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-20b8015b66c907007ed102f6e3faec3ec9b5cae4a0d43aae9b7f57f75118a51d | Triangle \( \triangle ABC \) is an isosceles triangle inscribed in a circle with center \( O \) and diameter \( AD \), where \( AB = AC \). The diameter intersects \( BC \) at \( E \) and \( F \) is the midpoint of \( OE \). Given that \( BD \parallel FC \) and \( BC = 2\sqrt{5} \), find the length of \( CD \). | [
"\\( \\sqrt{5} \\)"
] | [
"Triangle \\( \\triangle ABC \\) is an isosceles triangle inscribed in a circle with center \\( O \\) and diameter \\( AD \\), where \\( AB = AC \\). The diameter intersects \\( BC \\) at \\( E \\) and \\( F \\) is the midpoint of \\( OE \\). Given that \\( BD \\parallel FC \\) and \\( BC = 2\\sqrt{5} \\), find the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2211f5f592fcf7b6d43bff593dfe0eb5eb584279ec5fe21740d84822fea95799 | Triangle \( \triangle ABC \) is created from points \( A(0, 0) \), \( B(1, 0) \), and \( C(1/2, 2) \). Let \( q, r \), and \( s \) be numbers such that \( 0 < q < 1/2 < s < 1 \), and \( q < r < s \). Let \( D \) be the point on \( AC \) which has \( x \)-coordinate \( q \), \( E \) be the point on \( AB \) which has \(... | [
"2(2qs - qr - rs - q + r)"
] | [
"Triangle \\( \\triangle ABC \\) is created from points \\( A(0, 0) \\), \\( B(1, 0) \\), and \\( C(1/2, 2) \\). Let \\( q, r \\), and \\( s \\) be numbers such that \\( 0 < q < 1/2 < s < 1 \\), and \\( q < r < s \\). Let \\( D \\) be the point on \\( AC \\) which has \\( x \\)-coordinate \\( q \\), \\( E \\) be th... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-16da704f340081fb9c729ca8eef62eb98688a4a734f816436d4b746a591bc1c1 | Triangle \( \vartriangle ABC \) has vertices at \( A(0, 4) \), \( B(3, 0) \), and \( C(0, 0) \). There exists a line \( \ell \) crossing segments \( \overline{AB} \) and \( \overline{AC} \) at \( P \) and \( Q \), respectively, such that circles can be inscribed in \( \vartriangle APQ \) as well as in quadrilateral \( ... | [
"\\(\\frac{1}{3}\\)"
] | [
"Triangle \\( \\vartriangle ABC \\) has vertices at \\( A(0, 4) \\), \\( B(3, 0) \\), and \\( C(0, 0) \\). There exists a line \\( \\ell \\) crossing segments \\( \\overline{AB} \\) and \\( \\overline{AC} \\) at \\( P \\) and \\( Q \\), respectively, such that circles can be inscribed in \\( \\vartriangle APQ \\) a... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cecfdc41f8770754cc504f1f9aaeb71ca3d1fd43e3aabf39dc99c34c9214b81d | Triangle \(ABC\) (which is not rectangular) is inscribed in a circle \(o\) with center \(O\). Point \(H\) is the orthocenter of \(\Delta ABC\). Point \(M\) is the midpoint of \(BH\) and point \(N\) is the midpoint of \(CH\). We know that \(HOMN\) is a cyclic quadrilateral. \(BB'\) is the diameter of \(o\). Calculate th... | [
"\\dfrac{1}{2}"
] | [
"Triangle \\(ABC\\) (which is not rectangular) is inscribed in a circle \\(o\\) with center \\(O\\). Point \\(H\\) is the orthocenter of \\(\\Delta ABC\\). Point \\(M\\) is the midpoint of \\(BH\\) and point \\(N\\) is the midpoint of \\(CH\\). We know that \\(HOMN\\) is a cyclic quadrilateral. \\(BB'\\) is the dia... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1347a4bfd1d6a89814f1d466f5ec14c9cb8d9b93c9b39749233eb43d4a8b4add | Triangle \(ABC\) and \(ABD\) are isosceles with \(AB = AC = BD\), and \(BD\) intersects \(AC\) at \(E\) (point \(E\) is on segment \(AC\) and segment \(BD\)). If \(BD \perp AC\), then \(\angle C + \angle D\) is: | [
"135^\\circ"
] | [
"Triangle \\(ABC\\) and \\(ABD\\) are isosceles with \\(AB = AC = BD\\), and \\(BD\\) intersects \\(AC\\) at \\(E\\) (point \\(E\\) is on segment \\(AC\\) and segment \\(BD\\)). If \\(BD \\perp AC\\), then \\(\\angle C + \\angle D\\) is:"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f10fdbde1ce52382f895d49054d27e16167e1804724b85c51fc7b796849e18b2 | Triangle \(ABC\) has \(AB = 13\) and \(AC = 15\), and the altitude to \(\overline{BC}\) has length 12. What is the sum of the two possible values of \(BC\)? | [
"18"
] | [
"Triangle \\(ABC\\) has \\(AB = 13\\) and \\(AC = 15\\), and the altitude to \\(\\overline{BC}\\) has length 12. What is the sum of the two possible values of \\(BC\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0fa25a30d4f608f4e51fde88fe01d2bd76d29dec750900daacad6ae3caa5e98d | Triangle \(ABC\) has \(AB = 13\), \(BC = 14\), and \(AC = 15\). The internal angle bisector of \(A\) intersects \(BC\) at \(D\). A perpendicular from \(C\) intersects the extension of \(AD\) at \(E\). Find \(CE\). | [
"\\(\\frac{6\\sqrt{65}}{13}\\)"
] | [
"Triangle \\(ABC\\) has \\(AB = 13\\), \\(BC = 14\\), and \\(AC = 15\\). The internal angle bisector of \\(A\\) intersects \\(BC\\) at \\(D\\). A perpendicular from \\(C\\) intersects the extension of \\(AD\\) at \\(E\\). Find \\(CE\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c246b82fb23dc0acbf976a21e1163cff43b8e76d1c7f662dfc6de99c8afd91d5 | Triangle \(ABC\) has \(AB = 13\), \(BC = 14\), and \(CA = 15\). Point \(P\) lies on \(\overline{BC}\) such that the circumcircle \(\Omega\) of \(\triangle APC\) intersects \(\overline{AB}\) at \(Q \neq A\) and \(AQ = \frac{13}{4}\). Then \(BP = \frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers... | [
"563"
] | [
"Triangle \\(ABC\\) has \\(AB = 13\\), \\(BC = 14\\), and \\(CA = 15\\). Point \\(P\\) lies on \\(\\overline{BC}\\) such that the circumcircle \\(\\Omega\\) of \\(\\triangle APC\\) intersects \\(\\overline{AB}\\) at \\(Q \\neq A\\) and \\(AQ = \\frac{13}{4}\\). Then \\(BP = \\frac{m}{n}\\), where \\(m\\) and \\(n\\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-81e5a543468f1bf5c842325d8d491cd2d4afecc37cb29871e5657a18f4e25162 | Triangle \(ABC\) has \(AB = 14\), \(BC = 13\), \(CA = 15\). \(D\) is its orthocenter, \(E\) is its circumcenter, \(F\) is the intersection of its angle bisectors, and \(G\) is its centroid. What is the area of quadrilateral \(DEFG\)? | [
"\\(\\frac{1}{16}\\)"
] | [
"Triangle \\(ABC\\) has \\(AB = 14\\), \\(BC = 13\\), \\(CA = 15\\). \\(D\\) is its orthocenter, \\(E\\) is its circumcenter, \\(F\\) is the intersection of its angle bisectors, and \\(G\\) is its centroid. What is the area of quadrilateral \\(DEFG\\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-64a2949a5039a09eead04f4ec2d84bb2e6f7f4167351c2eadbc02de5321a91bc | Triangle \(ABC\) has \(AB = 15\), \(AC = 14\), and \(BC = 13\). Let its incenter be \(I\), and let its incircle be tangent to \(AB\), \(AC\), and \(BC\) at points \(Z\), \(Y\), and \(X\) respectively. There are two points, \(Y'\) and \(Z'\) which lie on rays \(\overrightarrow{IY}\) and \(\overrightarrow{IZ}\) such that... | [
"17"
] | [
"Triangle \\(ABC\\) has \\(AB = 15\\), \\(AC = 14\\), and \\(BC = 13\\). Let its incenter be \\(I\\), and let its incircle be tangent to \\(AB\\), \\(AC\\), and \\(BC\\) at points \\(Z\\), \\(Y\\), and \\(X\\) respectively. There are two points, \\(Y'\\) and \\(Z'\\) which lie on rays \\(\\overrightarrow{IY}\\) and... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2613951b4d681762d56970b25bfc6fb007a2ff724db96b8cc75eb872959e7708 | Triangle \(ABC\) has \(AB = 36\), \(AC = 72\), and \(\angle B = 90^\circ\). Find the area of the set of points \(P\) inside the triangle such that if creases are made by folding (and then unfolding) each of \(A\), \(B\), \(C\) to \(P\), then the creases do not overlap. | [
"162\\sqrt{3}"
] | [
"Triangle \\(ABC\\) has \\(AB = 36\\), \\(AC = 72\\), and \\(\\angle B = 90^\\circ\\). Find the area of the set of points \\(P\\) inside the triangle such that if creases are made by folding (and then unfolding) each of \\(A\\), \\(B\\), \\(C\\) to \\(P\\), then the creases do not overlap."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-cdc9308cb4f18c81b26bd22eaec9f637f4eea36e0fc21567cd046cbdc6c84c5e | Triangle \(ABC\) has \(AB = 40\), \(AC = 31\), and \(\sin A = \frac{1}{5}\). This triangle is inscribed in rectangle \(AQRS\) with \(B\) on \(\overline{QR}\) and \(C\) on \(\overline{RS}\). Find the maximum possible area of \(AQRS\). | [
"744"
] | [
"Triangle \\(ABC\\) has \\(AB = 40\\), \\(AC = 31\\), and \\(\\sin A = \\frac{1}{5}\\). This triangle is inscribed in rectangle \\(AQRS\\) with \\(B\\) on \\(\\overline{QR}\\) and \\(C\\) on \\(\\overline{RS}\\). Find the maximum possible area of \\(AQRS\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-6166889d789083c42582272ee1dc8b3dd6d33709c62af4b3d4b318c835911d4c | Triangle \(ABC\) has \(AB = 40\), \(AC = 31\), and \(\sin A = \frac{1}{5}\). This triangle is inscribed in rectangle \(AQRS\) with \(B\) on \(\overline{QR}\) and \(C\) on \(\overline{RS}\). Find the maximum possible area of rectangle \(AQRS\). | [
"744"
] | [
"Triangle \\(ABC\\) has \\(AB = 40\\), \\(AC = 31\\), and \\(\\sin A = \\frac{1}{5}\\). This triangle is inscribed in rectangle \\(AQRS\\) with \\(B\\) on \\(\\overline{QR}\\) and \\(C\\) on \\(\\overline{RS}\\). Find the maximum possible area of rectangle \\(AQRS\\)."
