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-5ee4256aa877b6974a0c7dac3379732f79dc3add869c8aabaaf4c54a1e9e4608 | Triangle ABC has side lengths AB = 13, BC = 14, and AC = 15. Point D is on side AC such that AD = 5. The incircle of triangle ABC intersects segment BD at a point E. What is the length of segment DE? | [
"3\\sqrt{2} - \\sqrt{14}"
] | [
"Triangle ABC has side lengths AB = 13, BC = 14, and AC = 15. Point D is on side AC such that AD = 5. The incircle of triangle ABC intersects segment BD at a point E. What is the length of segment DE?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-064a6592895f5196db788fa70bca9a1ccd5d20ac594d7582db9e63485b794e77 | Triangle ABC has side lengths AB = 15, AC = 15, and BC = 18. Find the distance from vertex A to the circumcenter of triangle ABC. | [
"\\(\\frac{21}{8}\\)"
] | [
"Triangle ABC has side lengths AB = 15, AC = 15, and BC = 18. Find the distance from vertex A to the circumcenter of triangle ABC."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6a1a500873c2d7325c403339316c2318b0209fcef538337acb5cbc7f865d4ecb | Triangle ABC has side lengths AB = 7, BC = 8, and AC = 9. The angle bisector of angle B intersects AC at point D. What is the length of BD? | [
"\\dfrac{8\\sqrt{14}}{5}"
] | [
"Triangle ABC has side lengths AB = 7, BC = 8, and AC = 9. The angle bisector of angle B intersects AC at point D. What is the length of BD?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01ff46670f9438bbeed3501636a012163767b836769336ca0e748561d5cea69f | Triangle ABC has sides AB = 137, AC = 241, and BC = 200. There is a point D on BC such that both the incircles of triangles ABD and ACD touch AD at the same point E. Determine the length of CD. | [
"152"
] | [
"Triangle ABC has sides AB = 137, AC = 241, and BC = 200. There is a point D on BC such that both the incircles of triangles ABD and ACD touch AD at the same point E. Determine the length of CD."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7477f486fb18ce1a0fb353ca04bae4a6b8f9512c78e93f10b4f37bafbb25ba8b | Triangle ABC has sides AB = 14, BC = 13, and CA = 15. It is inscribed in circle Γ, which has center O. Let M be the midpoint of AB, let B′ be the point on Γ diametrically opposite B, and let X be the intersection of AO and MB′. Find the length of AX. | [
"\\(\\frac{65}{12}\\)"
] | [
"Triangle ABC has sides AB = 14, BC = 13, and CA = 15. It is inscribed in circle Γ, which has center O. Let M be the midpoint of AB, let B′ be the point on Γ diametrically opposite B, and let X be the intersection of AO and MB′. Find the length of AX."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-b1d476624ebd458197209fb8852274a71938a2572c7aa4baca231ed3ef922150 | Triangle ABC has sides AC = 3, BC = 5, and AB = 7. A circle is drawn internally tangent to the circumcircle of triangle ABC at point C and tangent to side AB at point D. Find the value of BD - DA. | [
"\\(\\frac{16}{7}\\)"
] | [
"Triangle ABC has sides AC = 3, BC = 5, and AB = 7. A circle is drawn internally tangent to the circumcircle of triangle ABC at point C and tangent to side AB at point D. Find the value of BD - DA."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-946b591cdcbf00a5ff29f859e1cfff73f86ea64b2af1dbb9e13725b744997621 | Triangle ABC has sides \(a = \sqrt{15}\), \(b = \sqrt{6}\), and \(c = \sqrt{30}\). Let M be the midpoint of BC. Let D be the projection of B onto the extension of AM. Find the ratio of the areas of triangles ADB and ABC. | [
"\\dfrac{27}{38}"
] | [
"Triangle ABC has sides \\(a = \\sqrt{15}\\), \\(b = \\sqrt{6}\\), and \\(c = \\sqrt{30}\\). Let M be the midpoint of BC. Let D be the projection of B onto the extension of AM. Find the ratio of the areas of triangles ADB and ABC."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-c4ce90da1ff48b29344232892c24df630c3eb43524a2f5a326afef7eae179e27 | Triangle ABC has sides of length 13, 14, and 15. Triangle PQR is inside triangle ABC such that the corresponding sides are parallel and 2 units away from the sides of triangle ABC. Find the area of triangle PQR. | [
"21"
] | [
"Triangle ABC has sides of length 13, 14, and 15. Triangle PQR is inside triangle ABC such that the corresponding sides are parallel and 2 units away from the sides of triangle ABC. Find the area of triangle PQR."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-f2ab5686056f708f5853bf43a620fd17b283ac8c5f0b4c927af7822024414bff | Triangle ABC is a 45-45-90 triangle with hypotenuse AB of length $4\sqrt{2}$. Let P be the midpoint of BC and let Q and R be points on AC and AB, respectively. What is the smallest possible perimeter for triangle PQR? | [
"\\(2\\sqrt{10}\\)"
] | [
"Triangle ABC is a 45-45-90 triangle with hypotenuse AB of length $4\\sqrt{2}$. Let P be the midpoint of BC and let Q and R be points on AC and AB, respectively. What is the smallest possible perimeter for triangle PQR?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-e7467de38db072919023e972f2f43d05c137a24be359422e58a35d2b2dc0a0b7 | Triangle ABC is a right triangle at B with integer side lengths. A second triangle PQR is located inside triangle ABC such that its sides are parallel to the sides of triangle ABC and the distance between the parallel lines is 2. What is the area of triangle PQR, and how many different triangles ABC are possible such t... | [
"3"
] | [
"Triangle ABC is a right triangle at B with integer side lengths. A second triangle PQR is located inside triangle ABC such that its sides are parallel to the sides of triangle ABC and the distance between the parallel lines is 2. What is the area of triangle PQR, and how many different triangles ABC are possible s... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-790b5b5604a5fddd18d4358f07cf3ed86ca0e2a640952f05def7babea314952a | Triangle ABC is a right triangle with AB = 66 and BC = 77. If the hypotenuse AC is expressed in the simplified form \( x\sqrt{y} \), find the value of \( x + y \). | [
"96"
] | [
"Triangle ABC is a right triangle with AB = 66 and BC = 77. If the hypotenuse AC is expressed in the simplified form \\( x\\sqrt{y} \\), find the value of \\( x + y \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-0b3bfa04fecbe11ad5b6fb360d2c5e8671efcb75994481d0f85037da7385058b | Triangle ABC is a right triangle with angle C as the right angle and angle B as 55 degrees. Line AC is 120 inches. What is the length of line BC in centimeters? | [
"213.42 cm"
] | [
"Triangle ABC is a right triangle with angle C as the right angle and angle B as 55 degrees. Line AC is 120 inches. What is the length of line BC in centimeters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 26
} | 46 | 0 | 0 |
omr-ws-53bcb21d522b795b57e8300c3ee53e88523a0ec5342ac3dc75c2ff9cc05554f8 | Triangle ABC is a right triangle with angle C as the right angle and angle B as 65 degrees. Line AC is 120 inches. What is the length of line BC in centimeters? | [
"142.13 cm"
] | [
"Triangle ABC is a right triangle with angle C as the right angle and angle B as 65 degrees. Line AC is 120 inches. What is the length of line BC in centimeters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 24
} | 41 | 0 | 0 |
omr-ws-7b034e639eac1bf46b24c13234ffd866e52b853ac0f29c5f71a186ff63ca3c4e | Triangle ABC is a right triangle with the right angle at C. Points D and E are on side BC such that AC = CD = DE = EB and AE = \( 13\sqrt{5} \). What is the area of triangle ADB in square inches? | [
