\) is inscribed into a circle with center \(O\). Points \(P\) and \(Q\) are opposite to \(C\) and \(D\) respectively. Two tangents drawn to that circle at these points meet the line \(AB\) in points \(E\) and \(F\) (\(A\) is between \(E\) and \(B\), \(B\) is between \(A\) and \(F\)). The line \(EO... | [
"25"
] | [
"A quadrilateral \\(ABCD\\) is inscribed into a circle with center \\(O\\). Points \\(P\\) and \\(Q\\) are opposite to \\(C\\) and \\(D\\) respectively. Two tangents drawn to that circle at these points meet the line \\(AB\\) in points \\(E\\) and \\(F\\) (\\(A\\) is between \\(E\\) and \\(B\\), \\(B\\) is between ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-716143f48d0352797fd7c9a60e9fa781395be9d160eea31be4845819af1a4948 | A quadrilateral \(ABCD\) with \( \angle A + \angle D > 180^\circ \) is circumscribed about a circle with center \(I\). A line through \(I\) meets \(AB\) at \(X\) and \(CD\) at \(Y\) such that \(IX = IY\). If \(AX = 3\), \(BX = 7\), and \(CY = 5\), find the value of \(DY\). | [
"15"
] | [
"A quadrilateral \\(ABCD\\) with \\( \\angle A + \\angle D > 180^\\circ \\) is circumscribed about a circle with center \\(I\\). A line through \\(I\\) meets \\(AB\\) at \\(X\\) and \\(CD\\) at \\(Y\\) such that \\(IX = IY\\). If \\(AX = 3\\), \\(BX = 7\\), and \\(CY = 5\\), find the value of \\(DY\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-07c48a6dd4a7d125bc8ca2b03db635b71cd3e6f079ce23dfd013214baf9deabd | A quadrilateral \(ABCD\) with \(\angle B = \angle D = 90^\circ\) is inscribed in a circle \(C(O, R)\). Let \(H\) be the orthocenter of the acute triangle \(ABD\). The circle \(C_1(A, AH)\) meets circle \(C\) at points \(I\) and \(K\). If \(BD = 10\), calculate the value of \(CI^2 + CK^2\). | [
"200"
] | [
"A quadrilateral \\(ABCD\\) with \\(\\angle B = \\angle D = 90^\\circ\\) is inscribed in a circle \\(C(O, R)\\). Let \\(H\\) be the orthocenter of the acute triangle \\(ABD\\). The circle \\(C_1(A, AH)\\) meets circle \\(C\\) at points \\(I\\) and \\(K\\). If \\(BD = 10\\), calculate the value of \\(CI^2 + CK^2\\).... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-1c8e6d650c4619d2119130d5e082eb1c9c0ffe6d22122bc77820140c028a180f | A quadrilateral \(ABCD\) with perpendicular diagonals is inscribed in a circle \(\omega\). Two arcs \(\alpha\) and \(\beta\) with diameters \(AB\) and \(CD\) lie outside \(\omega\). Consider the two crescents formed by the circle \(\omega\) and the arcs \(\alpha\) and \(\beta\). Let \(r_1\) and \(r_2\) be the radii of ... | [
"\\sqrt{74}"
] | [
"A quadrilateral \\(ABCD\\) with perpendicular diagonals is inscribed in a circle \\(\\omega\\). Two arcs \\(\\alpha\\) and \\(\\beta\\) with diameters \\(AB\\) and \\(CD\\) lie outside \\(\\omega\\). Consider the two crescents formed by the circle \\(\\omega\\) and the arcs \\(\\alpha\\) and \\(\\beta\\). Let \\(r... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ea444661b8f11bf81771dff5ef0a5a4accaf9e5f4e05d78fb7bf45c8146bd861 | A quadrilateral \(ABCD\) without parallel sides is circumscribed around a circle with center \(O\). Given that \(O\) is the point of intersection of the middle lines of quadrilateral \(ABCD\) (i.e., barycenter of points \(A, B, C, D\)), calculate the value of \(\frac{OA \cdot OC}{OB \cdot OD}\). | [
"1"
] | [
"A quadrilateral \\(ABCD\\) without parallel sides is circumscribed around a circle with center \\(O\\). Given that \\(O\\) is the point of intersection of the middle lines of quadrilateral \\(ABCD\\) (i.e., barycenter of points \\(A, B, C, D\\)), calculate the value of \\(\\frac{OA \\cdot OC}{OB \\cdot OD}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4088046e181943478c418d32206b38579de7763e3b43e4069c490113cca9eed5 | A quadrilateral circumscribed about a circle has three of its side lengths 132, 162, and 201. What is the positive difference between the minimum and maximum lengths for the fourth side? | [