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 22,
"genselect": 5
} | 27 | 0 | 0 |
omr-ws-d0833d387f3c4a3293157139b7fc4826dee07a5f659ee95b376513e9abe27ef4 | Triangle \(ABC\) has \(AB = 4\), \(BC = 6\), and \(CA = 5\). Let \(M\) be the midpoint of \(BC\) and \(P\) be the point on the circumcircle of triangle \(ABC\) such that \(\angle MPA = 90^\circ\). Let points \(D\) and \(E\) lie on \(AC\) and \(AB\) respectively such that \(BD \perp AC\) and \(CE \perp AB\). Find \(\fra... | [
"\\dfrac{4}{3}"
] | [
"Triangle \\(ABC\\) has \\(AB = 4\\), \\(BC = 6\\), and \\(CA = 5\\). Let \\(M\\) be the midpoint of \\(BC\\) and \\(P\\) be the point on the circumcircle of triangle \\(ABC\\) such that \\(\\angle MPA = 90^\\circ\\). Let points \\(D\\) and \\(E\\) lie on \\(AC\\) and \\(AB\\) respectively such that \\(BD \\perp AC... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-15bf033914fb949e4d799ef0086e0f6d6a1945db851b38b5dcd8129e1163ee1e | Triangle \(ABC\) has \(AB = 5\) and \(BC = 12\). For some real numbers \(x \in [0, 1]\) and \(t\), if \(\cos(m\angle ABC) = x\), then \(CA^2 = t\), but if \(\sin(m\angle ABC) = x\), then \(CA^2 = t - 24\). Compute \(t\). | [
"97"
] | [
"Triangle \\(ABC\\) has \\(AB = 5\\) and \\(BC = 12\\). For some real numbers \\(x \\in [0, 1]\\) and \\(t\\), if \\(\\cos(m\\angle ABC) = x\\), then \\(CA^2 = t\\), but if \\(\\sin(m\\angle ABC) = x\\), then \\(CA^2 = t - 24\\). Compute \\(t\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-369376c25379b3f41c8799d9f7f704e16be2eadf8c32e855e89d2f8aac41a164 | Triangle \(ABC\) has \(AB = 75\), \(BC = 78\), and \(CA = 51\). Triangle \(DEF\) fully contains \(ABC\) and has \(DE\) parallel to \(AB\), \(EF\) parallel to \(BC\), and \(FD\) parallel to \(CA\) with a distance of 3 between corresponding sides. What is the area of \(DEF\)? | [
"2499"
] | [
"Triangle \\(ABC\\) has \\(AB = 75\\), \\(BC = 78\\), and \\(CA = 51\\). Triangle \\(DEF\\) fully contains \\(ABC\\) and has \\(DE\\) parallel to \\(AB\\), \\(EF\\) parallel to \\(BC\\), and \\(FD\\) parallel to \\(CA\\) with a distance of 3 between corresponding sides. What is the area of \\(DEF\\)?"
] | [
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-ae62181502803f10f6874976ffdbe614e8b2a67579a6f8f2d8e0a13207fe130b | Triangle \(ABC\) has \(AB = 7\), \(BC = 10\), and \(CA = 11\). Let \(t_1\) and \(t_2\) be the trisectors from \(A\) to \(BC\), \(t_3\) and \(t_4\) be the trisectors from \(B\) to \(CA\), and \(t_5\) and \(t_6\) be the trisectors from \(C\) to \(AB\). Find \(t_1^2 + t_2^2 + t_3^2 + t_4^2 + t_5^2 + t_6^2\). Clarification... | [
"420"
] | [
"Triangle \\(ABC\\) has \\(AB = 7\\), \\(BC = 10\\), and \\(CA = 11\\). Let \\(t_1\\) and \\(t_2\\) be the trisectors from \\(A\\) to \\(BC\\), \\(t_3\\) and \\(t_4\\) be the trisectors from \\(B\\) to \\(CA\\), and \\(t_5\\) and \\(t_6\\) be the trisectors from \\(C\\) to \\(AB\\). Find \\(t_1^2 + t_2^2 + t_3^2 + ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 15
} | 29 | 0 | 0 |
omr-ws-9d0ccca40e09a9ef64fc2ea7ff8f0f27a0391922b3a9ef2aad8b9c87a4892059 | Triangle \(ABC\) has \(AB > AC\) and \(\angle A = 60^\circ\). Let \(M\) be the midpoint of \(BC\), and \(N\) be the point on segment \(AB\) such that \(\angle BNM = 30^\circ\). Let \(D\) and \(E\) be points on \(AB\) and \(AC\) respectively. Let \(F\), \(G\), and \(H\) be the midpoints of \(BE\), \(CD\), and \(DE\) res... | [
"7"
] | [
"Triangle \\(ABC\\) has \\(AB > AC\\) and \\(\\angle A = 60^\\circ\\). Let \\(M\\) be the midpoint of \\(BC\\), and \\(N\\) be the point on segment \\(AB\\) such that \\(\\angle BNM = 30^\\circ\\). Let \\(D\\) and \\(E\\) be points on \\(AB\\) and \\(AC\\) respectively. Let \\(F\\), \\(G\\), and \\(H\\) be the midp... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0a043b25ed7c368508de1e0a7e0411c321065494b9c149382b71cc30c6514fb7 | Triangle \(ABC\) has \(AB=13\), \(BC=5\), and \(CA=12\). \(D\) is on \(AB\) such that \(CD\) is perpendicular to \(AB\), and \(E\) is on \(AC\) such that \(DE\) is perpendicular to \(AC\). The length \(DE\) can be expressed as \(\frac{m}{n}\) where \(\gcd(m,n)=1\). Find \(m+n\). | [
"229"
] | [
"Triangle \\(ABC\\) has \\(AB=13\\), \\(BC=5\\), and \\(CA=12\\). \\(D\\) is on \\(AB\\) such that \\(CD\\) is perpendicular to \\(AB\\), and \\(E\\) is on \\(AC\\) such that \\(DE\\) is perpendicular to \\(AC\\). The length \\(DE\\) can be expressed as \\(\\frac{m}{n}\\) where \\(\\gcd(m,n)=1\\). Find \\(m+n\\)."
... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-94e6fdfa8fd659feec56913ff6a60189248da65af06da8112daf238bf78c4318 | Triangle \(ABC\) has \(AB=15\), \(BC=14\), and \(AC=13\). The altitude from \(A\) to \(BC\) is extended to meet the circumcircle of \(ABC\) at \(D\). Find \(AD\). | [
"15.75"
] | [
"Triangle \\(ABC\\) has \\(AB=15\\), \\(BC=14\\), and \\(AC=13\\). The altitude from \\(A\\) to \\(BC\\) is extended to meet the circumcircle of \\(ABC\\) at \\(D\\). Find \\(AD\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-51a49ebf6c78481010be994667996091fcb2388c82d848cb756c9219fd6dd240 | Triangle \(ABC\) has \(AB=3\), \(BC=4\), and \(AC=5\). \(AB\) is extended beyond \(B\) to point \(R\) so that \(AR=8\), \(BC\) is extended beyond \(C\) to point \(S\) so that \(BS=8\), and \(AC\) is extended beyond \(A\) to point \(T\) so that \(CT=8\). What is the area of triangle \(RST\)? | [
"\\(\\frac{226}{5}\\)"
] | [
"Triangle \\(ABC\\) has \\(AB=3\\), \\(BC=4\\), and \\(AC=5\\). \\(AB\\) is extended beyond \\(B\\) to point \\(R\\) so that \\(AR=8\\), \\(BC\\) is extended beyond \\(C\\) to point \\(S\\) so that \\(BS=8\\), and \\(AC\\) is extended beyond \\(A\\) to point \\(T\\) so that \\(CT=8\\). What is the area of triangle ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-38e06855d7f5f952efbe9aa956536ef013bea394da65d0e6161f87df42006e7e | Triangle \(ABC\) has \(AB=4\), \(BC=3\), and \(AC=5\), with circumcenter \(O\) and incenter \(I\). What is the area of \(\triangle OIC\)? | [
"\\(\\frac{5}{4}\\)"
] | [
"Triangle \\(ABC\\) has \\(AB=4\\), \\(BC=3\\), and \\(AC=5\\), with circumcenter \\(O\\) and incenter \\(I\\). What is the area of \\(\\triangle OIC\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-4a7a000b6dca7b2a5f10a84288c246ae7951e7b15920a2e874394e948df87972 | Triangle \(ABC\) has \(AC = 3\), \(BC = 4\), and \(AB = 5\). Point \(D\) is on \(\overline{AB}\), and \(\overline{CD}\) bisects the right angle. The inscribed circles of \(\triangle ADC\) and \(\triangle BCD\) have radii \(r_a\) and \(r_b\), respectively. What is \(r_a/r_b\)? | [
"\\(\\frac{3}{28} \\left(10 - \\sqrt{2}\\right)\\)"
] | [
"Triangle \\(ABC\\) has \\(AC = 3\\), \\(BC = 4\\), and \\(AB = 5\\). Point \\(D\\) is on \\(\\overline{AB}\\), and \\(\\overline{CD}\\) bisects the right angle. The inscribed circles of \\(\\triangle ADC\\) and \\(\\triangle BCD\\) have radii \\(r_a\\) and \\(r_b\\), respectively. What is \\(r_a/r_b\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 39,
"genselect": 7,
"tir": 26
} | 72 | 0 | 0 |
omr-ws-abfe6b272f5790f8c80a9a301e19b3745e760b4837b38314343dbf2ebb78b628 | Triangle \(ABC\) has \(AC = 3\), \(BC = 5\), and \(AB = 7\). A circle is drawn internally tangent to the circumcircle of \(ABC\) at \(C\) and tangent to \(AB\). Let \(D\) be its point of tangency with \(AB\). Find \(BD - DA\). | [
"\\(\\frac{7}{4}\\)"
] | [
"Triangle \\(ABC\\) has \\(AC = 3\\), \\(BC = 5\\), and \\(AB = 7\\). A circle is drawn internally tangent to the circumcircle of \\(ABC\\) at \\(C\\) and tangent to \\(AB\\). Let \\(D\\) be its point of tangency with \\(AB\\). Find \\(BD - DA\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 17
} | 36 | 0 | 0 |
omr-ws-2f6fce21c197f7f91f6e91efb661dbe9bbdca111cb94dd719aa22abe8d19f490 | Triangle \(ABC\) has \(AC = BC = 13\) and \(AB = 10\). The triangle is inscribed in a circle with center \(O\). Segment \(BO\) is extended so that it divides \(\triangle ABC\) into two regions. Find the ratio of the area of the smaller region to the area of the larger region. Express your answer as a common fraction. | [
"\\(\\frac{13}{16}\\)"
] | [
"Triangle \\(ABC\\) has \\(AC = BC = 13\\) and \\(AB = 10\\). The triangle is inscribed in a circle with center \\(O\\). Segment \\(BO\\) is extended so that it divides \\(\\triangle ABC\\) into two regions. Find the ratio of the area of the smaller region to the area of the larger region. Express your answer as a ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c60fdcf4f5f3a1ef42a2220d94ff1cb1979a6e75dfd2b2d8ac311789f9f0865d | Triangle \(ABC\) has \(AC = BC\) and \(\angle ACB = 106^\circ\). Point \(M\) is inside the triangle such that \(\angle MAC = 7^\circ\) and \(\angle MCA = 23^\circ\). Find \(\angle CMB\). Explanation: The problem involves finding the measure of \(\angle CMB\) in an isosceles triangle \(ABC\) with given angles and a poin... | [
"83"
] | [