"253.5"
] | [
"Triangle ABC is a right triangle with the right angle at C. Points D and E are on side BC such that AC = CD = DE = EB and AE = \\( 13\\sqrt{5} \\). What is the area of triangle ADB in square inches?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b8b656c70081cf6ea54feff942bd443f87474562d17e07f6b090541e49f2256 | Triangle ABC is a right-angled triangle with AB = 12, AC = 16, and the right angle at A. DEFG is a rectangle with D on AB, E on AC, and F, G on BC. H is a point on the same side as C with respect to the line EF such that BH = CD and angle(HEF) = angle(EBC). If EH = 7, find DE. | [
"15"
] | [
"Triangle ABC is a right-angled triangle with AB = 12, AC = 16, and the right angle at A. DEFG is a rectangle with D on AB, E on AC, and F, G on BC. H is a point on the same side as C with respect to the line EF such that BH = CD and angle(HEF) = angle(EBC). If EH = 7, find DE."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 4
} | 10 | 0 | 0 |
omr-ws-6edb8cfc4fcee80211afbfaa161f1fe677f76873ccd66d73503c097d6f9b412e | Triangle ABC is acute with angle A = 30 degrees. A half-circle with diameter BC intersects AB at D and AC at E. Find the ratio of the area of triangle ADE to the area of quadrilateral BCED. | [
"\\(\\frac{3}{2}\\)"
] | [
"Triangle ABC is acute with angle A = 30 degrees. A half-circle with diameter BC intersects AB at D and AC at E. Find the ratio of the area of triangle ADE to the area of quadrilateral BCED."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-329ceabe35936b3279096fbeb3d3248dc81c329085838235f64e62547ec99a67 | Triangle ABC is an acute triangle. Let E be the foot of the perpendicular from B to AC, and let F be the foot of the perpendicular from C to AB. If \(2[AEF] = [ABC]\), what is the measure (in degrees) of \(\angle BAC\)? | [
"\\(45^\\circ\\)"
] | [
"Triangle ABC is an acute triangle. Let E be the foot of the perpendicular from B to AC, and let F be the foot of the perpendicular from C to AB. If \\(2[AEF] = [ABC]\\), what is the measure (in degrees) of \\(\\angle BAC\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-1a6e8b030fb106a8769d2179c74516f3b643e63b707bdfefe480770d0b9f37de | Triangle ABC is an equilateral triangle and O is the center of its inscribed circle. If the area of the circle is $4\pi$ sq cm, what is the area, in square centimeters, of triangle ABC? Express your answer in simplest radical form. Note: The area of a circle with radius $r$ is $\pi r^2.$ | [
"12\\sqrt 3"
] | [
"Triangle ABC is an equilateral triangle and O is the center of its inscribed circle. If the area of the circle is $4\\pi$ sq cm, what is the area, in square centimeters, of triangle ABC? Express your answer in simplest radical form.\n\nNote: The area of a circle with radius $r$ is $\\pi r^2.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-83095b25f2670433567dfc48d2b3c62f80c92b3a7a8186559a7231283c658ff6 | Triangle ABC is an equilateral triangle with a point O inside such that OA=3, OB=4, and OC=5. If the side length of the triangle AB is given by \( AB^2 = a + b\sqrt{c} \) where \( a, b, c \) are integers and \( c \) is not divisible by the square of any prime, what is the value of \( a + b + c \)? | [
"40"
] | [
"Triangle ABC is an equilateral triangle with a point O inside such that OA=3, OB=4, and OC=5. If the side length of the triangle AB is given by \\( AB^2 = a + b\\sqrt{c} \\) where \\( a, b, c \\) are integers and \\( c \\) is not divisible by the square of any prime, what is the value of \\( a + b + c \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9d949ac7aff32418a623a9061710f9c70d34cf17ea243233ffe7bdbf63645cc | Triangle ABC is an equilateral triangle with a side length of 20 units. An arc is drawn with the center at A and a radius of 5 units, intersecting AB and AC at points B′ and C′ respectively. If a circular wheel with a radius of 1 unit travels the whole path BB′C′C, how many times will the wheel rotate? | [
"\\(\\frac{15}{\\pi}\\)"
] | [
"Triangle ABC is an equilateral triangle with a side length of 20 units. An arc is drawn with the center at A and a radius of 5 units, intersecting AB and AC at points B′ and C′ respectively. If a circular wheel with a radius of 1 unit travels the whole path BB′C′C, how many times will the wheel rotate?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-db4ba659ef5d65714a097c4956ea7df0ece5665deff28a898a5c0db6c88005a1 | Triangle ABC is an equilateral triangle. There is a point O within ABC such that OA=3, OB=4, and OC=5. If AB^2 = a + b√c where a, b, and c are integers and c is not divisible by the square of any prime, what is the value of a + b + c? | [
"40"
] | [
"Triangle ABC is an equilateral triangle. There is a point O within ABC such that OA=3, OB=4, and OC=5. If AB^2 = a + b√c where a, b, and c are integers and c is not divisible by the square of any prime, what is the value of a + b + c?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9a8b1f9e54326a76aa6c43f12b9fe6068be6ff6435a69e047f965f46c5c45dad | Triangle ABC is an isosceles right triangle with hypotenuse AB. Point D is drawn on AB such that BD = BA. Find the measure of angle DBC for each of the two possible positions of D. | [
"135^\\circ"
] | [
"Triangle ABC is an isosceles right triangle with hypotenuse AB. Point D is drawn on AB such that BD = BA. Find the measure of angle DBC for each of the two possible positions of D."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b6fa054f3720eedde6e185ba9cf19db3b3e3a6c39fd09a341447d851c770fbea | Triangle ABC is circumscribed in a circle with AB = AC = 4√3 and height AD = 4. Find the diameter of the circle. | [
"12"
] | [
"Triangle ABC is circumscribed in a circle with AB = AC = 4√3 and height AD = 4. Find the diameter of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-fee38e0004c5a31466cfc09737f23ee7141f264e88a7ab47a9b8c49ad3d46ebf | Triangle ABC is equilateral. Let P, Q, and R be points on AB, BC, and AC respectively such that $\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RA}=2/1$. Calculate the ratio of the area of the triangle formed by the segments AQ, BR, and CP to the area of triangle ABC. | [
"\\dfrac{1}{7}"
] | [
"Triangle ABC is equilateral. Let P, Q, and R be points on AB, BC, and AC respectively such that $\\frac{AP}{PB}=\\frac{BQ}{QC}=\\frac{CR}{RA}=2/1$. Calculate the ratio of the area of the triangle formed by the segments AQ, BR, and CP to the area of triangle ABC."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a5c147fab73c05c48dd143a814fe16412b6eb1eef107d5b45497026dc4bf0d8 | Triangle ABC is graphed in the first quadrant of the coordinate plane, with one vertex located at the origin and \( \angle B = 90^\circ \). The incenter of the triangle is located at (1.5, 0.5). Let \( P \) be the point of tangency between the radius and segment \( BC \). Given that \( \frac{BP}{BC} = \frac{1}{2} \), f... | [
"\\frac{3}{4}"
] | [
"Triangle ABC is graphed in the first quadrant of the coordinate plane, with one vertex located at the origin and \\( \\angle B = 90^\\circ \\). The incenter of the triangle is located at (1.5, 0.5). Let \\( P \\) be the point of tangency between the radius and segment \\( BC \\). Given that \\( \\frac{BP}{BC} = \\... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-98aa5043283483758940f4205fd4011697e3f77708dcf7cecb258e1c0bce0ebd | Triangle ABC is in plane P with AC=3, CB=4, and AB=5. A point S, not in P, is such that SB is perpendicular to plane P and angle ASC is 30 degrees. Compute the length of SB. | [