"138"
] | [
"A quadrilateral circumscribed about a circle has three of its side lengths 132, 162, and 201. What is the positive difference between the minimum and maximum lengths for the fourth side?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e8eb86025e756387eed7383e8fc5bfd491af296d23f9b90789c19c28d0843d36 | A quadrilateral has a side with length $1$ and another side with length $4$. A diagonal with length $2$ divides the quadrilateral into two isosceles triangles. What is the perimeter of the quadrilateral? | [
"11"
] | [
"A quadrilateral has a side with length $1$ and another side with length $4$. A diagonal with length $2$ divides the quadrilateral into two isosceles triangles. What is the perimeter of the quadrilateral?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-208120da23b917faf0fd5aca34df32d6c7b49a42c9078375eeabc84807bbabdd | A quadrilateral has all integer side lengths, a perimeter of 26, and one side of length 4. What is the greatest possible length of one side of this quadrilateral? | [
"12"
] | [
"A quadrilateral has all integer side lengths, a perimeter of 26, and one side of length 4. What is the greatest possible length of one side of this quadrilateral?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-481667a8ddca88ad14f40c5139f7fd55a81e6cba417f47111ab401dcc8148a15 | A quadrilateral has all of its vertices on the sides of a square with side length \(a\). What is the minimum value of the perimeter of this quadrilateral? Determine its nature. | [
"\\(2a\\sqrt{2}\\)"
] | [
"A quadrilateral has all of its vertices on the sides of a square with side length \\(a\\). What is the minimum value of the perimeter of this quadrilateral? Determine its nature."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-a2c68cb65bdf6f27e72beb0776e23855e5e63bd8af6478afd23af5e636a1a907 | A quadrilateral has distinct integer side lengths. If the second-largest side has a length of 10, what is the maximum possible length of the largest side? | [
"26"
] | [
"A quadrilateral has distinct integer side lengths. If the second-largest side has a length of 10, what is the maximum possible length of the largest side?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9473a789880617a8e814ad6a13ac362f550e44b29207cd1791b4d8742921828e | A quadrilateral has perpendicular diagonals and three of its sides have lengths of 2, 3, and 4. There are two possible lengths for the fourth side: \( \sqrt{m} \) or \( \sqrt{n} \). What is \( m + n \)? | [
"32"
] | [
"A quadrilateral has perpendicular diagonals and three of its sides have lengths of 2, 3, and 4. There are two possible lengths for the fourth side: \\( \\sqrt{m} \\) or \\( \\sqrt{n} \\). What is \\( m + n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2cbaa95663aa488b62ce8f40e9d78592350b35ee84ccd139e97f3e021b1402ce | A quadrilateral has perpendicular diagonals. Three of its side lengths are \(a\), \(b\), and \(c\). Find the maximum possible area of the quadrilateral. | [
"\\(\\frac{1}{2} (ac + b \\sqrt{a^2 + c^2 - b^2})\\)"
] | [
"A quadrilateral has perpendicular diagonals. Three of its side lengths are \\(a\\), \\(b\\), and \\(c\\). Find the maximum possible area of the quadrilateral."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-ba9a921346e5658bddbc38b1794be184f343122907dc9a4e98a927220688037c | A quadrilateral has side lengths 3, 3, \(x\), and \(y\). The values of \(x\) and \(y\) are integers. After \(x\) is chosen, let \(N\) be the number of possible values for \(y\). What is the maximum value of \(N\)? | [
"11"
] | [