"Triangle \\(ABC\\) has \\(AC = BC\\) and \\(\\angle ACB = 106^\\circ\\). Point \\(M\\) is inside the triangle such that \\(\\angle MAC = 7^\\circ\\) and \\(\\angle MCA = 23^\\circ\\). Find \\(\\angle CMB\\).\n\nExplanation: The problem involves finding the measure of \\(\\angle CMB\\) in an isosceles triangle \\(A... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-d1f64d826fd9a0807316324ecc16df9aaf88af73afc184c625ce5a0798d46114 | Triangle \(ABC\) has \(BC = 20\). The incircle of the triangle evenly trisects the median \(AD\). If the area of the triangle is \(K\), find the remainder when \(10K\) is divided by 1000. | [
"732"
] | [
"Triangle \\(ABC\\) has \\(BC = 20\\). The incircle of the triangle evenly trisects the median \\(AD\\). If the area of the triangle is \\(K\\), find the remainder when \\(10K\\) is divided by 1000."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-34a0fb2ba039cda0397e994a2644482cf0fe3029bf22bfbc4bcff50ac5bdde07 | Triangle \(ABC\) has \(BC = 20\). The incircle of the triangle evenly trisects the median \(AD\). If the area of the triangle is \(m \sqrt{n}\) where \(m\) and \(n\) are integers and \(n\) is not divisible by the perfect square of a prime, find \(m + n\). | [
"103"
] | [
"Triangle \\(ABC\\) has \\(BC = 20\\). The incircle of the triangle evenly trisects the median \\(AD\\). If the area of the triangle is \\(m \\sqrt{n}\\) where \\(m\\) and \\(n\\) are integers and \\(n\\) is not divisible by the perfect square of a prime, find \\(m + n\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-5f74ae4b4aef67b872fc9e335b9f62992d4323caff8fcdb06a210954dd28c6c9 | Triangle \(ABC\) has \(BC = 7\), \(AC = 13\), \(AB = 9\), incenter \(I\), and incircle \(\omega\). Let \(\omega\) touch \(BC\) at \(D\), and let \(E\) be the point on \(\omega\) such that \(\angle AED = 90^\circ\). Let \(F\) be the intersection of lines \(DE\) and \(AI\), and let \(G\) be the point on \(BC\) such that ... | [
"170"
] | [
"Triangle \\(ABC\\) has \\(BC = 7\\), \\(AC = 13\\), \\(AB = 9\\), incenter \\(I\\), and incircle \\(\\omega\\). Let \\(\\omega\\) touch \\(BC\\) at \\(D\\), and let \\(E\\) be the point on \\(\\omega\\) such that \\(\\angle AED = 90^\\circ\\). Let \\(F\\) be the intersection of lines \\(DE\\) and \\(AI\\), and let... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 5,
"tir": 1
} | 8 | 0 | 0 |
omr-ws-00a46591bef1b8328381421fb1f31cbb46eb88d7ccecdf2f09ef94d650334f86 | Triangle \(ABC\) has \(\angle A = 15^\circ\), \(\angle B = 120^\circ\), and \(AC = 8\). Compute the area of \(\triangle ABC\). | [
"\\(16 - \\frac{16\\sqrt{3}}{3}\\)"
] | [
"Triangle \\(ABC\\) has \\(\\angle A = 15^\\circ\\), \\(\\angle B = 120^\\circ\\), and \\(AC = 8\\). Compute the area of \\(\\triangle ABC\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-692597b870a4167ded726baf4cd53921713efdf1e472537d0d6bbe2a34fab693 | Triangle \(ABC\) has \(\angle A = 60^\circ\) and is inscribed in a circle with radius 10. Point \(D\) is the center of the arc \(BC\) (not containing \(A\)). Point \(Q\) is the incenter of triangle \(ABC\). Find the length of \(DQ\). | [
"10"
] | [
"Triangle \\(ABC\\) has \\(\\angle A = 60^\\circ\\) and is inscribed in a circle with radius 10. Point \\(D\\) is the center of the arc \\(BC\\) (not containing \\(A\\)). Point \\(Q\\) is the incenter of triangle \\(ABC\\). Find the length of \\(DQ\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-5d9ef587334be29ecc79d5383693ec384662cd7030d8d8bce6c243509fbfde75 | Triangle \(ABC\) has \(\angle B = 60^\circ\), \(\angle C = 90^\circ\), and \(AB = 1\). Triangles \(BCP\), \(CAQ\), and \(ABR\) are equilateral triangles external to \(ABC\). The segments \(QR\) and \(AB\) intersect at the point \(T\). Find the area of triangle \(PRT\). | [
"\\dfrac{9\\sqrt{3}}{32}"
] | [
"Triangle \\(ABC\\) has \\(\\angle B = 60^\\circ\\), \\(\\angle C = 90^\\circ\\), and \\(AB = 1\\). Triangles \\(BCP\\), \\(CAQ\\), and \\(ABR\\) are equilateral triangles external to \\(ABC\\). The segments \\(QR\\) and \\(AB\\) intersect at the point \\(T\\). Find the area of triangle \\(PRT\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cf2698d315e9e3b29bcc8e2990f597ec3352c43313ec5803fe61602b42831ed9 | Triangle \(ABC\) has \(\angle BAC = 60^\circ\), \(\angle CBA \le 90^\circ\), \(BC = 1\), and \(AC \ge AB\). Let \(H\), \(I\), and \(O\) be the orthocenter, incenter, and circumcenter of \(\triangle ABC\), respectively. Assume that the area of the pentagon \(BCOIH\) is the maximum possible. What is \(\angle CBA\)? | [
"80^\\circ"
] | [
"Triangle \\(ABC\\) has \\(\\angle BAC = 60^\\circ\\), \\(\\angle CBA \\le 90^\\circ\\), \\(BC = 1\\), and \\(AC \\ge AB\\). Let \\(H\\), \\(I\\), and \\(O\\) be the orthocenter, incenter, and circumcenter of \\(\\triangle ABC\\), respectively. Assume that the area of the pentagon \\(BCOIH\\) is the maximum possibl... | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-6745450069939f7b165665ece76b1c22caedb0264a5a980c8184778e6d76d411 | Triangle \(ABC\) has \(\angle BAC = 60^\circ\), \(\angle CBA \leq 90^\circ\), \(BC = 1\), and \(AC \geq AB\). Let \(H\), \(I\), and \(O\) be the orthocenter, incenter, and circumcenter of \(\triangle ABC\), respectively. Assume that the area of pentagon \(BCOIH\) is the maximum possible. What is \(\angle CBA\)? | [
"\\(80^\\circ\\)"
] | [
"Triangle \\(ABC\\) has \\(\\angle BAC = 60^\\circ\\), \\(\\angle CBA \\leq 90^\\circ\\), \\(BC = 1\\), and \\(AC \\geq AB\\). Let \\(H\\), \\(I\\), and \\(O\\) be the orthocenter, incenter, and circumcenter of \\(\\triangle ABC\\), respectively. Assume that the area of pentagon \\(BCOIH\\) is the maximum possible.... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-c866d136b88518f7908c4049d77fb2a02e9ec25a4a2725a52edecfc5c574ab34 | Triangle \(ABC\) has a circumcircle \(\gamma\). Let \(\omega\) be a circle tangent to \(AB\) and \(AC\) that intersects \(\gamma\) at two points \(X\) and \(Y\). Let \(P = BC \cap XY\) and \(L\) be a point on \(\omega\) such that \(PL\) is tangent to \(\omega\). Let \(S = AL \cap BC\) and \(D\) be the tangency point of... | [
"1"
] | [
"Triangle \\(ABC\\) has a circumcircle \\(\\gamma\\). Let \\(\\omega\\) be a circle tangent to \\(AB\\) and \\(AC\\) that intersects \\(\\gamma\\) at two points \\(X\\) and \\(Y\\). Let \\(P = BC \\cap XY\\) and \\(L\\) be a point on \\(\\omega\\) such that \\(PL\\) is tangent to \\(\\omega\\). Let \\(S = AL \\cap ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9462f1a52d25b98c47aea2745c42969ec62bbf6c87cc721a7d6a4a3dafa67a04 | Triangle \(ABC\) has a circumcircle with center \(O\). A new circle \(k\) is drawn with \(O\) as its center and is tangent to \(AB\) at \(H\). Circle \(k\) intersects \(BC\) at points \(D\) and \(E\) and intersects \(AC\) at points \(F\) and \(G\). It is given that \(CE < CD\) and \(CF < CG\). Calculate the value of \(... | [
"1"
] | [
"Triangle \\(ABC\\) has a circumcircle with center \\(O\\). A new circle \\(k\\) is drawn with \\(O\\) as its center and is tangent to \\(AB\\) at \\(H\\). Circle \\(k\\) intersects \\(BC\\) at points \\(D\\) and \\(E\\) and intersects \\(AC\\) at points \\(F\\) and \\(G\\). It is given that \\(CE < CD\\) and \\(CF... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c8a3e4372188d1f8f5edee35ede48ab0d0fd79699f4fffa6abf0be7a2c80cf3e | Triangle \(ABC\) has a perimeter of 1. Its three altitudes form the side lengths of another triangle. Find the set of all possible values of \(\min(AB, BC, CA)\). | [
"\\left( \\dfrac{1}{5}, \\dfrac{1}{3} \\right]"
] | [
"Triangle \\(ABC\\) has a perimeter of 1. Its three altitudes form the side lengths of another triangle. Find the set of all possible values of \\(\\min(AB, BC, CA)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-0181454beb2a7ff5881e508c77a0be9be66fa2fa9b151f4aaebb82cbc1364de7 | Triangle \(ABC\) has a right angle at \(A\). Among all points \(P\) on the perimeter of the triangle, find the position of \(P\) such that \(AP + BP + CP\) is minimized. | [
"\\(P = A\\)"
] | [
"Triangle \\(ABC\\) has a right angle at \\(A\\). Among all points \\(P\\) on the perimeter of the triangle, find the position of \\(P\\) such that \\(AP + BP + CP\\) is minimized."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-27c0537f9eb812242f1c2d6984d405dd5411c5be610be478f3070270ce6f9d27 | Triangle \(ABC\) has a right angle at \(B\), and points \(D\) and \(E\) are on \(AC\) such that \(AD=3\), \(CE=5\), and \(\angle DBE = 45^\circ\). Find the length of \(DE\). | [
"15"
] | [
"Triangle \\(ABC\\) has a right angle at \\(B\\), and points \\(D\\) and \\(E\\) are on \\(AC\\) such that \\(AD=3\\), \\(CE=5\\), and \\(\\angle DBE = 45^\\circ\\). Find the length of \\(DE\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-559be38165b9b11b6ae6602c0903b4b9b5e1ddb97371e8bee0652f672e0969b2 | Triangle \(ABC\) has a right angle at \(B\). Point \(D\) is the foot of the altitude from \(B\), \(AD = 3\), and \(DC = 4\). What is the area of \(\triangle ABC\)? | [