"\\(\\sqrt{11}\\)"
] | [
"Triangle ABC is in plane P with AC=3, CB=4, and AB=5. A point S, not in P, is such that SB is perpendicular to plane P and angle ASC is 30 degrees. Compute the length of SB."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-a06c065af786f150b66ff3a757186a320e7200574cec39e626783725f2e3061a | Triangle ABC is inscribed in a circle (O) with diameter BC. The incircle (I) touches sides AB and AC at points D and E, respectively. M and N are the midpoints of AB and AC. Line MN intersects circle (O) at points P and Q. If the radius of circle (O) is 10 units, find the length of segment DE given that points D, E, P,... | [
"20 - 10\\sqrt{2}"
] | [
"Triangle ABC is inscribed in a circle (O) with diameter BC. The incircle (I) touches sides AB and AC at points D and E, respectively. M and N are the midpoints of AB and AC. Line MN intersects circle (O) at points P and Q. If the radius of circle (O) is 10 units, find the length of segment DE given that points D, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-d6d15884368ef3f6e531bb2efd55326fbee6837856a8d0f6df1bef6f067f08b7 | Triangle ABC is inscribed in a circle centered at Q. A circle with center O is inscribed in triangle ABC. AO is drawn and extended to intersect the larger circle at D. If the radius of the larger circle is 10 units and the radius of the inscribed circle is 3 units, calculate the length of segment OD. | [
"10 + 2\\sqrt{10}"
] | [
"Triangle ABC is inscribed in a circle centered at Q. A circle with center O is inscribed in triangle ABC. AO is drawn and extended to intersect the larger circle at D. If the radius of the larger circle is 10 units and the radius of the inscribed circle is 3 units, calculate the length of segment OD."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-2b36ccd3b9dccd61c0ce590f7166bbd00cc24666f6b4a19c4ce5db097a26a89d | Triangle ABC is inscribed in a circle k. The height from C intersects AB and k at points D and E, respectively. F is the midpoint of segment BD. A line l through D, perpendicular to EF, intersects segment AC at point P. If AP = x and CP = y, find the value of $\frac{y}{x}$. | [
"2"
] | [
"Triangle ABC is inscribed in a circle k. The height from C intersects AB and k at points D and E, respectively. F is the midpoint of segment BD. A line l through D, perpendicular to EF, intersects segment AC at point P. If AP = x and CP = y, find the value of $\\frac{y}{x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-83320c05e3add0f8df1db7d4ec1d3bdcd2d5c112ff075a9b8a2bf14236d24654 | Triangle ABC is inscribed in a circle with a diameter of 12 meters. The angles of triangle ABC are A = 45°, B = 60°, and C = 75°. Find the exact lengths of the arcs BC, AC, and AB. | [
"BC = 3, AC = 4, AB = 5"
] | [
"Triangle ABC is inscribed in a circle with a diameter of 12 meters. The angles of triangle ABC are A = 45°, B = 60°, and C = 75°. Find the exact lengths of the arcs BC, AC, and AB."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 10
} | 11 | 0 | 0 |
omr-ws-5971676a8cc186d581042c56a0bfb76ddabff97caf4c04bc2b52dcc8778f9f46 | Triangle ABC is inscribed in a circle with diameter AD. A tangent to the circle at D cuts AB extended at E and AC extended at F. Given that AB = 4, AC = 6, and BE = 8, find the length of CF. | [
"2"
] | [
"Triangle ABC is inscribed in a circle with diameter AD. A tangent to the circle at D cuts AB extended at E and AC extended at F. Given that AB = 4, AC = 6, and BE = 8, find the length of CF."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f5f6338b672f6c0a1329e0b5b49a58bccd964ae72f97795a6ad867a7536e3a96 | Triangle ABC is inscribed in a circle, and the bisector of angle ABC meets the circle at P. Given that AB = 6, BC = 8, and AC = 7, compute the length of BP. | [
"8"
] | [
"Triangle ABC is inscribed in a circle, and the bisector of angle ABC meets the circle at P. Given that AB = 6, BC = 8, and AC = 7, compute the length of BP."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d79d6155673920822020a91593580184c5c05cb6b70c8a6767dd095612448201 | Triangle ABC is inscribed in a fixed circle (O). The medians from A, B, and C meet (O) at A', B', and C', respectively. Which triangle minimizes the value of the expression \( p = \frac{AA'^2 + BB'^2 + CC'^2}{AB^2 + BC^2 + CA^2} \)? | [
"\\text{equilateral triangle}"
] | [
"Triangle ABC is inscribed in a fixed circle (O). The medians from A, B, and C meet (O) at A', B', and C', respectively. Which triangle minimizes the value of the expression \\( p = \\frac{AA'^2 + BB'^2 + CC'^2}{AB^2 + BC^2 + CA^2} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-c3f9f22558a1cc575cde6df3c19a5346cf9a7a8fef1b71afa00f351f01e172f3 | Triangle ABC is inscribed in a unit circle. The angle bisectors of angles A, B, and C are extended to intersect the circle at points \( A_1 \), \( B_1 \), and \( C_1 \) respectively. Find the value of \( \frac{AA_1 \cos \frac{A}{2} + BB_1 \cos \frac{B}{2} + CC_1 \cos \frac{C}{2}}{\sin A + \sin B + \sin C} \). | [
"2"
] | [
"Triangle ABC is inscribed in a unit circle. The angle bisectors of angles A, B, and C are extended to intersect the circle at points \\( A_1 \\), \\( B_1 \\), and \\( C_1 \\) respectively. Find the value of \\( \\frac{AA_1 \\cos \\frac{A}{2} + BB_1 \\cos \\frac{B}{2} + CC_1 \\cos \\frac{C}{2}}{\\sin A + \\sin B + ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-0e894637bc58789d9898a78a091c446eaf238ef378218929919dae38042ed93c | Triangle ABC is inscribed in circle $\Gamma$. A circle with center O is drawn, tangent to side BC at a point P, and internally tangent to the arc BC of $\Gamma$ not containing A at point Q. If $\angle BAO = \angle CAO$, calculate the ratio $\frac{\angle PAO}{\angle QAO}$. | [
"\\dfrac{1}{2}"
] | [
"Triangle ABC is inscribed in circle $\\Gamma$. A circle with center O is drawn, tangent to side BC at a point P, and internally tangent to the arc BC of $\\Gamma$ not containing A at point Q. If $\\angle BAO = \\angle CAO$, calculate the ratio $\\frac{\\angle PAO}{\\angle QAO}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-5f2277ca2a102a47323d92bf6e75aade8b12ba91ceb6c996c6b12d8333988220 | Triangle ABC is inscribed in circle O, with T being the antipode of A on the circle. Let H be the orthocenter of triangle ABC, and points D and E lie on segments AB and AC, respectively, such that AD = AE, and D, H, E are collinear. Let DE intersect circle O at points K and L. KT and LT intersect BC at points X and Y, ... | [
"4\\sqrt{6}"
] | [
"Triangle ABC is inscribed in circle O, with T being the antipode of A on the circle. Let H be the orthocenter of triangle ABC, and points D and E lie on segments AB and AC, respectively, such that AD = AE, and D, H, E are collinear. Let DE intersect circle O at points K and L. KT and LT intersect BC at points X an... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-c18cf5f202d52d2fa06b53f7c1859f8f46c2c5e00a83eafcf2eb5b55e505de6b | Triangle ABC is inscribed in circle O, with the A-symmedian intersecting side BC at D and the circle at K. Let N be the point on segment AK such that BN is isogonal to BC. If BC is the B-symmedian of triangle BAK, find the ratio AN:NK when N is the midpoint of segment AK. | [
"1"
] | [