"A quadrilateral has side lengths 3, 3, \\(x\\), and \\(y\\). The values of \\(x\\) and \\(y\\) are integers. After \\(x\\) is chosen, let \\(N\\) be the number of possible values for \\(y\\). What is the maximum value of \\(N\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-648c2f007f37885fcdb7cb408105fc2c0d33ec2532324c2b25614f6d2e7de2d2 | A quadrilateral has sides of lengths $24, 25, 26,$ and $27$, and its area is $180x$, where $x$ is a real number. Find $x^2$. | [
"13"
] | [
"A quadrilateral has sides of lengths $24, 25, 26,$ and $27$, and its area is $180x$, where $x$ is a real number. Find $x^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e1f02bff0edeefe6c0f5d8e4b7d931e1be13680b905c2ea223cd2daf4d15527 | A quadrilateral has sides of lengths 24, 25, 26, and 27, and its area is given by \(180x\), where \(x\) is a real number. Find \(x^2\). | [
"663"
] | [
"A quadrilateral has sides of lengths 24, 25, 26, and 27, and its area is given by \\(180x\\), where \\(x\\) is a real number. Find \\(x^2\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f3a5c7b277461697bd94c4440705f831f7f5bd6980646c5f1a8ee4f6dc7907df | A quadrilateral has two sides of lengths $1$ and $4$. A diagonal of length $2$ divides it into two isosceles triangles. Find the perimeter of this quadrilateral. | [
"11"
] | [
"A quadrilateral has two sides of lengths $1$ and $4$. A diagonal of length $2$ divides it into two isosceles triangles. Find the perimeter of this quadrilateral."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f96f0124852673225a0b4e466478b89bfe8d303e47cc2ce919e3fa9a7e137e76 | A quadrilateral has vertices $PQRS$, labelled anticlockwise, with internal angles all less than $180^{\circ}$. Squares with centers $X, Y, Z$, and $T$ are constructed externally on sides $PQ, QR, RS$, and $SP$, respectively. If $XYZT$ is a square, find the vector expression for $\overrightarrow{PQ} + \overrightarrow{SR... | [
"\\overrightarrow{QR} + \\overrightarrow{SP}"
] | [
"A quadrilateral has vertices $PQRS$, labelled anticlockwise, with internal angles all less than $180^{\\circ}$. Squares with centers $X, Y, Z$, and $T$ are constructed externally on sides $PQ, QR, RS$, and $SP$, respectively. If $XYZT$ is a square, find the vector expression for $\\overrightarrow{PQ} + \\overright... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-0c58b38100fbdd6225a07ec2a8e3a9289dfd456b21421da3e577cf02c5fb6e10 | A quadrilateral has vertices \( P(a,b) \), \( Q(b,a) \), \( R(-a, -b) \), and \( S(-b, -a) \), where \( a \) and \( b \) are integers with \( a > b > 0 \). The area of \( PQRS \) is 16. What is \( a + b \)? | [
"4"
] | [
"A quadrilateral has vertices \\( P(a,b) \\), \\( Q(b,a) \\), \\( R(-a, -b) \\), and \\( S(-b, -a) \\), where \\( a \\) and \\( b \\) are integers with \\( a > b > 0 \\). The area of \\( PQRS \\) is 16. What is \\( a + b \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b1c5820d7f26a8123b0aa25290add9d51d1ae76a49687246c5192e7df185658 | A quadrilateral has vertices at $(0,1)$, $(3,4)$, $(4,3)$ and $(3,0)$. Its perimeter can be expressed in the form $a\sqrt2+b\sqrt{10}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$? | [
"6"
] | [
"A quadrilateral has vertices at $(0,1)$, $(3,4)$, $(4,3)$ and $(3,0)$. Its perimeter can be expressed in the form $a\\sqrt2+b\\sqrt{10}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2ec4e300fb2a3de0524d15500fa305b71de3c815ff1a8d2fb5303a1e03108538 | A quadrilateral in the plane has vertices $(1, 3)$, $(1, 1)$, $(2, 1)$ and $(2006, 2007)$. How many square units is the area of the quadrilateral? | [
"3008"
] | [
"A quadrilateral in the plane has vertices $(1, 3)$, $(1, 1)$, $(2, 1)$ and $(2006, 2007)$. How many square units is the area of the quadrilateral?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e288c4d59746e2d51c61f7f0eaf85f7eaf9a0f427420ac2c8b0ef5ac9954975b | A quadrilateral inscribed in a circle has side lengths whose squares are 17, 99, 19, and 97. Find the area of the circle. | [