"\\(7\\sqrt{3}\\)"
] | [
"Triangle \\(ABC\\) has a right angle at \\(B\\). Point \\(D\\) is the foot of the altitude from \\(B\\), \\(AD = 3\\), and \\(DC = 4\\). What is the area of \\(\\triangle ABC\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-8646fb058fdfa41c3cd034ca4b3cd1a5803999c7a57fdef658dd1713d8f66b8d | Triangle \(ABC\) has a right angle at \(C\), with \(AC = 3\) and \(BC = 4\). Triangle \(ABD\) has a right angle at \(A\) and \(AD = 12\). Points \(C\) and \(D\) are on opposite sides of \(\overline{AB}\). The line through \(D\) parallel to \(\overline{AC}\) meets \(\overline{CB}\) extended at \(E\). If \(\frac{DE}{DB} ... | [
"128"
] | [
"Triangle \\(ABC\\) has a right angle at \\(C\\), with \\(AC = 3\\) and \\(BC = 4\\). Triangle \\(ABD\\) has a right angle at \\(A\\) and \\(AD = 12\\). Points \\(C\\) and \\(D\\) are on opposite sides of \\(\\overline{AB}\\). The line through \\(D\\) parallel to \\(\\overline{AC}\\) meets \\(\\overline{CB}\\) exte... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 2
} | 50 | 0 | 0 |
omr-ws-09653f1bafedc7255ab5f9bf9704e11f778b334fcf0dc01b39496327920c9ba1 | Triangle \(ABC\) has an incircle \((I)\) that is tangent to \(BC\) and \(CA\) at points \(D\) and \(E\) respectively. The circle centered at \(A\) with radius \(AE\) intersects the altitude \(AH\) at point \(M\). The line \(DM\) intersects the circle \((A, AE)\) again at point \(N\). Let \(P\) be the intersection point... | [
"\\dfrac{5}{2}"
] | [
"Triangle \\(ABC\\) has an incircle \\((I)\\) that is tangent to \\(BC\\) and \\(CA\\) at points \\(D\\) and \\(E\\) respectively. The circle centered at \\(A\\) with radius \\(AE\\) intersects the altitude \\(AH\\) at point \\(M\\). The line \\(DM\\) intersects the circle \\((A, AE)\\) again at point \\(N\\). Let ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-89896fc3052d136c3ec08d5ece94863ab31efadc172c00c3fe5d0ba486648fb6 | Triangle \(ABC\) has an incircle \(\omega_1\). Point \(A\) is reflected over the incenter to get \(A'\). Similarly, \(B'\) and \(C'\) are constructed. The lines \(AB'\), \(BC'\), and \(CA'\) are drawn. The circumcircle \(\omega_2\) of \(\triangle A'B'C'\) intersects these lines at \(D\), \(E\), and \(F\), respectively,... | [
"21"
] | [
"Triangle \\(ABC\\) has an incircle \\(\\omega_1\\). Point \\(A\\) is reflected over the incenter to get \\(A'\\). Similarly, \\(B'\\) and \\(C'\\) are constructed. The lines \\(AB'\\), \\(BC'\\), and \\(CA'\\) are drawn. The circumcircle \\(\\omega_2\\) of \\(\\triangle A'B'C'\\) intersects these lines at \\(D\\),... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-30b5ef151d401e61ce5754947d71f8c724baaa59cfd23e305c533c178c9983fa | Triangle \(ABC\) has base \(BC = a\) and height from \(A\) to \(BC\) equal to \(h\), with \(h < \frac{\sqrt{3}}{2}a\). Point \(P\) is inside the triangle such that \(\angle PAB = \angle PBA = \angle PCB\). If \(\angle PAB = \theta\), find the value of \(\tan(\theta)\) in terms of \(a\) and \(h\). | [
"\\dfrac{h}{a + \\sqrt{a^2 - h^2}}"
] | [
"Triangle \\(ABC\\) has base \\(BC = a\\) and height from \\(A\\) to \\(BC\\) equal to \\(h\\), with \\(h < \\frac{\\sqrt{3}}{2}a\\). Point \\(P\\) is inside the triangle such that \\(\\angle PAB = \\angle PBA = \\angle PCB\\). If \\(\\angle PAB = \\theta\\), find the value of \\(\\tan(\\theta)\\) in terms of \\(a\... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-2600eca5bfe4b6a7528ecbc043226011fdcc161ebf266fc216e4c2006a62a709 | Triangle \(ABC\) has base \(BC = a\) and height from \(A\) to \(BC\) equal to \(h\), with \(h < \frac{\sqrt{3}}{2}a\). Point \(P\) is inside the triangle such that \(\angle PAB = \angle PBA = \angle PCB\). Let \(\theta\) be the measure of \(\angle PAB\), \(\angle PBA\), and \(\angle PCB\). If \(a = 10\) and \(h = 4\), ... | [
"\\dfrac{1}{3}",
"\\dfrac{5 - \\sqrt{21}}{2}"
] | [
"Triangle \\(ABC\\) has base \\(BC = a\\) and height from \\(A\\) to \\(BC\\) equal to \\(h\\), with \\(h < \\frac{\\sqrt{3}}{2}a\\). Point \\(P\\) is inside the triangle such that \\(\\angle PAB = \\angle PBA = \\angle PCB\\). Let \\(\\theta\\) be the measure of \\(\\angle PAB\\), \\(\\angle PBA\\), and \\(\\angle... | [
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-b72c590a69ee1442353d9d1ed3d92e83e0996d889f440c93fcf541e1e94bba9c | Triangle \(ABC\) has circumcenter \(O\) and incenter \(I\). \(AI\) intersects \(BC\) at \(D\), and define \(E, F\) similarly. The circle with center \(D\) and radius \(DI\) is denoted as \(\omega_D\), define \(\omega_E, \omega_F\) similarly. The intersection of \(\omega_E\) and \(\omega_F\) not equal to \(I\) is \(X\),... | [
"5"
] | [
"Triangle \\(ABC\\) has circumcenter \\(O\\) and incenter \\(I\\). \\(AI\\) intersects \\(BC\\) at \\(D\\), and define \\(E, F\\) similarly. The circle with center \\(D\\) and radius \\(DI\\) is denoted as \\(\\omega_D\\), define \\(\\omega_E, \\omega_F\\) similarly. The intersection of \\(\\omega_E\\) and \\(\\ome... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-10a510f2d8e08298f81b18598dc61e72cefe9e589814670dd194fe50a686a174 | Triangle \(ABC\) has circumcircle \((O)\) and incircle \((I)\). Point \(D\) lies on \(BC\). Lines \(l_1\), \(l_2\), and \(l_3\) are such that \(l_1\) and \(l_2\) are parallel to \(AD\), and \(l_3\) is parallel to \(BC\), with all three lines being tangent to \((O)\). Let \(l_1\) intersect \(l_3\) at \(M\), and \(l_2\) ... | [
"\\dfrac{3}{4}"
] | [
"Triangle \\(ABC\\) has circumcircle \\((O)\\) and incircle \\((I)\\). Point \\(D\\) lies on \\(BC\\). Lines \\(l_1\\), \\(l_2\\), and \\(l_3\\) are such that \\(l_1\\) and \\(l_2\\) are parallel to \\(AD\\), and \\(l_3\\) is parallel to \\(BC\\), with all three lines being tangent to \\((O)\\). Let \\(l_1\\) inter... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-d3c1ce59d5f0b76e3837ed8fdc9d45ec28e5d22dc0b688b13e609842fb5f48c4 | Triangle \(ABC\) has circumcircle \((O, R)\) and orthocenter \(H\). The symmedians through \(A\), \(B\), and \(C\) meet the perpendicular bisectors of \(BC\), \(CA\), and \(AB\) at \(D\), \(E\), and \(F\) respectively. Let \(M\), \(N\), and \(P\) be the perpendicular projections of \(H\) on the lines \(OD\), \(OE\), an... | [
"3"
] | [
"Triangle \\(ABC\\) has circumcircle \\((O, R)\\) and orthocenter \\(H\\). The symmedians through \\(A\\), \\(B\\), and \\(C\\) meet the perpendicular bisectors of \\(BC\\), \\(CA\\), and \\(AB\\) at \\(D\\), \\(E\\), and \\(F\\) respectively. Let \\(M\\), \\(N\\), and \\(P\\) be the perpendicular projections of \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-287227c1cca0849501d84b2aa04c03b9a9a74c8b56b780dea8ca76d923698bdf | Triangle \(ABC\) has circumcircle \(\Gamma\). Let \(P\) be a point inside \(\triangle ABC\) and \(K\) be a point on (possibly extended) \(BC\). \(AP\) meets \(\Gamma\) again at \(R\). The lines \(KA\) and \(KR\) meet \(\Gamma\) again at \(L\) and \(M\) respectively. \(KP\) meets the sides \(AB\) and \(AC\) at \(X\) and... | [
"1"
] | [
"Triangle \\(ABC\\) has circumcircle \\(\\Gamma\\). Let \\(P\\) be a point inside \\(\\triangle ABC\\) and \\(K\\) be a point on (possibly extended) \\(BC\\). \\(AP\\) meets \\(\\Gamma\\) again at \\(R\\). The lines \\(KA\\) and \\(KR\\) meet \\(\\Gamma\\) again at \\(L\\) and \\(M\\) respectively. \\(KP\\) meets t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-70a4e3f994e7b2cd44d3914c74236aaeabc90ef81af32f201aededcf0b7bcfd1 | Triangle \(ABC\) has circumcircle \(\Omega\) and circumcenter \(O\). A circle \(\Gamma\) with center \(A\) intersects the segment \(BC\) at points \(D\) and \(E\), such that \(B\), \(D\), \(E\), and \(C\) are all different and lie on line \(BC\) in this order. Let \(F\) and \(G\) be the points of intersection of \(\Gam... | [
"2"
] | [
"Triangle \\(ABC\\) has circumcircle \\(\\Omega\\) and circumcenter \\(O\\). A circle \\(\\Gamma\\) with center \\(A\\) intersects the segment \\(BC\\) at points \\(D\\) and \\(E\\), such that \\(B\\), \\(D\\), \\(E\\), and \\(C\\) are all different and lie on line \\(BC\\) in this order. Let \\(F\\) and \\(G\\) be... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cd4f82a279781188fb0368f2b9a009ad038134cb7c3a93c8e292a1684265ac94 | Triangle \(ABC\) has circumcircle \(\Omega\) and incircle \(\omega\). Let \(M\) be the midpoint of \(\widehat{BAC}\) in \(\Omega\) and let \(\omega\) and \(BC\) meet at \(D\). Let \(P\) be the unique point on \(\omega\) such that \(\angle MPD = 90^\circ\). The circumcircle of \(\triangle MAP\) intersects \(\omega\) aga... | [
"2R"
] | [