"Triangle ABC is inscribed in circle O, with the A-symmedian intersecting side BC at D and the circle at K. Let N be the point on segment AK such that BN is isogonal to BC. If BC is the B-symmedian of triangle BAK, find the ratio AN:NK when N is the midpoint of segment AK."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f50f810675defbde8e3c7f48f1eea1ab2bddf516b8fdce53615b6d74dd5aec1b | Triangle ABC is inscribed in circle O. Point P is inside triangle ABC, and PD, PE, PF are the perpendiculars from P to sides BC, AC, and AB, respectively, with D, E, and F being the feet of these perpendiculars. AP intersects circle O at point L, and LT is perpendicular to BC with T being the foot of this perpendicular... | [
"90^\\circ"
] | [
"Triangle ABC is inscribed in circle O. Point P is inside triangle ABC, and PD, PE, PF are the perpendiculars from P to sides BC, AC, and AB, respectively, with D, E, and F being the feet of these perpendiculars. AP intersects circle O at point L, and LT is perpendicular to BC with T being the foot of this perpendi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1fd0c044bdbc4d8d0fe26c18d66055db447a281f9198a0ae9d27304f88f965b5 | Triangle ABC is inscribed in circle W with AB = 5, BC = 7, and AC = 3. The bisector of angle A meets side BC at D and circle W at second point E. Let X be the circle with diameter DE. Circles X and W meet at E and at a second point F. Calculate $AF^2$. | [
"$\\frac{900}{19}$"
] | [
"Triangle ABC is inscribed in circle W with AB = 5, BC = 7, and AC = 3. The bisector of angle A meets side BC at D and circle W at second point E. Let X be the circle with diameter DE. Circles X and W meet at E and at a second point F. Calculate $AF^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 7,
"tir": 13
} | 56 | 0 | 0 |
omr-ws-fa4e520effbce1e6ae11d8af25093333ee228a46fe883933dedb17c0a6b2ec2b | Triangle ABC is inscribed in circle W with AB=5, BC=7, AC=3. The angle bisector of A meets side BC at D and circle W at a second point E. Let C be the circle with diameter DE. Circles W and C meet at E and at a second point F. Then \(AF^2 = n \cdot m\) where m and n are relatively prime positive integers. Find \(m+n\). | [
"919"
] | [
"Triangle ABC is inscribed in circle W with AB=5, BC=7, AC=3. The angle bisector of A meets side BC at D and circle W at a second point E. Let C be the circle with diameter DE. Circles W and C meet at E and at a second point F. Then \\(AF^2 = n \\cdot m\\) where m and n are relatively prime positive integers. Find ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 6
} | 27 | 0 | 0 |
omr-ws-fafd58c26b18d7060dc28c171e9b4edf98ff2e0e45022373cb25c677879d438b | Triangle ABC is inscribed in the unit circle. Let P be an arbitrary point inside triangle ABC. Calculate the maximum possible value of \( |PA| \cdot |PB| \cdot |PC| \). | [
"1"
] | [
"Triangle ABC is inscribed in the unit circle. Let P be an arbitrary point inside triangle ABC. Calculate the maximum possible value of \\( |PA| \\cdot |PB| \\cdot |PC| \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-48f7654efe9e0c4c56a6e37915e4c3cbbcb8fd40edd33d1401bec08c6b831018 | Triangle ABC is isosceles with AB = AC = 15 and BC = 18. A circle with center O is tangent to BC at X and to AC at Y. If XC = 2, find the radius of the circle. | [
"1"
] | [
"Triangle ABC is isosceles with AB = AC = 15 and BC = 18. A circle with center O is tangent to BC at X and to AC at Y. If XC = 2, find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-20e4f9aaae737808152f55b49b205864da48fa5f52fe26856b2999c0a38d3a4f | Triangle ABC is isosceles with AB = AC and BC = 1. Point D is on AB such that AD = 1 and ∠DAC = 180/7. Find the length of CD. | [
"\\(\\frac{\\sin\\left(\\frac{\\pi}{7}\\right)}{\\sin\\left(\\frac{3\\pi}{7}\\right)} + 2\\cos\\left(\\frac{2\\pi}{7}\\right)\\)"
] | [
"Triangle ABC is isosceles with AB = AC and BC = 1. Point D is on AB such that AD = 1 and ∠DAC = 180/7. Find the length of CD."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-177d5f2e2e7af0cdd729c1b655a2251c7f1299ef467fe9c7955c7f5472040560 | Triangle ABC is isosceles with AB = AC, BC = 10, and sinA = 18/25. If \( AB^2 \) can be expressed in the form \( \frac{a}{b - 2\sqrt{c}} \), find \( 100b + c \). | [
"2801"
] | [
"Triangle ABC is isosceles with AB = AC, BC = 10, and sinA = 18/25. If \\( AB^2 \\) can be expressed in the form \\( \\frac{a}{b - 2\\sqrt{c}} \\), find \\( 100b + c \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 2,
"tir": 5
} | 15 | 0 | 0 |
omr-ws-c36fcbf2a3f41b4ae52e3b1de87f3560f22d293cea0782141da0a7ea01d0dfe7 | Triangle ABC is isosceles with AB = AC. D is the midpoint of AB. If $<BCD = <BAC = x$, find $\cos{x}$. | [
"\\(\\frac{3}{4}\\)"
] | [
"Triangle ABC is isosceles with AB = AC. D is the midpoint of AB. If $<BCD = <BAC = x$, find $\\cos{x}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-18f0e50aae879cc4de5613d4843d0643c7ed47f7a52c066a60a1b503a0f150d1 | Triangle ABC is isosceles with AB = AC. E is the midpoint of BC. Points F and G are on sides AB and AC, respectively, such that triangle EFG is equilateral. Given AB = 13 and BC = 10, find the length of FG. | [
"\\dfrac{480 - 200\\sqrt{3}}{23}"
] | [
"Triangle ABC is isosceles with AB = AC. E is the midpoint of BC. Points F and G are on sides AB and AC, respectively, such that triangle EFG is equilateral. Given AB = 13 and BC = 10, find the length of FG."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-b9dba2daa98cafa09843915f279dda139729cabfef4d733fd1c2c2be8e4746b6 | Triangle ABC is isosceles with AB = AC. Point D on AB is such that ∠BCD = 15° and \( BC = \sqrt{6}AD \). Let \( \theta \) be the measure of ∠CAB in degrees. Calculate \( \theta \). | [
"90"
] | [
"Triangle ABC is isosceles with AB = AC. Point D on AB is such that ∠BCD = 15° and \\( BC = \\sqrt{6}AD \\). Let \\( \\theta \\) be the measure of ∠CAB in degrees. Calculate \\( \\theta \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0adf050d25b2474f4b76ba2f4fa906ae4cff22132cddef38e8ff6f1512aab976 | Triangle ABC is isosceles with AB = AC. Point P is chosen on AC such that AP = BC. Given that ∠BPC = 40°, find the angle measures of triangle ABC. | [
"40^\\circ"
] | [
"Triangle ABC is isosceles with AB = AC. Point P is chosen on AC such that AP = BC. Given that ∠BPC = 40°, find the angle measures of triangle ABC."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-226b0cd5784c3242a51c6d77c926f382de00e9825c442233579af349a2c0da37 | Triangle ABC is isosceles with AC=BC. Point H is on AB such that CH is perpendicular to AB. The angle bisector of angle CAB meets BC at point P. If AP=CH, find the measure of angle ACB. | [
"60^\\circ"
] | [
"Triangle ABC is isosceles with AC=BC. Point H is on AB such that CH is perpendicular to AB. The angle bisector of angle CAB meets BC at point P. If AP=CH, find the measure of angle ACB."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-a8b54e9fbbe98229aa98d7d991e42c923c711839242d804912a1b7bbf71ec5a8 | Triangle ABC is isosceles with base AC. Points P and Q are respectively on CB and AB such that AC = AP = PQ = QB. Find angle B. | [
"60^\\circ"
] | [
"Triangle ABC is isosceles with base AC. Points P and Q are respectively on CB and AB such that AC = AP = PQ = QB. Find angle B."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-e1e4d0f4f712a0d66abf493fa6ecab84c9861ddcb68a5815dc742aefcc04679a | Triangle ABC is isosceles with base AC. Points P and Q are respectively on CB and AB such that AC = AP = PQ = QB. Find the measure of angle B. | [
"\\( \\frac{180}{7} \\) degrees"
] | [