"29\\pi",
"\\( 29\\pi \\)"
] | [
"A quadrilateral inscribed in a circle has side lengths whose squares are 17, 99, 19, and 97. Find the area of the circle."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 8,
"tir": 21
} | 59 | 0 | 0 |
omr-ws-ffb63b234dfa1675ce534292065d07b5397104175c0c32402495aa75ed9120cc | A quadrilateral inscribed in a circle has sides of length \(\sqrt{2002}\), \(\sqrt{2005}\), \(\sqrt{2006}\), and \(\sqrt{2009}\) in that order. The diameter of the circle can be expressed in the form \(\sqrt{x}\) where \(x\) is a squarefree integer. Find the sum of the digits of \(x\). | [
"6"
] | [
"A quadrilateral inscribed in a circle has sides of length \\(\\sqrt{2002}\\), \\(\\sqrt{2005}\\), \\(\\sqrt{2006}\\), and \\(\\sqrt{2009}\\) in that order. The diameter of the circle can be expressed in the form \\(\\sqrt{x}\\) where \\(x\\) is a squarefree integer. Find the sum of the digits of \\(x\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-ef555cc98dc79a9d19bf3a0110864f835eaf9127b9070df2f7d1e5ec64c568ad | A quadrilateral inscribed in a circle has sides of lengths 7, 15, 20, and 24 (in order). Determine the area of the quadrilateral, the lengths of the two diagonals, and the radius of the circle. | [
"Area = 234, Diagonals = 25 and 20, Radius = 25/2"
] | [
"A quadrilateral inscribed in a circle has sides of lengths 7, 15, 20, and 24 (in order). Determine the area of the quadrilateral, the lengths of the two diagonals, and the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 30
} | 30 | 0 | 0 |
omr-ws-b7cebd5b69cca9a1f46a5d380886285947c43feb71c312c0e4c32b6837aebaba | A quadrilateral is circumscribed about a circle with three of its sides measuring 9, 15, and 8. What is the smallest possible value of the fourth side? | [
"2"
] | [
"A quadrilateral is circumscribed about a circle with three of its sides measuring 9, 15, and 8. What is the smallest possible value of the fourth side?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-55db55f14627642f63dbb231289740514fe3af36223eb6b7242364f9b790c97b | A quadrilateral is cut from a piece of gift wrapping paper with equally wide white and gray stripes. The gray stripes in the quadrilateral have a combined area of $10$. Determine the area of the quadrilateral. | [
"20"
] | [
"A quadrilateral is cut from a piece of gift wrapping paper with equally wide white and gray stripes. The gray stripes in the quadrilateral have a combined area of $10$. Determine the area of the quadrilateral."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fd2e26bda432791b4f170c459f4eb2269416ba84becfc1aec01535339d964175 | A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side? | [
"500"
] | [
"A quadrilateral is inscribed in a circle of radius $200\\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 64,
"tir": 72
} | 136 | 0 | 0 |
omr-ws-d6745c217f6e1c44187180b9632fdae4d1c3dc36c5d1453e1bff629239eb6571 | A quadrilateral is inscribed in a circle of radius \(200\sqrt{2}\). Three of the sides of this quadrilateral have length 200. What is the length of the fourth side? | [
"500"
] | [
"A quadrilateral is inscribed in a circle of radius \\(200\\sqrt{2}\\). Three of the sides of this quadrilateral have length 200. What is the length of the fourth side?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-baec6927ea9ffd60dcfa89c09501fc0ecbbd2fad595c2f3095447c353c639ffd | A quadrilateral is inscribed in a circle of radius \(200\sqrt{2}\). Three of the sides of this quadrilateral have length \(200\). What is the length of the fourth side? | [
"E (500)[/hide]\n\nTherefore, the length of the fourth side is \\(500\\)."