"Triangle \\(ABC\\) has circumcircle \\(\\Omega\\) and incircle \\(\\omega\\). Let \\(M\\) be the midpoint of \\(\\widehat{BAC}\\) in \\(\\Omega\\) and let \\(\\omega\\) and \\(BC\\) meet at \\(D\\). Let \\(P\\) be the unique point on \\(\\omega\\) such that \\(\\angle MPD = 90^\\circ\\). The circumcircle of \\(\\t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-218cee1f46084f360876c12d08d0da6759790c8e86b1b8c7745d2615b5bd0f41 | Triangle \(ABC\) has incenter \(I\), \(A\)-excenter \(I_A\), and \(D\) is the foot of \(I\) onto \(BC\). \(M\) is the midpoint of arc \(\widehat{BAC}\) in \(\odot(ABC)\), \(MA\) intersects \(BC\) at \(E\), and line \(I_AD\) intersects \(\odot(AID)\) again at \(F \neq D\). Let \(O\) be the circumcenter of \(\triangle AM... | [
"1"
] | [
"Triangle \\(ABC\\) has incenter \\(I\\), \\(A\\)-excenter \\(I_A\\), and \\(D\\) is the foot of \\(I\\) onto \\(BC\\). \\(M\\) is the midpoint of arc \\(\\widehat{BAC}\\) in \\(\\odot(ABC)\\), \\(MA\\) intersects \\(BC\\) at \\(E\\), and line \\(I_AD\\) intersects \\(\\odot(AID)\\) again at \\(F \\neq D\\). Let \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-217b846b8ce65e83a2a3e3877f5b67e95c6fc585ce119bf4d6a77d0558857590 | Triangle \(ABC\) has incenter \(I\). Fix a line \(\ell\) tangent to the incircle of triangle \(ABC\) (not containing any of the sides). Let \(A'\), \(B'\), \(C'\) be points on \(\ell\) such that \(\angle AIA' = \angle BIB' = \angle CIC' = 90^\circ\). Let \(P\) be the point of concurrency of \(AA'\), \(BB'\), and \(CC'\... | [
"(2, 3)"
] | [
"Triangle \\(ABC\\) has incenter \\(I\\). Fix a line \\(\\ell\\) tangent to the incircle of triangle \\(ABC\\) (not containing any of the sides). Let \\(A'\\), \\(B'\\), \\(C'\\) be points on \\(\\ell\\) such that \\(\\angle AIA' = \\angle BIB' = \\angle CIC' = 90^\\circ\\). Let \\(P\\) be the point of concurrency ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-d9ecbdc35a16f306453a5105eff337d4eff524a674f6fb3fe82c73877b471927 | Triangle \(ABC\) has incenter \(I\). The incircle touches \(BC\), \(CA\), and \(AB\) at points \(D\), \(E\), and \(F\) respectively. \(J\) is the midpoint of \(BC\). \(AI\) intersects \(DE\) and \(DF\) at points \(M\) and \(N\) respectively. Let \(K\) be the center of the circumcircle of triangle \(JMN\). If \(O\) is t... | [
"\\dfrac{1}{2}"
] | [
"Triangle \\(ABC\\) has incenter \\(I\\). The incircle touches \\(BC\\), \\(CA\\), and \\(AB\\) at points \\(D\\), \\(E\\), and \\(F\\) respectively. \\(J\\) is the midpoint of \\(BC\\). \\(AI\\) intersects \\(DE\\) and \\(DF\\) at points \\(M\\) and \\(N\\) respectively. Let \\(K\\) be the center of the circumcirc... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-3e9764f07d2d71eab5417f3bb977d8fc154cd9da5cc820e0e900f5b13ac2f44d | Triangle \(ABC\) has integer side lengths. Extend \(\overline{AC}\) beyond \(C\) to point \(D\) such that \(CD = 120\). Extend \(\overline{CB}\) beyond \(B\) to point \(E\) such that \(BE = 112\). Extend \(\overline{BA}\) beyond \(A\) to point \(F\) such that \(AF = 104\). If triangles \(CBD\), \(BAE\), and \(ACF\) all... | [
"\\(6720\\)"
] | [
"Triangle \\(ABC\\) has integer side lengths. Extend \\(\\overline{AC}\\) beyond \\(C\\) to point \\(D\\) such that \\(CD = 120\\). Extend \\(\\overline{CB}\\) beyond \\(B\\) to point \\(E\\) such that \\(BE = 112\\). Extend \\(\\overline{BA}\\) beyond \\(A\\) to point \\(F\\) such that \\(AF = 104\\). If triangles... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4739153aff56b522ab4db958262553855cd0e5c82a3da6ff0351f511ab80a730 | Triangle \(ABC\) has its base on line segment \(PN\) and vertex \(A\) on line \(PM\). Circles with centers \(O\) and \(Q\), having radii \(r_1\) and \(r_2\) respectively, are tangent to the triangle \(ABC\) externally and to each of \(PM\) and \(PN\). Let \(T\) be the point of intersection of \(BC\) with the incircle o... | [
"120"
] | [
"Triangle \\(ABC\\) has its base on line segment \\(PN\\) and vertex \\(A\\) on line \\(PM\\). Circles with centers \\(O\\) and \\(Q\\), having radii \\(r_1\\) and \\(r_2\\) respectively, are tangent to the triangle \\(ABC\\) externally and to each of \\(PM\\) and \\(PN\\). Let \\(T\\) be the point of intersection ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-327dbaadd0cf55a23ec87d52229b32fa9e02147682d76b55593b1fac0cb1371f | Triangle \(ABC\) has orthocenter \(H\). Let \(D\) be a point distinct from the vertices on the circumcircle of \(ABC\). Suppose that circle \(BHD\) meets \(AB\) at \(P \ne B\), and circle \(CHD\) meets \(AC\) at \(Q \ne C\). Let \(R\) be the reflection of \(D\) across line \(PQ\). If \(D\) moves on the circumcircle of ... | [
"R_0"
] | [
"Triangle \\(ABC\\) has orthocenter \\(H\\). Let \\(D\\) be a point distinct from the vertices on the circumcircle of \\(ABC\\). Suppose that circle \\(BHD\\) meets \\(AB\\) at \\(P \\ne B\\), and circle \\(CHD\\) meets \\(AC\\) at \\(Q \\ne C\\). Let \\(R\\) be the reflection of \\(D\\) across line \\(PQ\\). If \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-10612f6a6f7e031a65355e29881ff1f689bbbab75483b2eb3a87bb060269f0fa | Triangle \(ABC\) has points \(D\), \(E\), and \(F\) on \(AB\), \(BC\), and \(CA\) respectively, such that \(AD = 2\) and \(DB = 3\), and the area of \(ABC\) is 10. If the area of \(ABE\) equals the area of \(DBEF\), find the area of \(ABE\). | [
"6"
] | [
"Triangle \\(ABC\\) has points \\(D\\), \\(E\\), and \\(F\\) on \\(AB\\), \\(BC\\), and \\(CA\\) respectively, such that \\(AD = 2\\) and \\(DB = 3\\), and the area of \\(ABC\\) is 10. If the area of \\(ABE\\) equals the area of \\(DBEF\\), find the area of \\(ABE\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-5e0ef304cfbaef7987f37be09e6ad123262430fdd20c2da992bddf50e2b60a47 | Triangle \(ABC\) has positive integer side lengths with \(AB = AC\). Let \(I\) be the intersection of the angle bisectors of \(\angle B\) and \(\angle C\). Suppose \(BI = 8\). Find the smallest possible perimeter of \(\triangle ABC\). | [
"108"
] | [
"Triangle \\(ABC\\) has positive integer side lengths with \\(AB = AC\\). Let \\(I\\) be the intersection of the angle bisectors of \\(\\angle B\\) and \\(\\angle C\\). Suppose \\(BI = 8\\). Find the smallest possible perimeter of \\(\\triangle ABC\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-e7e5be020f5a4e5b68c009d9ff6399f6f1092ab99ef34aad3870d6fbbc297c27 | Triangle \(ABC\) has positive integer side lengths with \(AB = AC\). Let \(I\) be the intersection of the bisectors of \(\angle B\) and \(\angle C\). Suppose \(BI = 8\). Find the smallest possible perimeter of \(\triangle ABC\). | [
"108"
] | [
"Triangle \\(ABC\\) has positive integer side lengths with \\(AB = AC\\). Let \\(I\\) be the intersection of the bisectors of \\(\\angle B\\) and \\(\\angle C\\). Suppose \\(BI = 8\\). Find the smallest possible perimeter of \\(\\triangle ABC\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 8
} | 22 | 0 | 0 |
omr-ws-5104be15db9ce49e89547ad18da8d1c877f4f283f419e84280c88a957e875f5b | Triangle \(ABC\) has side lengths \(13\), \(14\), and \(15\). A point \(Q\) is chosen uniformly at random in the interior of \(\triangle ABC\). A random ray with endpoint \(Q\) intersects the perimeter of \(\triangle ABC\) at \(P\). What is the expected value of \(QP^2\)? | [
"\\(\\frac{84}{\\pi}\\)"
] | [
"Triangle \\(ABC\\) has side lengths \\(13\\), \\(14\\), and \\(15\\). A point \\(Q\\) is chosen uniformly at random in the interior of \\(\\triangle ABC\\). A random ray with endpoint \\(Q\\) intersects the perimeter of \\(\\triangle ABC\\) at \\(P\\). What is the expected value of \\(QP^2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 5
} | 18 | 0 | 0 |
omr-ws-fde77c64a35cd674b36f77ff8b779da634ed13163b8185d7cff10ffe5d936e46 | Triangle \(ABC\) has side lengths \(AB = 10\), \(AC = 17\), \(BC = 21\), and circumcircle \(\omega\). Let \(\omega\) with center \(O\) be a circle internally tangent to \(\omega\) at \(A\), and tangent to \(\overline{BC}\) at \(D\), intersecting \(\overline{AB}\) and \(\overline{AC}\) again at points \(E\) and \(F\), r... | [
"85"
] | [
"Triangle \\(ABC\\) has side lengths \\(AB = 10\\), \\(AC = 17\\), \\(BC = 21\\), and circumcircle \\(\\omega\\). Let \\(\\omega\\) with center \\(O\\) be a circle internally tangent to \\(\\omega\\) at \\(A\\), and tangent to \\(\\overline{BC}\\) at \\(D\\), intersecting \\(\\overline{AB}\\) and \\(\\overline{AC}\... | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-1a4185bfaa66718519d24e17f9118c43d60be49029568a410060db16c72b09d9 | Triangle \(ABC\) has side lengths \(AB = 13\), \(BC = 14\), and \(AC = 15\). A point \(P\) is randomly chosen within triangle \(ABC\). What is the probability that the area of \(\triangle ABP\) is greater than the area of \(\triangle ACP\) and the area of \(\triangle BCP\)? | [
"\\dfrac{1}{6}"
] | [
"Triangle \\(ABC\\) has side lengths \\(AB = 13\\), \\(BC = 14\\), and \\(AC = 15\\). A point \\(P\\) is randomly chosen within triangle \\(ABC\\). What is the probability that the area of \\(\\triangle ABP\\) is greater than the area of \\(\\triangle ACP\\) and the area of \\(\\triangle BCP\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-cc125638fccf08d5926ba551cd3846b14db7f48d12487d31e05eaed350e39bd4 | Triangle \(ABC\) has side lengths \(AB = 15\), \(BC = 17\), and \(CA = 19\). Let \(X\) and \(Y\) be the feet of the altitudes from \(B\) and \(C\) to \(CA\) and \(AB\), respectively. Let \(Z = XY \cap BC\) and \(H\) be the orthocenter of \(\triangle ABC\). Find the area of quadrilateral \(BHYZ\). | [