"Triangle ABC is isosceles with base AC. Points P and Q are respectively on CB and AB such that AC = AP = PQ = QB. Find the measure of angle B."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-7f764cf09a537dbda59423696e0dc2da20f45829271f30467a32b77ad7a539e1 | Triangle ABC is right-angled at C. AQ is drawn parallel to BC with Q and B on opposite sides of AC. When BQ is drawn, intersecting AC at P, we have PQ = 2AB. Let the measure of angle ABC be $\theta$ and the measure of angle PBC be $\phi$. Calculate the value of $\frac{\theta}{\phi}$. | [
"3"
] | [
"Triangle ABC is right-angled at C. AQ is drawn parallel to BC with Q and B on opposite sides of AC. When BQ is drawn, intersecting AC at P, we have PQ = 2AB. Let the measure of angle ABC be $\\theta$ and the measure of angle PBC be $\\phi$. Calculate the value of $\\frac{\\theta}{\\phi}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-5496c956f96f5b081d09bcaf055d406ececfbe083c5ab179c05e9949a618da2f | Triangle ABC is similar to triangle MNP. Given BC = 60, AB = 12, MP = 8, and BC > AC > AB, compute the sum of all possible integer values for the ratio of the area of triangle ABC to the area of triangle MNP. | [
"930"
] | [
"Triangle ABC is similar to triangle MNP. Given BC = 60, AB = 12, MP = 8, and BC > AC > AB, compute the sum of all possible integer values for the ratio of the area of triangle ABC to the area of triangle MNP."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 1
} | 19 | 0 | 0 |
omr-ws-4a07a1b20497736d8c5e559323aff630a17a93e551b847040b6b781813d7ad0d | Triangle ABC is similar to triangle MNP. Given that BC = 60, AB = 12, MP = 8, and BC > AC > AB, compute the sum of all possible integer values of the ratio of the area of triangle ABC to triangle MNP. | [
"49"
] | [
"Triangle ABC is similar to triangle MNP. Given that BC = 60, AB = 12, MP = 8, and BC > AC > AB, compute the sum of all possible integer values of the ratio of the area of triangle ABC to triangle MNP."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-955710f973d17270122760b47935ac198dc5ca124e45f9e52652cf78abdb7963 | Triangle ABC with \(\angle B = 90^\circ\) is rotated about AB to form a cone with volume \(800\pi\). When rotated about BC, it forms a cone with volume \(1920\pi\). What is the length of AC? | [
"18"
] | [
"Triangle ABC with \\(\\angle B = 90^\\circ\\) is rotated about AB to form a cone with volume \\(800\\pi\\). When rotated about BC, it forms a cone with volume \\(1920\\pi\\). What is the length of AC?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-251361e05ab0c0158a6723d9e699fd7162a7ed5387bbb745be8857a90e8b6594 | Triangle ACB is similar to triangle DEF. If AB=14, BC=9, AC=12, and DE=8, then what is the perimeter of triangle DEF? | [
"\\(\\frac{70}{3}\\)"
] | [
"Triangle ACB is similar to triangle DEF. If AB=14, BC=9, AC=12, and DE=8, then what is the perimeter of triangle DEF?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-edf311f9aebfdf316ce4504e888d221951128cbc1e760d5fa29d12ba80718852 | Triangle ADE and Triangle BDC are triangles with right angles at :ang: EAB and :ang: ABC, respectively. Given AB = 4, BC = 6, and AE = 8, and that segments AC and BE intersect at D, what is the difference between the areas of triangle ADE and triangle BDC? | [
"4"
] | [
"Triangle ADE and Triangle BDC are triangles with right angles at :ang: EAB and :ang: ABC, respectively. Given AB = 4, BC = 6, and AE = 8, and that segments AC and BE intersect at D, what is the difference between the areas of triangle ADE and triangle BDC?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-6e510b8eb76886809addb17106d30978b6fbd3678992311c6b96b1e76ee98b64 | Triangle AEF is a right triangle with AE = 4 and EF = 3. The triangle is inscribed inside square ABCD. What is the area of the square? | [
"\\( \\frac{256}{17} \\)"
] | [
"Triangle AEF is a right triangle with AE = 4 and EF = 3. The triangle is inscribed inside square ABCD. What is the area of the square?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-1bf6e13d33cdab3c024c9c4db339542e7b590706691de5f559faa5a6653c9198 | Triangle APM has ∠A = 90 degrees and a perimeter of 152. A circle centered at O (on AP) has a radius of 19 and touches AM at A and PM at T. Find the length of OP. | [
"\\(\\frac{95}{3}\\)"
] | [
"Triangle APM has ∠A = 90 degrees and a perimeter of 152. A circle centered at O (on AP) has a radius of 19 and touches AM at A and PM at T. Find the length of OP."
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 22
} | 68 | 0 | 0 |
omr-ws-1ab372124bdf4a66da307487be07682abb50995217142e36b15d1693be20b26b | Triangle APM has ∠A = 90° and a perimeter of 152. A circle centered at O (on AP) has a radius of 19 and touches AM at A and PM at T. Find the length of OP. | [
"\\(\\frac{95}{3}\\)"
] | [
"Triangle APM has ∠A = 90° and a perimeter of 152. A circle centered at O (on AP) has a radius of 19 and touches AM at A and PM at T. Find the length of OP."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-38afaa0422604cadb11a962eb30a19383c5db75160cc5954153047544bd66439 | Triangle CEF is an equilateral triangle. Points A and B are on line EF such that EA=BF=2AB. Points P and Q are outside triangle CEF, and point D is inside triangle CEF. Given that PA=QB=PC=QC=PD=QD=1, find the length of BD. | [
"\\(\\frac{\\sqrt{19}}{19}\\)"
] | [
"Triangle CEF is an equilateral triangle. Points A and B are on line EF such that EA=BF=2AB. Points P and Q are outside triangle CEF, and point D is inside triangle CEF. Given that PA=QB=PC=QC=PD=QD=1, find the length of BD."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-08a0ee9d8e9ba8dddc2305219b5347eaebc9ab02a8544f6c01056b136a16be0e | Triangle DEF has angle D = 45 degrees, side d = 11, and side f = 13. Find and label the smallest angle possible. | [
"11.68583993645°"
] | [
"Triangle DEF has angle D = 45 degrees, side d = 11, and side f = 13. Find and label the smallest angle possible."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 32
} | 53 | 0 | 0 |
omr-ws-9028f015e4b4b8813f160de30dcec120a32df8da514fd0d588c63645ec9713e3 | Triangle PQR has vertices P(0,0), Q(3,12), and R(6,0). The triangle is rotated counterclockwise such that the median lies on the y-axis. Find the new coordinates of point Q. | [
"\\(\\left(-\\frac{24}{5}, \\frac{57}{5}\\right)\\)"
] | [
"Triangle PQR has vertices P(0,0), Q(3,12), and R(6,0). The triangle is rotated counterclockwise such that the median lies on the y-axis. Find the new coordinates of point Q."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 6,
"tir": 24
} | 66 | 0 | 0 |
omr-ws-f67f8324a0bb349315de0495070c5af17cef87e2f2646ed62b55f4a9d0f0f4fa | Triangle PQR has vertices P=(-8,5), Q=(-15,-19), and R=(1,-7). The equation of the angle bisector of ∠P can be written in the form \(ax + 2y + c = 0\). Find \(a + c\). | [
"89"
] | [
"Triangle PQR has vertices P=(-8,5), Q=(-15,-19), and R=(1,-7). The equation of the angle bisector of ∠P can be written in the form \\(ax + 2y + c = 0\\). Find \\(a + c\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-bc4df2085afcb523988679eb4dbc590d5259f7ec184b5655bfcf7faf395b316e | Triangle R'P'T' has the same side lengths as triangle RPT (R'T'=RT, P'T'=PT) and the same angle ∠P'R'T' = ∠RTP. Find the measure of ∠R'T'P'. | [
"60^\\circ"
] | [