] | [
"A quadrilateral is inscribed in a circle of radius \\(200\\sqrt{2}\\). Three of the sides of this quadrilateral have length \\(200\\). What is the length of the fourth side?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-28f8908278e2a0e3a23b0153249fe25aa2eae98536f6c826192176c166910d64 | A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, what is the sum of these four angles, expressed in degrees? | [
"540"
] | [
"A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, what is the sum of these four angles, expressed in degrees?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-767f58aa409f1d019a5c879181da3dfe764e2d9bc91a28be699af611d0b8deec | A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, what is the sum of these angles? | [
"540^\\circ"
] | [
"A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, what is the sum of these angles?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-67fb784bd2aaa71267413ec0b06bff5f8913b11004d24067638b5f60e97dcb69 | A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, what is the sum of these inscribed angles? Note: The original post suggests the answer is \(180^\circ\), but the solution on the AoPS Wiki suggests \(540^\circ\). | [
"540 degrees"
] | [
"A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, what is the sum of these inscribed angles?\n\nNote: The original post suggests the answer is \\(180^\\circ\\), but the solution on the AoPS Wiki suggests \\(540^\\circ\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-574d2ee6d648cb557d3ca89c8f72f97aab2466236ff77e0c2dcd3e97fefd28fb | A quadrilateral is inscribed inside a circle of radius $10,$ and the lengths of three of its sides are $10, 16,$ and $20$. Given that the length of the remaining side is $p\sqrt{q}-r,$ where $p, q, r$ are positive integers and $q$ is not divisible by the square of any prime, what is $p+q+r$? | [
"17"
] | [
"A quadrilateral is inscribed inside a circle of radius $10,$ and the lengths of three of its sides are $10, 16,$ and $20$. Given that the length of the remaining side is $p\\sqrt{q}-r,$ where $p, q, r$ are positive integers and $q$ is not divisible by the square of any prime, what is $p+q+r$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-b84fe4b185b1b5ce21ca6725df034fdad6025963a7f744b8e51c9ebc84a17663 | A quadrilateral plot of land was divided into four smaller lots by two straight fences joining the midpoints of the sides. The areas of three of the lots are given. Calculate the area of the fourth lot. | [
"20"
] | [
"A quadrilateral plot of land was divided into four smaller lots by two straight fences joining the midpoints of the sides. The areas of three of the lots are given. Calculate the area of the fourth lot."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-3ec7818c8a8f917f3051b67420945a8d0150cb8559454ad8d887121a61a0c880 | A quadrilateral pyramid and a regular tetrahedron have edges that are all equal in length. They are glued together so that they share one equilateral triangle. Calculate the total number of faces in the resulting body. | [
"7"
] | [
"A quadrilateral pyramid and a regular tetrahedron have edges that are all equal in length. They are glued together so that they share one equilateral triangle. Calculate the total number of faces in the resulting body."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e125b5e4567427792128b13008bd3c52cc4be55ad7da3de95ea40f0c1d82d885 | A quadrilateral with consecutive sides $6, 10, 5, 12$ is inscribed in a circle. The diagonal of this quadrilateral that forms the third side of a triangle with the other two side lengths being $5$ and $10$ is equal to $12$. Rounded to the nearest whole number, what is the angle between the sides of length $6$ and $10$? | [
"100"
] | [