"126"
] | [
"Triangle \\(ABC\\) has side lengths \\(AB = 15\\), \\(BC = 17\\), and \\(CA = 19\\). Let \\(X\\) and \\(Y\\) be the feet of the altitudes from \\(B\\) and \\(C\\) to \\(CA\\) and \\(AB\\), respectively. Let \\(Z = XY \\cap BC\\) and \\(H\\) be the orthocenter of \\(\\triangle ABC\\). Find the area of quadrilateral... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-302f085f0c462d33bfcb40e9827e7da4c9d8a15b80b391cd4a70897a102debd4 | Triangle \(ABC\) has side lengths \(AB = 4\), \(BC = 5\), and \(CA = 6\). Points \(A'\), \(B'\), and \(C'\) are such that \(B'C'\) is tangent to the circumcircle of \(\triangle ABC\) at \(A\), \(C'A'\) is tangent to the circumcircle at \(B\), and \(A'B'\) is tangent to the circumcircle at \(C\). If the length \(B'C' = ... | [
"83"
] | [
"Triangle \\(ABC\\) has side lengths \\(AB = 4\\), \\(BC = 5\\), and \\(CA = 6\\). Points \\(A'\\), \\(B'\\), and \\(C'\\) are such that \\(B'C'\\) is tangent to the circumcircle of \\(\\triangle ABC\\) at \\(A\\), \\(C'A'\\) is tangent to the circumcircle at \\(B\\), and \\(A'B'\\) is tangent to the circumcircle a... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-93403d65468627ee6836aaf1342462e2d84e2522c946de653861853b15587085 | Triangle \(ABC\) has side lengths \(AB = x\), \(AC = x + 1\), and \(BC = x + 2\). Suppose the measure of angle \(A\) is twice that of angle \(B\). Find \(x\). | [
"\\( \\frac{1 + \\sqrt{13}}{2} \\)",
"\\(\\frac{1 + \\sqrt{13}}{2}\\)"
] | [
"Triangle \\(ABC\\) has side lengths \\(AB = x\\), \\(AC = x + 1\\), and \\(BC = x + 2\\). Suppose the measure of angle \\(A\\) is twice that of angle \\(B\\). Find \\(x\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 57
} | 89 | 0 | 0 |
omr-ws-a80df590b96e6a22bfebb652260a255a9361e518cf5a2b77fde94cd7c734e09b | Triangle \(ABC\) has side lengths \(AB=13\), \(BC=14\), and \(CA=15\). Let \(\Gamma\) denote the circumcircle of \(\triangle ABC\). Let \(H\) be the orthocenter of \(\triangle ABC\). Let \(AH\) intersect \(\Gamma\) at a point \(D\) other than \(A\). Let \(BH\) intersect \(AC\) at \(F\) and \(\Gamma\) at point \(G\) oth... | [
"24"
] | [
"Triangle \\(ABC\\) has side lengths \\(AB=13\\), \\(BC=14\\), and \\(CA=15\\). Let \\(\\Gamma\\) denote the circumcircle of \\(\\triangle ABC\\). Let \\(H\\) be the orthocenter of \\(\\triangle ABC\\). Let \\(AH\\) intersect \\(\\Gamma\\) at a point \\(D\\) other than \\(A\\). Let \\(BH\\) intersect \\(AC\\) at \\... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 3
} | 26 | 0 | 0 |
omr-ws-1306964319035282e9e528276b59969f377c2f126b6f7af089922649a4d550a8 | Triangle \(ABC\) has side lengths \(AB=18\), \(BC=36\), and \(CA=24\). The circle \(\Gamma\) passes through point \(C\) and is tangent to segment \(AB\) at point \(A\). Let \(X\), distinct from \(C\), be the second intersection of \(\Gamma\) with \(BC\). Moreover, let \(Y\) be the point on \(\Gamma\) such that segment ... | [
"69"
] | [
"Triangle \\(ABC\\) has side lengths \\(AB=18\\), \\(BC=36\\), and \\(CA=24\\). The circle \\(\\Gamma\\) passes through point \\(C\\) and is tangent to segment \\(AB\\) at point \\(A\\). Let \\(X\\), distinct from \\(C\\), be the second intersection of \\(\\Gamma\\) with \\(BC\\). Moreover, let \\(Y\\) be the point... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-6fc3ac22970794b16c00cc6a8b48ced81d6d401d2f954819baa7df384cd9cc94 | Triangle \(ABC\) has side lengths \(AB=7\), \(BC=8\), and \(CA=9\). Points \(J\) and \(K\) lie on side \(BC\) such that \(J\) lies on the incircle of \(\triangle ABC\) and \(\angle BAJ = \angle CAK\). The circumcircle of \(\triangle AJK\) meets sides \(AB\) and \(AC\) again at \(X\) and \(Y\) respectively. Line \(JY\) ... | [
"\\(\\frac{484 \\sqrt{5}}{147}\\)"
] | [
"Triangle \\(ABC\\) has side lengths \\(AB=7\\), \\(BC=8\\), and \\(CA=9\\). Points \\(J\\) and \\(K\\) lie on side \\(BC\\) such that \\(J\\) lies on the incircle of \\(\\triangle ABC\\) and \\(\\angle BAJ = \\angle CAK\\). The circumcircle of \\(\\triangle AJK\\) meets sides \\(AB\\) and \\(AC\\) again at \\(X\\)... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e11e0554b577d843ee52694e716a0d7a7a06e61d451f050a448c015b05e29c02 | Triangle \(ABC\) has side lengths \(a\), \(b\), and \(c\), where \(a\), \(b\), and \(c\) are consecutive integers with \(a < b < c\). A median drawn from one side of \(\triangle ABC\) divides \(\triangle ABC\) into two triangles, at least one of which is isosceles. Calculate the sum of all possible values of \(c\). | [
"18"
] | [
"Triangle \\(ABC\\) has side lengths \\(a\\), \\(b\\), and \\(c\\), where \\(a\\), \\(b\\), and \\(c\\) are consecutive integers with \\(a < b < c\\). A median drawn from one side of \\(\\triangle ABC\\) divides \\(\\triangle ABC\\) into two triangles, at least one of which is isosceles. Calculate the sum of all po... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-e35a63de8dff46a4ebc19776c34ecb9252c5f2ebd351845a2ea841cdd7d4b47c | Triangle \(ABC\) has side-lengths \(AB = 12\), \(BC = 24\), and \(AC = 18\). The line through the incenter of \(\triangle ABC\) parallel to \(\overline{BC}\) intersects \(\overline{AB}\) at \(M\) and \(\overline{AC}\) at \(N\). What is the perimeter of \(\triangle AMN\)? | [
"18"
] | [
"Triangle \\(ABC\\) has side-lengths \\(AB = 12\\), \\(BC = 24\\), and \\(AC = 18\\). The line through the incenter of \\(\\triangle ABC\\) parallel to \\(\\overline{BC}\\) intersects \\(\\overline{AB}\\) at \\(M\\) and \\(\\overline{AC}\\) at \\(N\\). What is the perimeter of \\(\\triangle AMN\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7d17e48ff2a248e1824ee4ddcd7a50541789835ce3262bccbb527f41fa77ceae | Triangle \(ABC\) has side-lengths \(AB=12\), \(BC=24\), and \(AC=18\). The line through the incenter of \(\triangle ABC\) parallel to \(\overline{BC}\) intersects \(\overline{AB}\) at \(M\) and \(\overline{AC}\) at \(N\). What is the perimeter of \(\triangle AMN\)? | [
"30"
] | [
"Triangle \\(ABC\\) has side-lengths \\(AB=12\\), \\(BC=24\\), and \\(AC=18\\). The line through the incenter of \\(\\triangle ABC\\) parallel to \\(\\overline{BC}\\) intersects \\(\\overline{AB}\\) at \\(M\\) and \\(\\overline{AC}\\) at \\(N\\). What is the perimeter of \\(\\triangle AMN\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-ea7e54711db36bbdff544fb0845d9591ca8bc0863acbb3272254168d5cc35525 | Triangle \(ABC\) has sides \(AB = 13\), \(BC = 15\), and \(AC = 9\). A line \(r\) parallel to \(BC\) is drawn through \(A\). The bisector of \(\angle ABC\) intersects \(r\) at \(E\), and the bisector of \(\angle ACB\) intersects \(r\) at \(F\). Calculate \(EF\). | [
"22"
] | [
"Triangle \\(ABC\\) has sides \\(AB = 13\\), \\(BC = 15\\), and \\(AC = 9\\). A line \\(r\\) parallel to \\(BC\\) is drawn through \\(A\\). The bisector of \\(\\angle ABC\\) intersects \\(r\\) at \\(E\\), and the bisector of \\(\\angle ACB\\) intersects \\(r\\) at \\(F\\). Calculate \\(EF\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d348ae00da4cb3763d4e60df6ec0383ba14b392100bbeb4838b2d270b2533c63 | Triangle \(ABC\) has sides \(AB = 17\), \(BC = 13\), and \(AC = 23\). Point \(P\) is on side \(BC\) such that \(BP = \frac{BC}{3}\). The bisector of \(\angle B\) intersects side \(AC\) at \(D\), and the bisector of \(\angle C\) intersects side \(AB\) at \(E\). The line perpendicular to \(BD\) through \(P\) meets side \... | [
"27"
] | [
"Triangle \\(ABC\\) has sides \\(AB = 17\\), \\(BC = 13\\), and \\(AC = 23\\). Point \\(P\\) is on side \\(BC\\) such that \\(BP = \\frac{BC}{3}\\). The bisector of \\(\\angle B\\) intersects side \\(AC\\) at \\(D\\), and the bisector of \\(\\angle C\\) intersects side \\(AB\\) at \\(E\\). The line perpendicular to... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-cc49d9e65c20fc3aee453320a63c73afd9c8547997463d536a95c0c262b62eb0 | Triangle \(ABC\) has sides \(AB = 8\), \(BC = 10\), and \(CA = 11\). Let \(L\) be the locus of points in the interior of triangle \(ABC\) which are within one unit of either \(A\), \(B\), or \(C\). Find the area of \(L\). | [
"\\dfrac{\\pi}{2}"
] | [
"Triangle \\(ABC\\) has sides \\(AB = 8\\), \\(BC = 10\\), and \\(CA = 11\\). Let \\(L\\) be the locus of points in the interior of triangle \\(ABC\\) which are within one unit of either \\(A\\), \\(B\\), or \\(C\\). Find the area of \\(L\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0e3703846666784e2f73bb04292962500b873ae16ee51ec93b1c21a0ae0da960 | Triangle \(ABC\) has sides of length 5, 6, and 7, with one vertex on the positive \(x\)-axis, one on the positive \(y\)-axis, and one on the positive \(z\)-axis. Let \(O\) be the origin. What is the volume of tetrahedron \(OABC\)? | [