"Triangle R'P'T' has the same side lengths as triangle RPT (R'T'=RT, P'T'=PT) and the same angle ∠P'R'T' = ∠RTP. Find the measure of ∠R'T'P'."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-76c617faa954ecdda3ada677ebd8a0125443ed2d635b94a951edc506efda330d | Triangle \( A'B'C' \) is inscribed in triangle \( ABC \), similar to it with coefficient \( k < 1 \), with the names of the vertices corresponding. In this case, vertex \( A' \) lies on \( BC \), \( B' \) on \( AC \), \( C' \) on \( AB \), \( \angle BC'A' = \theta \), and \( \angle CAB = \alpha \). If \( k \cos \frac{\... | [
"\\dfrac{\\sqrt{6} - \\sqrt{2}}{2}"
] | [
"Triangle \\( A'B'C' \\) is inscribed in triangle \\( ABC \\), similar to it with coefficient \\( k < 1 \\), with the names of the vertices corresponding. In this case, vertex \\( A' \\) lies on \\( BC \\), \\( B' \\) on \\( AC \\), \\( C' \\) on \\( AB \\), \\( \\angle BC'A' = \\theta \\), and \\( \\angle CAB = \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e10e944c097180b73d491497fbfcbfe51827f3aaf251dea2b1cc88151ae6022f | Triangle \( ABC \) has \( AB = 4 \), \( AC = 5 \), and \( BC = 3 \). Point \( D \) lies on \( \overline{BC} \) such that \( \overline{AD} \) bisects \( \angle A \), and the perpendicular bisector of \( \overline{AD} \) meets \( \overline{AB} \) and \( \overline{AC} \) at \( M \) and \( N \) respectively. Compute the ar... | [
"\\(\\frac{16}{3}\\)"
] | [
"Triangle \\( ABC \\) has \\( AB = 4 \\), \\( AC = 5 \\), and \\( BC = 3 \\). Point \\( D \\) lies on \\( \\overline{BC} \\) such that \\( \\overline{AD} \\) bisects \\( \\angle A \\), and the perpendicular bisector of \\( \\overline{AD} \\) meets \\( \\overline{AB} \\) and \\( \\overline{AC} \\) at \\( M \\) and \... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6034c8c622a149a33d76cb1626d1e717bf45f723a6dbd29caaf4f268400c91b6 | Triangle \( ABC \) has \( AB = 6 \), \( BC = 11 \), and \( CA = 7 \). Let \( M \) be the midpoint of \( BC \). Points \( E \) and \( O \) are on \( AC \) and \( AB \), respectively, and point \( N \) lies on line \( AM \). Given that quadrilateral \( NEMO \) is a rectangle, find its area. | [
"\\dfrac{77\\sqrt{10}}{27}"
] | [
"Triangle \\( ABC \\) has \\( AB = 6 \\), \\( BC = 11 \\), and \\( CA = 7 \\). Let \\( M \\) be the midpoint of \\( BC \\). Points \\( E \\) and \\( O \\) are on \\( AC \\) and \\( AB \\), respectively, and point \\( N \\) lies on line \\( AM \\). Given that quadrilateral \\( NEMO \\) is a rectangle, find its area.... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-b9a8e7377d89093c658cfa9065390b5674dadba7080708a3375a0f5f69dff8e3 | Triangle \( ABC \) has \( AB = 7 \), \( BC = 8 \), and \( CA = 9 \). Points \( D \) and \( E \) lie on \( \overline{AB} \) and \( \overline{AC} \), respectively, with \( AD = AE \). Given that the area of \( \triangle ADE \) is 1, compute \( DE^2 \). Express your answer in simplest radical form. | [
"\\sqrt{5}"
] | [
"Triangle \\( ABC \\) has \\( AB = 7 \\), \\( BC = 8 \\), and \\( CA = 9 \\). Points \\( D \\) and \\( E \\) lie on \\( \\overline{AB} \\) and \\( \\overline{AC} \\), respectively, with \\( AD = AE \\). Given that the area of \\( \\triangle ADE \\) is 1, compute \\( DE^2 \\). Express your answer in simplest radical... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de78c053146fb103f34ade6286e544a259bfe339c80514ff5a1f41bf66b69c49 | 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 \( ... | [
"75^\\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 t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c3e3ae448a4a59fd6208963682b22fe4ee32c877ec8f87e100b785067f7f8426 | Triangle \( ABC \) has a right angle at \( B \). A circle centered at \( B \) with radius \( BA \) intersects side \( AC \) at a point \( D \) different from \( A \). Given that \( AD = 20 \) and \( DC = 16 \), find the length of \( BA \). | [
"60"
] | [
"Triangle \\( ABC \\) has a right angle at \\( B \\). A circle centered at \\( B \\) with radius \\( BA \\) intersects side \\( AC \\) at a point \\( D \\) different from \\( A \\). Given that \\( AD = 20 \\) and \\( DC = 16 \\), find the length of \\( BA \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-efbd0afdd2a7f061b45c8c965d0aa5d8ec1185d7c74e779818c4addd9a3b746e | Triangle \( ABC \) has a right angle at \( B \). Point \( D \) lies on side \( BC \) such that \( 3 \angle BAD = \angle BAC \). Given that \( AC = 2 \) and \( CD = 1 \), compute \( BD \). | [
"\\(\\frac{3}{8}\\)"
] | [
"Triangle \\( ABC \\) has a right angle at \\( B \\). Point \\( D \\) lies on side \\( BC \\) such that \\( 3 \\angle BAD = \\angle BAC \\). Given that \\( AC = 2 \\) and \\( CD = 1 \\), compute \\( BD \\)."
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-419623d82f31e54b5996b69f690e67b6afdd6d628c6bf20df1a6a09521459b6e | Triangle \( ABC \) has an incenter \( I \). Its incircle touches the side \( BC \) at \( T \). The line through \( T \) parallel to \( IA \) meets the incircle again at \( S \), and the tangent to the incircle at \( S \) meets \( AB \) and \( AC \) at points \( C' \) and \( B' \) respectively. If the ratio of the area ... | [
"10"
] | [
"Triangle \\( ABC \\) has an incenter \\( I \\). Its incircle touches the side \\( BC \\) at \\( T \\). The line through \\( T \\) parallel to \\( IA \\) meets the incircle again at \\( S \\), and the tangent to the incircle at \\( S \\) meets \\( AB \\) and \\( AC \\) at points \\( C' \\) and \\( B' \\) respective... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-91e81d571e5a0e25c3ffdb72f48093beb2503500a72ca2dafa6d6567ce7ad5b8 | Triangle \( ABC \) has an incenter \( I \). Let points \( X, Y \) be located on the line segments \( AB \) and \( AC \) respectively, such that \( BX \cdot AB = IB^2 \) and \( CY \cdot AC = IC^2 \). Given that the points \( X, I, Y \) lie on a straight line, find the possible values of the measure of angle \( A \). | [
"\\( 60^\\circ \\)"
] | [
"Triangle \\( ABC \\) has an incenter \\( I \\). Let points \\( X, Y \\) be located on the line segments \\( AB \\) and \\( AC \\) respectively, such that \\( BX \\cdot AB = IB^2 \\) and \\( CY \\cdot AC = IC^2 \\). Given that the points \\( X, I, Y \\) lie on a straight line, find the possible values of the measur... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 10
} | 30 | 0 | 0 |
omr-ws-272e81ae3bead46ede05259761c6c91b601f1ab2ad05891b87146f42af9f0b7b | Triangle \( ABC \) has an inradius of 5 and a circumradius of 16. If \( 2\cos{B} = \cos{A} + \cos{C} \), then the area of triangle \( ABC \) can be expressed as \( \frac{a\sqrt{b}}{c} \), where \( a, b \), and \( c \) are positive integers such that \( a \) and \( c \) are relatively prime and \( b \) is not divisible ... | [
"141"
] | [
"Triangle \\( ABC \\) has an inradius of 5 and a circumradius of 16. If \\( 2\\cos{B} = \\cos{A} + \\cos{C} \\), then the area of triangle \\( ABC \\) can be expressed as \\( \\frac{a\\sqrt{b}}{c} \\), where \\( a, b \\), and \\( c \\) are positive integers such that \\( a \\) and \\( c \\) are relatively prime and... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ae61e708bc8d7406dd230b27675b6bcc37ec93fd1d71e99c513d698504b6886 | Triangle \( ABC \) has orthocenter \( H \) with \( AB = 19 \), \( BC = 26 \), \( AC = 34 \). Let the perpendicular bisectors of \( HA \), \( HB \), and \( HC \) be lines \( \ell_a \), \( \ell_b \), and \( \ell_c \) respectively. Find the perimeter of the triangle formed by the intersections of \( \ell_a \), \( \ell_b \... | [
"79"
] | [