"A quadrilateral with consecutive sides $6, 10, 5, 12$ is inscribed in a circle. The diagonal of this quadrilateral that forms the third side of a triangle with the other two side lengths being $5$ and $10$ is equal to $12$. Rounded to the nearest whole number, what is the angle between the sides of length $6$ and ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-80430b7942d5b426094cf8858f9ef90cdfbdfea47ea4806276290c01c5b67a5c | A quadrilateral with consecutive sides of length 70, 90, 130, and 110 is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length \( x \) and \( y \). Find \( |x - y| \). | [
"13"
] | [
"A quadrilateral with consecutive sides of length 70, 90, 130, and 110 is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length \\( x \\) and \\( y \\). Find \\( |x - y| \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-5cd5a65e047b68f91b873dec9a57514e3b6585ff994f37f05a680968593746ad | A quadrilateral with consecutive sides of lengths $70, 90, 130,$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length $130$ divides that side into segments of lengths $x$ and $y$. Find $|x-y|$. | [
"13"
] | [
"A quadrilateral with consecutive sides of lengths $70, 90, 130,$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length $130$ divides that side into segments of lengths $x$ and $y$. Find $|x-y|$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 30,
"genselect": 5
} | 35 | 0 | 0 |
omr-ws-0daad4688b4ec4682be46117e5a9bd59a68ddff14a8d4c597923f2a7a6a9a702 | A quadrilateral with consecutive sides of lengths 70, 90, 130, and 110 is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length \( x \) and \( y \). If \( y \geq x \), find \( y - x \). | [
"20"
] | [
"A quadrilateral with consecutive sides of lengths 70, 90, 130, and 110 is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length \\( x \\) and \\( y \\). If \\( y \\geq x \\), find \\( y - x \\).... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-7db2ed4b293a23b0e6ca13cd9c1dd71d5b4f4ed91264031f289288a292cc64f8 | A quadrilateral with consecutive sides of lengths 70, 90, 130, and 110 is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length \(x\) and \(y\). Find \(|x - y|\). | [
"13"
] | [
"A quadrilateral with consecutive sides of lengths 70, 90, 130, and 110 is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length \\(x\\) and \\(y\\). Find \\(|x - y|\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 31,
"genselect": 6
} | 37 | 0 | 0 |
omr-ws-f95664560b8ce8c227ec95c7bf7edf51fdcc1b56bec809f76c7f6e0d5153996c | A quadrilateral with consecutive sides of lengths 70, 90, 130, and 110 is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of lengths \(x\) and \(y\). Find \(|x - y|\). | [
"13"
] | [
"A quadrilateral with consecutive sides of lengths 70, 90, 130, and 110 is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of lengths \\(x\\) and \\(y\\). Find \\(|x - y|\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-f3cf1595a592eaf2e08595380f2fa6eb56011400a14ff6d513b4f581ce86d430 | A quadrilateral with mutually perpendicular diagonals is inscribed in a square such that exactly one vertex of the quadrilateral lies on each side of the square. The sides and diagonals of the quadrilateral divide the square into eight triangles. Each of these triangles is colored in one of two available colors such th... | [
"0"
] | [
"A quadrilateral with mutually perpendicular diagonals is inscribed in a square such that exactly one vertex of the quadrilateral lies on each side of the square. The sides and diagonals of the quadrilateral divide the square into eight triangles. Each of these triangles is colored in one of two available colors su... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-d684c9779c9b63e27187af2d146000997e5485d66504da27c1120f2c2b6fb913 | A quadrilateral with perpendicular diagonals is inscribed in a circle with radius 13 cm. One of the diagonals has a length of 18 cm, and the distance between the center of the circle and the intersection of the diagonals is \(4\sqrt{6}\) cm. Find the area of the quadrilateral. | [
"\\( 18\\sqrt{161} \\)"
] | [