"\\(\\sqrt{95}\\)"
] | [
"Triangle \\(ABC\\) has sides of length 5, 6, and 7, with one vertex on the positive \\(x\\)-axis, one on the positive \\(y\\)-axis, and one on the positive \\(z\\)-axis. Let \\(O\\) be the origin. What is the volume of tetrahedron \\(OABC\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-82e9c7c327c3773e6ca16c7b8f4b12d269d0a3e2d199b9b187716305be36541d | Triangle \(ABC\) has sides of length \(AB=6\), \(AC=5\), and \(BC=4\). Compute \[\frac{\cos{\frac{1}{2}(A-B)}}{\sin{\frac{1}{2}C}}-\frac{\sin{\frac{1}{2}(A-B)}}{\cos{\frac{1}{2}C}}.\] | [
"\\(\\frac{5}{3}\\)"
] | [
"Triangle \\(ABC\\) has sides of length \\(AB=6\\), \\(AC=5\\), and \\(BC=4\\). Compute \\[\\frac{\\cos{\\frac{1}{2}(A-B)}}{\\sin{\\frac{1}{2}C}}-\\frac{\\sin{\\frac{1}{2}(A-B)}}{\\cos{\\frac{1}{2}C}}.\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-f0ae48c5296b9e688ea28d4246527882d1e6f701a1c46e8bb59058de5bddc521 | Triangle \(ABC\) has sides of lengths 3, 4, and 5. A line divides \(ABC\) into a triangle \(T_1\) and a trapezoid \(R_1\). A rectangle \(DEFG\) has dimensions 6 by 7 and is divided by another line into a triangle \(T_2\) and a trapezoid \(R_2\), such that \(T_1\) and \(T_2\) are similar, and \(R_1\) and \(R_2\) are sim... | [
"\\dfrac{21}{8}"
] | [
"Triangle \\(ABC\\) has sides of lengths 3, 4, and 5. A line divides \\(ABC\\) into a triangle \\(T_1\\) and a trapezoid \\(R_1\\). A rectangle \\(DEFG\\) has dimensions 6 by 7 and is divided by another line into a triangle \\(T_2\\) and a trapezoid \\(R_2\\), such that \\(T_1\\) and \\(T_2\\) are similar, and \\(R... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-446400b60ec295506dc6f6db61c18364f547ced02f3f3030adcc7edff92974de | Triangle \(ABC\) has sides of lengths 8, 15, and 17. A point \(P\) is inside the triangle. Find the minimum value of \(PA^2 + PB^2 + PC^2\). | [
"\\(\\frac{578}{3}\\)"
] | [
"Triangle \\(ABC\\) has sides of lengths 8, 15, and 17. A point \\(P\\) is inside the triangle. Find the minimum value of \\(PA^2 + PB^2 + PC^2\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-7af46cd8036785da1312bc2a81544f4c08ce51a687652f0a47b5e205109fbcbb | Triangle \(ABC\) has three circles \(\omega_1\), \(\omega_2\), and \(\omega_3\) inscribed in it such that \(\omega_1\) has radius \(\frac{1}{2}\) and is tangent to \(AB\) and \(BC\), \(\omega_2\) has radius \(1\) and is tangent to \(AB\) and \(AC\), \(\omega_3\) has radius \(1\) and is tangent to \(AC\) and \(BC\), and... | [
"\\(\\frac{1}{2} \\times (3 + \\sqrt{2} + \\sqrt{5} + \\sqrt{10}) \\times \\left( \\frac{2\\sqrt{2} + \\sqrt{5}}{3} + 1 \\right)\\)"
] | [
"Triangle \\(ABC\\) has three circles \\(\\omega_1\\), \\(\\omega_2\\), and \\(\\omega_3\\) inscribed in it such that \\(\\omega_1\\) has radius \\(\\frac{1}{2}\\) and is tangent to \\(AB\\) and \\(BC\\), \\(\\omega_2\\) has radius \\(1\\) and is tangent to \\(AB\\) and \\(AC\\), \\(\\omega_3\\) has radius \\(1\\) ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66d3cb42ebbb064fd430cd3aec562dc1b7bd95cc29fa47843b82a87be1e44313 | Triangle \(ABC\) is a 30-60-90 right triangle with the right angle at \(C\), \(\angle ABC = 60^\circ\), and hypotenuse of length 2. Let \(P\) be a point chosen randomly inside triangle \(ABC\), and extend ray \(BP\) to hit side \(AC\) at \(D\). What is the probability that \(BD < \sqrt{2}\)? | [
"\\(\\frac{\\sqrt{3}}{3}\\)"
] | [
"Triangle \\(ABC\\) is a 30-60-90 right triangle with the right angle at \\(C\\), \\(\\angle ABC = 60^\\circ\\), and hypotenuse of length 2. Let \\(P\\) be a point chosen randomly inside triangle \\(ABC\\), and extend ray \\(BP\\) to hit side \\(AC\\) at \\(D\\). What is the probability that \\(BD < \\sqrt{2}\\)?"
... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-d08ba6220164c9ee21a4c6d7dbc712d8e412ca3e34d121fe26048689f97100a3 | Triangle \(ABC\) is a regular triangle with side length 6. Line \(t\) passes through vertex \(B\) without crossing the triangle. Each circle touches the straight lines \(m\) and \(t\) and one side of the triangle \(ABC\). Calculate the sum of the radii of these circles. | [
"3\\sqrt{3}"
] | [
"Triangle \\(ABC\\) is a regular triangle with side length 6. Line \\(t\\) passes through vertex \\(B\\) without crossing the triangle. Each circle touches the straight lines \\(m\\) and \\(t\\) and one side of the triangle \\(ABC\\). Calculate the sum of the radii of these circles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-346f0330e580285b0a2c54c3cfae35bd05573ee5a0edd27e290494bd5ef71e24 | Triangle \(ABC\) is a right triangle with \(\angle ACB\) as its right angle, \(m\angle ABC = 60^\circ\), and \(AB = 10\). Let \(P\) be a randomly chosen point inside \(\triangle ABC\), and extend \(BP\) to meet \(AC\) at \(D\). What is the probability that \(BD > 5\sqrt{2}\)? | [
"\\(1 - \\frac{1}{\\sqrt{3}}\\)"
] | [
"Triangle \\(ABC\\) is a right triangle with \\(\\angle ACB\\) as its right angle, \\(m\\angle ABC = 60^\\circ\\), and \\(AB = 10\\). Let \\(P\\) be a randomly chosen point inside \\(\\triangle ABC\\), and extend \\(BP\\) to meet \\(AC\\) at \\(D\\). What is the probability that \\(BD > 5\\sqrt{2}\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-93f5abb9de8495323279740c94f75b6c291724496b9ffe35f37165c314f9d92f | Triangle \(ABC\) is a right triangle with \(\angle ACB\) as its right angle, \(m\angle ABC = 60^\circ\), and \(AB = 10\). Let \(P\) be randomly chosen inside \(\triangle ABC\), and extend \(\overline{BP}\) to meet \(\overline{AC}\) at \(D\). What is the probability that \(BD > 5\sqrt{2}\)? | [
"\\(\\frac{3 - \\sqrt{3}}{3}\\)"
] | [
"Triangle \\(ABC\\) is a right triangle with \\(\\angle ACB\\) as its right angle, \\(m\\angle ABC = 60^\\circ\\), and \\(AB = 10\\). Let \\(P\\) be randomly chosen inside \\(\\triangle ABC\\), and extend \\(\\overline{BP}\\) to meet \\(\\overline{AC}\\) at \\(D\\). What is the probability that \\(BD > 5\\sqrt{2}\\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-90e2ce759da516ff2023a008b306e8a0caa62cdac673c8053bbc550a43287155 | Triangle \(ABC\) is acute. Point \(P\) satisfies \(\angle APB = \angle BPC = \angle CPA\). Let \(O_1\), \(O_2\), and \(O_3\) be the circumcenters of triangles \(APB\), \(BPC\), and \(CPA\) respectively. Let \(H_1\), \(H_2\), and \(H_3\) be the orthocenters of triangles \(APB\), \(BPC\), and \(CPA\) respectively. Calcul... | [
"\\dfrac{2\\sqrt{3}}{3}"
] | [
"Triangle \\(ABC\\) is acute. Point \\(P\\) satisfies \\(\\angle APB = \\angle BPC = \\angle CPA\\). Let \\(O_1\\), \\(O_2\\), and \\(O_3\\) be the circumcenters of triangles \\(APB\\), \\(BPC\\), and \\(CPA\\) respectively. Let \\(H_1\\), \\(H_2\\), and \\(H_3\\) be the orthocenters of triangles \\(APB\\), \\(BPC\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9a7734fad0f2cb1a717f3d74720e221389f1b0e4450d8f68dbe1e22dd313f02 | Triangle \(ABC\) is an acute triangle with \(BC = 6\) and \(AC = 7\). Let \(D\), \(E\), and \(F\) be the feet of the altitudes from \(A\), \(B\), and \(C\) respectively. \(AD\) bisects angle \(FDE\). Let \(m\) be the maximum possible value of \(FD + ED\). Find \(m^2\). | [
"36"
] | [
"Triangle \\(ABC\\) is an acute triangle with \\(BC = 6\\) and \\(AC = 7\\). Let \\(D\\), \\(E\\), and \\(F\\) be the feet of the altitudes from \\(A\\), \\(B\\), and \\(C\\) respectively. \\(AD\\) bisects angle \\(FDE\\). Let \\(m\\) be the maximum possible value of \\(FD + ED\\). Find \\(m^2\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-f6d72b5492b5e67af7d5954a86e0f1cb34c10d668522177f12de1a51b3252fd6 | Triangle \(ABC\) is an equilateral triangle with side length 1. Let \(X_0, X_1, \ldots\) be an infinite sequence of points such that: 1. \(X_0\) is the center of \(ABC\). 2. For all \(i \geq 0\), \(X_{2i+1}\) lies on segment \(AB\) and \(X_{2i+2}\) lies on segment \(AC\). 3. For all \(i \geq 0\), \(\angle X_iX_{i+1}X_{... | [
"\\dfrac{2}{\\sqrt{3}}"
] | [
"Triangle \\(ABC\\) is an equilateral triangle with side length 1. Let \\(X_0, X_1, \\ldots\\) be an infinite sequence of points such that:\n1. \\(X_0\\) is the center of \\(ABC\\).\n2. For all \\(i \\geq 0\\), \\(X_{2i+1}\\) lies on segment \\(AB\\) and \\(X_{2i+2}\\) lies on segment \\(AC\\).\n3. For all \\(i \\g... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-cac96646abe2023bfaf362365a980598cca7640b9377d9b9d6096c67d9b1d742 | Triangle \(ABC\) is an equilateral triangle with side length 6. Points \(D\), \(E\), and \(F\) are on \(AB\), \(BC\), and \(AC\) respectively such that \(BE = 2\), \(CF = 2\), and \(AD = 3\). Find the area of the triangle formed by the intersection of lines \(AE\), \(BF\), and \(CD\). | [
"\\(\\frac{81\\sqrt{3}}{64}\\)"
] | [