"Triangle \\( ABC \\) has orthocenter \\( H \\) with \\( AB = 19 \\), \\( BC = 26 \\), \\( AC = 34 \\). Let the perpendicular bisectors of \\( HA \\), \\( HB \\), and \\( HC \\) be lines \\( \\ell_a \\), \\( \\ell_b \\), and \\( \\ell_c \\) respectively. Find the perimeter of the triangle formed by the intersection... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-795cb98eb182d558528ba2d819b1f817c6431ad51a48aa3bbfbe221f75bf3129 | Triangle \( ABC \) has side lengths \( AB = 10 \), \( BC = 7 \), and \( CA = 12 \). If each side is extended to three times its original length, then the triangle formed has an area of how many times the area of \( ABC \)? | [
"9"
] | [
"Triangle \\( ABC \\) has side lengths \\( AB = 10 \\), \\( BC = 7 \\), and \\( CA = 12 \\). If each side is extended to three times its original length, then the triangle formed has an area of how many times the area of \\( ABC \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-ce38489bdf4f7574fe8d4431094c441315d5256b2f759e190fbc5924edac879d | Triangle \( ABC \) has side lengths \( AB = 13 \), \( BC = 14 \), and \( AC = 15 \). A circle \( w \) with radius \( 4 \) is tangent to \( AB \) and \( AC \). Line \( l_{1} \) is tangent to \( w \) and parallel to \( AB \), and line \( l_{2} \) is tangent to \( w \) and parallel to \( AC \). Lines \( l_{1} \) and \( l_... | [
"\\dfrac{28}{3}"
] | [
"Triangle \\( ABC \\) has side lengths \\( AB = 13 \\), \\( BC = 14 \\), and \\( AC = 15 \\). A circle \\( w \\) with radius \\( 4 \\) is tangent to \\( AB \\) and \\( AC \\). Line \\( l_{1} \\) is tangent to \\( w \\) and parallel to \\( AB \\), and line \\( l_{2} \\) is tangent to \\( w \\) and parallel to \\( AC... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-792c088b2591b0780710c9165eeffd9f3df684aee3e67c8e5c162abf91897e99 | Triangle \( ABC \) has sides 3, 4, and 5. Rectangle \( DEFG \) has dimensions 6 x 7. A line divides \( ABC \) into a triangle \( T1 \) and a trapezoid \( R1 \). Another line divides the rectangle into a triangle \( T2 \) and a trapezoid \( R2 \) such that \( T1 \) and \( T2 \) are similar, and \( R1 \) and \( R2 \) are... | [
"\\(\\frac{3}{32}\\)"
] | [
"Triangle \\( ABC \\) has sides 3, 4, and 5. Rectangle \\( DEFG \\) has dimensions 6 x 7. A line divides \\( ABC \\) into a triangle \\( T1 \\) and a trapezoid \\( R1 \\). Another line divides the rectangle into a triangle \\( T2 \\) and a trapezoid \\( R2 \\) such that \\( T1 \\) and \\( T2 \\) are similar, and \\... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c9e033c630bfb0dab0746dd7b7b6eff4531b507b3d87d3293ac43f85a8f71389 | Triangle \( ABC \) has sides \( AB = 13 \), \( BC = 14 \), and \( AC = 15 \). Points \( E \) and \( F \) are on \( AB \) and \( AC \) respectively. Triangle \( AEF \) is folded along crease \( EF \) such that point \( A \) lies on \( BC \) and \( EFCB \) is a cyclic quadrilateral after the fold. What is the length of \... | [
"\\( \\frac{65}{9} \\)"
] | [
"Triangle \\( ABC \\) has sides \\( AB = 13 \\), \\( BC = 14 \\), and \\( AC = 15 \\). Points \\( E \\) and \\( F \\) are on \\( AB \\) and \\( AC \\) respectively. Triangle \\( AEF \\) is folded along crease \\( EF \\) such that point \\( A \\) lies on \\( BC \\) and \\( EFCB \\) is a cyclic quadrilateral after th... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 3,
"tir": 6
} | 10 | 0 | 0 |
omr-ws-4a7faa3519192a40919669bb5cda56ac8f37cbf47f1ed05fbcb4df3ff6ef98d9 | Triangle \( ABC \) is a right-angled triangle at \( C \) with \( AB = 1 \). The bisector of \( \angle ACB \) intersects medians \( BE \) (where \( E \in AC \)) and \( AF \) (where \( F \in BC \)) at points \( P \) and \( M \) respectively. If \( N = AF \cap BE \), determine the maximum value for the area of \( \triangl... | [
"\\dfrac{1}{150}"
] | [
"Triangle \\( ABC \\) is a right-angled triangle at \\( C \\) with \\( AB = 1 \\). The bisector of \\( \\angle ACB \\) intersects medians \\( BE \\) (where \\( E \\in AC \\)) and \\( AF \\) (where \\( F \\in BC \\)) at points \\( P \\) and \\( M \\) respectively. If \\( N = AF \\cap BE \\), determine the maximum va... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-5511e9029431480924bb09478f5ca4ba1cd75be3339cc4b4b05190e5c6a52c30 | Triangle \( ABC \) is an equilateral triangle of side length 1. Let point \( M \) be the midpoint of side \( AC \). Another equilateral triangle \( DEF \) of side length 1 is drawn such that the circumcenter of \( DEF \) is \( M \) and point \( D \) rests on side \( AB \). The length of \( AD \) is of the form \( \frac... | [
"36"
] | [
"Triangle \\( ABC \\) is an equilateral triangle of side length 1. Let point \\( M \\) be the midpoint of side \\( AC \\). Another equilateral triangle \\( DEF \\) of side length 1 is drawn such that the circumcenter of \\( DEF \\) is \\( M \\) and point \\( D \\) rests on side \\( AB \\). The length of \\( AD \\) ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-f8c5c0fcc69e14934a98af44c1169f011e2eca8a102d62f26eb0c07d3c692123 | 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, so that \( AI > AE \) and \( AIME \) is a cyclic quadrilateral. Given that triangle ... | [
"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, so that \\( AI > AE \\) and \\( AIME \\) is a cyclic quadril... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-af42e354bd5cf87d801afe7e69dd9b8cd30c55a825450a2d6b63af459ac78234 | Triangle \( ABC \) is inscribed in circle \( w \). The line containing the median from \( A \) meets \( w \) again at \( M \), the line containing the angle bisector of \( \angle B \) meets \( w \) again at \( R \), and the line containing the altitude from \( C \) meets \( w \) again at \( L \). Given that quadrilater... | [
"\\(36^\\circ\\)"
] | [
"Triangle \\( ABC \\) is inscribed in circle \\( w \\). The line containing the median from \\( A \\) meets \\( w \\) again at \\( M \\), the line containing the angle bisector of \\( \\angle B \\) meets \\( w \\) again at \\( R \\), and the line containing the altitude from \\( C \\) meets \\( w \\) again at \\( L... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 5,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-8985689a97c9a5c74386e4f9a7a10cc01a21abe895a86a1f957448945206fa17 | Triangle \( ABC \) is inscribed in the circle \( \Gamma \). Let \( \Gamma_a \) denote the circle internally tangent to \( \Gamma \) and also tangent to sides \( AB \) and \( AC \). Let \( A' \) denote the point of tangency of \( \Gamma \) and \( \Gamma_a \). Define \( B' \) and \( C' \) similarly. If the coordinates of... | [
"\\left(2, \\dfrac{2\\sqrt{3}}{3}\\right)"
] | [
"Triangle \\( ABC \\) is inscribed in the circle \\( \\Gamma \\). Let \\( \\Gamma_a \\) denote the circle internally tangent to \\( \\Gamma \\) and also tangent to sides \\( AB \\) and \\( AC \\). Let \\( A' \\) denote the point of tangency of \\( \\Gamma \\) and \\( \\Gamma_a \\). Define \\( B' \\) and \\( C' \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b4d93a509f24718e2e8926fb88a0cc14549bc941ed45436c2921a94eafe0831b | Triangle \( ABC \) is inscribed in the ellipse with equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\). If its centroid is at the origin \((0,0)\), find its area. | [
"\\(\\frac{9\\sqrt{3}}{2}\\)"
] | [
"Triangle \\( ABC \\) is inscribed in the ellipse with equation \\(\\frac{x^2}{4} + \\frac{y^2}{9} = 1\\). If its centroid is at the origin \\((0,0)\\), find its area."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 25