"A quadrilateral with perpendicular diagonals is inscribed in a circle with radius 13 cm. One of the diagonals has a length of 18 cm, and the distance between the center of the circle and the intersection of the diagonals is \\(4\\sqrt{6}\\) cm. Find the area of the quadrilateral."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 34,
"genselect": 8,
"tir": 23
} | 65 | 0 | 0 |
omr-ws-965e820b8563238d6e068aac66e7ebf1dcc323f2d55aeb499a92e0f5d07cc674 | A quadrilateral with perpendicular diagonals of lengths $39$ and $41$ is inscribed in a circle of area $1690\pi$. Given that the perimeter of the quadrilateral is $a \sqrt{b} + c \sqrt{d}$, where $a$, $b$, $c$, and $d$ are positive integers and neither $b$ nor $d$ is divisible by the square of any prime, find $a + b + ... | [
"164"
] | [
"A quadrilateral with perpendicular diagonals of lengths $39$ and $41$ is inscribed in a circle of area $1690\\pi$. Given that the perimeter of the quadrilateral is $a \\sqrt{b} + c \\sqrt{d}$, where $a$, $b$, $c$, and $d$ are positive integers and neither $b$ nor $d$ is divisible by the square of any prime, find $... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1608dbb577f41aaa843f305bb7a176daf2fadb55eafafda04986c6477a287ad8 | A quadrilateral with sides $15$, $15$, $15$, and $20$ is inscribed in a circle. The sides of a square are tangent to that circle. Find the area of the square. | [
"540"
] | [
"A quadrilateral with sides $15$, $15$, $15$, and $20$ is inscribed in a circle. The sides of a square are tangent to that circle. Find the area of the square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 22
} | 45 | 0 | 0 |
omr-ws-f697af117e795f4a2e6ca43166ee4164b3a83e101ffedaf734ce498bf638b781 | A quadrilateral with sides $15, 15, 15,$ and $20$ is inscribed in a circle. A square is drawn around this circle such that each side of the square is tangent to the circle. What is the area, in square units, of this square? | [
"400"
] | [
"A quadrilateral with sides $15, 15, 15,$ and $20$ is inscribed in a circle. A square is drawn around this circle such that each side of the square is tangent to the circle. What is the area, in square units, of this square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-52d2a1b62c126da45544da2a8c2d516063480755dc47ee28595634c34dd10584 | A quadrilateral with sides 15, 15, 15, and 20 is inscribed in a circle which is tangential to a square. Find the area of the square. Do not use Brahmagupta's formula. | [
"540"
] | [
"A quadrilateral with sides 15, 15, 15, and 20 is inscribed in a circle which is tangential to a square. Find the area of the square. Do not use Brahmagupta's formula."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 8
} | 22 | 0 | 0 |
omr-ws-40b566a2490adcd8175edf168cdf4585d4be90aaf015d1dc9739c5d4511d0404 | A quadrilateral with sides 15, 15, 15, and 20 is inscribed in a circle. A square is drawn around this circle such that each side of the square is tangent to the circle. What is the area, in square units, of the square? | [
"540"
] | [
"A quadrilateral with sides 15, 15, 15, and 20 is inscribed in a circle. A square is drawn around this circle such that each side of the square is tangent to the circle. What is the area, in square units, of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 29
} | 51 | 0 | 0 |
omr-ws-d913c17d4485f28c004abf845e88cf14e232783ea2ffc5e04a26d929ef3a1c88 | A quadrilateral with sides \(a, b, c, d\) is inscribed in a circle of radius \(R\). If \(a^2 + b^2 + c^2 + d^2 = 8R^2\), calculate the sum of all possible values of \(\cos(\theta)\) where \(\theta\) is the angle between the diagonals of the quadrilateral. | [
"0"
] | [
"A quadrilateral with sides \\(a, b, c, d\\) is inscribed in a circle of radius \\(R\\). If \\(a^2 + b^2 + c^2 + d^2 = 8R^2\\), calculate the sum of all possible values of \\(\\cos(\\theta)\\) where \\(\\theta\\) is the angle between the diagonals of the quadrilateral."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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