"Triangle \\(ABC\\) is an equilateral triangle with side length 6. Points \\(D\\), \\(E\\), and \\(F\\) are on \\(AB\\), \\(BC\\), and \\(AC\\) respectively such that \\(BE = 2\\), \\(CF = 2\\), and \\(AD = 3\\). Find the area of the triangle formed by the intersection of lines \\(AE\\), \\(BF\\), and \\(CD\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f0e331499ab0d42e082e35d31845a0a23f5bb4333688458e0bbe87c06c4e185 | Triangle \(ABC\) is an isosceles right triangle with \(AB = AC = 3\). Let \(M\) be the midpoint of hypotenuse \(\overline{BC}\). Points \(I\) and \(E\) lie on sides \(\overline{AC}\) and \(\overline{AB}\), respectively, such that \(AI > AE\) and \(AIME\) is a cyclic quadrilateral. Given that triangle \(EMI\) has area \... | [
"12"
] | [
"Triangle \\(ABC\\) is an isosceles right triangle with \\(AB = AC = 3\\). Let \\(M\\) be the midpoint of hypotenuse \\(\\overline{BC}\\). Points \\(I\\) and \\(E\\) lie on sides \\(\\overline{AC}\\) and \\(\\overline{AB}\\), respectively, such that \\(AI > AE\\) and \\(AIME\\) is a cyclic quadrilateral. Given that... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3ee8a47d28214d175d1d3ef9a91c010770bf3021953d68df4b00b1c0c0d10e8b | Triangle \(ABC\) is an isosceles triangle with \(AC = BC\). Point \(H\) is on \(AB\) such that \(CH\) is perpendicular to \(AB\), and the angle bisector of \(\angle CAB\) meets \(BC\) at point \(P\). If \(AP = CH = 2\), find the measure, in degrees, of \(\angle ACB\). | [
"108"
] | [
"Triangle \\(ABC\\) is an isosceles triangle with \\(AC = BC\\). Point \\(H\\) is on \\(AB\\) such that \\(CH\\) is perpendicular to \\(AB\\), and the angle bisector of \\(\\angle CAB\\) meets \\(BC\\) at point \\(P\\). If \\(AP = CH = 2\\), find the measure, in degrees, of \\(\\angle ACB\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-467f34027499b6076792202424c3f501c83eb04824ec23abad3edc52e6a4d232 | Triangle \(ABC\) is constructed such that \(AB > BC > AC\). Points \(E\), \(F\), and \(G\) are drawn from \(A\), \(B\), and \(C\) to their opposite sides respectively. If \(\frac{BE}{EC} = \frac{7}{12}\), the ratio of \(\frac{AF}{FC}\) can be represented as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime po... | [
"19"
] | [
"Triangle \\(ABC\\) is constructed such that \\(AB > BC > AC\\). Points \\(E\\), \\(F\\), and \\(G\\) are drawn from \\(A\\), \\(B\\), and \\(C\\) to their opposite sides respectively. If \\(\\frac{BE}{EC} = \\frac{7}{12}\\), the ratio of \\(\\frac{AF}{FC}\\) can be represented as \\(\\frac{m}{n}\\), where \\(m\\) ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-5dd6df206b741a688868b7102c2d6da62ac5c8d675f206eb3bb143b4bbfbb1b5 | Triangle \(ABC\) is divided by its medians into 6 smaller triangles. Suppose the radii of the inscribed circles in four of these triangles are equal to \(r\). If the side lengths of triangle \(ABC\) are \(a\), \(b\), and \(c\), find the value of \(\frac{a^2 + b^2 + c^2}{ab + bc + ca}\). | [
"2"
] | [
"Triangle \\(ABC\\) is divided by its medians into 6 smaller triangles. Suppose the radii of the inscribed circles in four of these triangles are equal to \\(r\\). If the side lengths of triangle \\(ABC\\) are \\(a\\), \\(b\\), and \\(c\\), find the value of \\(\\frac{a^2 + b^2 + c^2}{ab + bc + ca}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-960845af10824744675a6a471c8ca39331f64708e20b3478d6ab7675f8cebf64 | Triangle \(ABC\) is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are given as 84, 35, 30, and 40. Find the area of triangle \(ABC\). | [
"315"
] | [
"Triangle \\(ABC\\) is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are given as 84, 35, 30, and 40. Find the area of triangle \\(ABC\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 2
} | 21 | 0 | 0 |
omr-ws-a7272a51ee572cfd534aded68d024778485fcfa2ed27dc3b78d85b7856b86c1f | Triangle \(ABC\) is drawn such that \(\angle A = 80^\circ\), \(\angle B = 60^\circ\), and \(\angle C = 40^\circ\). Let the circumcenter of \(\triangle ABC\) be \(O\), and let \(\omega\) be the circle with diameter \(AO\). Circle \(\omega\) intersects side \(AC\) at point \(P\). Let \(M\) be the midpoint of side \(BC\),... | [
"120^\\circ"
] | [
"Triangle \\(ABC\\) is drawn such that \\(\\angle A = 80^\\circ\\), \\(\\angle B = 60^\\circ\\), and \\(\\angle C = 40^\\circ\\). Let the circumcenter of \\(\\triangle ABC\\) be \\(O\\), and let \\(\\omega\\) be the circle with diameter \\(AO\\). Circle \\(\\omega\\) intersects side \\(AC\\) at point \\(P\\). Let \... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9db50225b7be80ebed9734d4375461b6a892614881f1db96622daac12d5ff6e2 | Triangle \(ABC\) is equilateral with \(AB = 1\). Points \(E\) and \(G\) are on \(\overline{AC}\) and points \(D\) and \(F\) are on \(\overline{AB}\) such that both \(\overline{DE}\) and \(\overline{FG}\) are parallel to \(\overline{BC}\). Furthermore, triangle \(ADE\) and trapezoids \(DFGE\) and \(FBCG\) all have the s... | [
"\\(\\frac{21}{13}\\)"
] | [
"Triangle \\(ABC\\) is equilateral with \\(AB = 1\\). Points \\(E\\) and \\(G\\) are on \\(\\overline{AC}\\) and points \\(D\\) and \\(F\\) are on \\(\\overline{AB}\\) such that both \\(\\overline{DE}\\) and \\(\\overline{FG}\\) are parallel to \\(\\overline{BC}\\). Furthermore, triangle \\(ADE\\) and trapezoids \\... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 20
} | 68 | 0 | 0 |
omr-ws-5572552a45c235b48aaee68b0b5e5c2b53d49cbf4baf9c5ce1bb9a7fd165864d | Triangle \(ABC\) is equilateral with side length 1. Triangle \(BDC\) is isosceles with \(BD = DC\) and \(\angle BDC = 120^\circ\), with \(D\) outside triangle \(ABC\). Points \(M\) and \(N\) are on \(AB\) and \(AC\) respectively such that \(\angle MDN = 60^\circ\). Find the perimeter of triangle \(AMN\). | [
"2"
] | [
"Triangle \\(ABC\\) is equilateral with side length 1. Triangle \\(BDC\\) is isosceles with \\(BD = DC\\) and \\(\\angle BDC = 120^\\circ\\), with \\(D\\) outside triangle \\(ABC\\). Points \\(M\\) and \\(N\\) are on \\(AB\\) and \\(AC\\) respectively such that \\(\\angle MDN = 60^\\circ\\). Find the perimeter of t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-9303cb5c60d689198cb9d931f34e2b352bdb2881de0fa71793b5f80a2bf9aa3e | Triangle \(ABC\) is equilateral. Points \(D\) and \(E\) are the midpoints of segments \(BC\) and \(AC\) respectively. \(F\) is the point on segment \(AB\) such that \(2BF = AF\). Let \(P\) denote the intersection of \(AD\) and \(EF\). The value of \(EP/PF\) can be expressed as \(m/n\) where \(m\) and \(n\) are relative... | [
"7"
] | [
"Triangle \\(ABC\\) is equilateral. Points \\(D\\) and \\(E\\) are the midpoints of segments \\(BC\\) and \\(AC\\) respectively. \\(F\\) is the point on segment \\(AB\\) such that \\(2BF = AF\\). Let \\(P\\) denote the intersection of \\(AD\\) and \\(EF\\). The value of \\(EP/PF\\) can be expressed as \\(m/n\\) whe... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-96cd8e6ed4720fade627f5490322c9f6c2b90aabbbc4bd4718a104e931296116 | Triangle \(ABC\) is equilateral. There are 3 distinct points \(X\), \(Y\), \(Z\) inside \(\triangle ABC\) that each satisfy the property that the distances from the point to the three sides of the triangle are in ratio \(1 : 1 : 2\) in some order. Find the ratio of the area of \(\triangle ABC\) to that of \(\triangle X... | [
"9"
] | [
"Triangle \\(ABC\\) is equilateral. There are 3 distinct points \\(X\\), \\(Y\\), \\(Z\\) inside \\(\\triangle ABC\\) that each satisfy the property that the distances from the point to the three sides of the triangle are in ratio \\(1 : 1 : 2\\) in some order. Find the ratio of the area of \\(\\triangle ABC\\) to ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0df0ee17f69d77fe441f00708c3a75beb85bf6097d3f3b29d59ed42035437e24 | Triangle \(ABC\) is given. A line parallel to \(AC\) intersects side \(AB\) at point \(P\), median \(AM\) at point \(N\), and side \(BC\) at point \(Q\). Find the length of side \(AC\) if \(PN = 3\) and \(NQ = 5\). | [
"11"
] | [
"Triangle \\(ABC\\) is given. A line parallel to \\(AC\\) intersects side \\(AB\\) at point \\(P\\), median \\(AM\\) at point \\(N\\), and side \\(BC\\) at point \\(Q\\). Find the length of side \\(AC\\) if \\(PN = 3\\) and \\(NQ = 5\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-9474d4f0ca71a2ba615ba4aac4ed737f2d71bc5621fc300c379b1a87477c8655 | Triangle \(ABC\) is given. All possible circles \(S\) are drawn through the vertices \(A\) and \(C\), which intersect the sides \(AB\) and \(BC\) of the triangle at the points \(K\) and \(L\), respectively. Let \(P\) be the intersection point of the lines \(CK\) and \(AL\), and let \(O\) be the center of the circle \(S... | [
"(0, \\dfrac{5}{3})"
] | [
"Triangle \\(ABC\\) is given. All possible circles \\(S\\) are drawn through the vertices \\(A\\) and \\(C\\), which intersect the sides \\(AB\\) and \\(BC\\) of the triangle at the points \\(K\\) and \\(L\\), respectively. Let \\(P\\) be the intersection point of the lines \\(CK\\) and \\(AL\\), and let \\(O\\) be... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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