} | 73 | 0 | 0 |
omr-ws-310b300e0810c86db6823627f8a88553cffceb805b607bbe550b9615b0bca493 | Triangle \( ABC \) is scalene with angle \( A \) having a measure greater than 90 degrees. Determine the set of points \( D \) that lie on the extended line \( BC \), for which \( |AD| = \sqrt{|BD| \cdot |CD|} \), where \( |XY| \) refers to the (positive) distance between \( X \) and \( Y \). | [
"\\( \\{B, C\\} \\)"
] | [
"Triangle \\( ABC \\) is scalene with angle \\( A \\) having a measure greater than 90 degrees. Determine the set of points \\( D \\) that lie on the extended line \\( BC \\), for which \\( |AD| = \\sqrt{|BD| \\cdot |CD|} \\), where \\( |XY| \\) refers to the (positive) distance between \\( X \\) and \\( Y \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a71c2f0bbddcfa00a211248487f512533f74fd164cbeb3d16920d63eaa01eed2 | Triangle \( ABC \) with \( AB = BC = 6 \) and \( \angle ABC = 120^\circ \) is rotated about \( A \), and the images of points \( B \) and \( C \) are \( B' \) and \( C' \), respectively. If \( A \), \( B' \), and \( C \) are collinear in that order, find the area of triangle \( B'CC' \) expressed as \( a - b\sqrt{c} \)... | [
"39"
] | [
"Triangle \\( ABC \\) with \\( AB = BC = 6 \\) and \\( \\angle ABC = 120^\\circ \\) is rotated about \\( A \\), and the images of points \\( B \\) and \\( C \\) are \\( B' \\) and \\( C' \\), respectively. If \\( A \\), \\( B' \\), and \\( C \\) are collinear in that order, find the area of triangle \\( B'CC' \\) e... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-85f9d8602dcbe7b69e9b48fd679991ab624c97a259b77daeee89d3ff6449b34b | Triangle \( AOC \) has \( AO = 16 \), \( AC = 25 \), and \( \angle AOC = 120^\circ \). Points \( D, E, \) and \( F \) lie on \( \overline{AO}, \overline{AC}, \) and \( \overline{OC} \) respectively, such that \( AD = \frac{1}{4}AO \), \( AE = \frac{1}{4}AC \), and \( OF = \frac{1}{4}OC \). The area of \( \triangle DEF ... | [
"1369"
] | [
"Triangle \\( AOC \\) has \\( AO = 16 \\), \\( AC = 25 \\), and \\( \\angle AOC = 120^\\circ \\). Points \\( D, E, \\) and \\( F \\) lie on \\( \\overline{AO}, \\overline{AC}, \\) and \\( \\overline{OC} \\) respectively, such that \\( AD = \\frac{1}{4}AO \\), \\( AE = \\frac{1}{4}AC \\), and \\( OF = \\frac{1}{4}OC... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b81e0154bb55a3e72962e163685400efbbd04c189c89258883311a433f08a069 | Triangle \( A_1 A_2 A_3 \) has a right angle at \( A_3 \). A sequence of points is defined by the following iterative process: from \( A_n \) (\( n \geq 3 \)), a perpendicular line is drawn to meet \( A_{n-2}A_{n-1} \) at \( A_{n+1} \). If this process is continued indefinitely, let \( P \) be the unique point interior... | [
"\\left( \\dfrac{1}{5}, \\dfrac{2}{5} \\right)"
] | [
"Triangle \\( A_1 A_2 A_3 \\) has a right angle at \\( A_3 \\). A sequence of points is defined by the following iterative process: from \\( A_n \\) (\\( n \\geq 3 \\)), a perpendicular line is drawn to meet \\( A_{n-2}A_{n-1} \\) at \\( A_{n+1} \\). If this process is continued indefinitely, let \\( P \\) be the u... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e0a8b3fae47f350b9301ad75cbae8b4853ab3ef77b545cd407b36a6c0187fde2 | Triangle \( MNP \) has \( m \angle M = 3 \times m \angle N \). Find the least possible perimeter of \( \triangle MNP \) if the side lengths are integers. | [
"21"
] | [
"Triangle \\( MNP \\) has \\( m \\angle M = 3 \\times m \\angle N \\). Find the least possible perimeter of \\( \\triangle MNP \\) if the side lengths are integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-6ff9676b752f64835769419d2c2ca797566e9c0bf1bab1ba8447ed7a99a01942 | Triangle \( OAB \) has \( O = (0,0) \), \( B = (5,0) \), and \( A \) in the first quadrant. Given that \( \angle ABO = 90^\circ \) and \( \angle AOB = 30^\circ \), suppose that \( \overline{OA} \) is rotated \( 90^\circ \) counterclockwise about \( O \). What are the coordinates of the image of \( A \)? | [
"\\(\\left( -\\frac{5}{3}\\sqrt{3}, 5 \\right)\\)"
] | [
"Triangle \\( OAB \\) has \\( O = (0,0) \\), \\( B = (5,0) \\), and \\( A \\) in the first quadrant. Given that \\( \\angle ABO = 90^\\circ \\) and \\( \\angle AOB = 30^\\circ \\), suppose that \\( \\overline{OA} \\) is rotated \\( 90^\\circ \\) counterclockwise about \\( O \\). What are the coordinates of the imag... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-4d95acb300371c39b0c978fc564a9134fe3dae550f37b6d04c56db2ee0d1e416 | Triangle \( PAB \) is formed by three tangents to circle \( O \) and \( \angle APB = 40^\circ \). What is the measure of \( \angle AOB \)? | [
"\\( 70^\\circ \\)"
] | [
"Triangle \\( PAB \\) is formed by three tangents to circle \\( O \\) and \\( \\angle APB = 40^\\circ \\). What is the measure of \\( \\angle AOB \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d67d9512d6baaf9f302a97cf7957d3d649d78874d2f1b728e65118ef33e6a321 | Triangle \( PQR \) has \( PQ = 3 \), \( QR = 4 \), \( RP = 5 \), and a right angle at \( Q \). Let \( S \) lie on \( \overline{PR} \) so that \( \overline{QS} \) is an altitude of \( \triangle PQR \). Compute the smallest possible distance between two points lying on the inscribed circles of \( \triangle PQS \) and \( ... | [
"\\(\\sqrt{2} - \\frac{7}{5}\\)"
] | [
"Triangle \\( PQR \\) has \\( PQ = 3 \\), \\( QR = 4 \\), \\( RP = 5 \\), and a right angle at \\( Q \\). Let \\( S \\) lie on \\( \\overline{PR} \\) so that \\( \\overline{QS} \\) is an altitude of \\( \\triangle PQR \\). Compute the smallest possible distance between two points lying on the inscribed circles of \... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-5f049ae58d4e640ece094ce41aa039c9812f4b6f9a5c426564f1836155316d98 | Triangle \( T_1 \) has sides of lengths 4, 5, and 6. A second triangle \( T_2 \) is formed by using the excenters of \( T_1 \) as vertices. If \([T_i]\) denotes the area of triangle \( T_i \), then the value of \(\frac{[T_1]}{[T_2]}\) can be written in the form \(\frac{m}{n}\), where \( m \) and \( n \) are relatively ... | [
"39"
] | [
"Triangle \\( T_1 \\) has sides of lengths 4, 5, and 6. A second triangle \\( T_2 \\) is formed by using the excenters of \\( T_1 \\) as vertices. If \\([T_i]\\) denotes the area of triangle \\( T_i \\), then the value of \\(\\frac{[T_1]}{[T_2]}\\) can be written in the form \\(\\frac{m}{n}\\), where \\( m \\) and ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 23
} | 45 | 0 | 0 |
omr-ws-decf1a03288cffdabf043daf120b5f0bec752eaffc7f12c3d65426b0cc8dde9b | Triangle \( XYZ \) has semiperimeter \( s \) and area \( A \). A square \( PQRS \) with side length \( a \) is inscribed in \( XYZ \) with \( P \) and \( Q \) on \( YZ \), \( R \) on \( ZX \), and \( S \) on \( XY \). Similarly, \( b \) and \( c \) are the side lengths of squares with two vertices on \( ZX \) and \( XY... | [
"\\dfrac{121}{96}"
] | [
"Triangle \\( XYZ \\) has semiperimeter \\( s \\) and area \\( A \\). A square \\( PQRS \\) with side length \\( a \\) is inscribed in \\( XYZ \\) with \\( P \\) and \\( Q \\) on \\( YZ \\), \\( R \\) on \\( ZX \\), and \\( S \\) on \\( XY \\). Similarly, \\( b \\) and \\( c \\) are the side lengths of squares with... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-c866bb723e7593dc4983fadd1cc98d35ea2efbbb4c90837bdd639365f3c7970a | Triangle \( XYZ \) is equilateral, and circle \( w \) is its circumcircle. Point \( P \) is in the interior of \( XYZ \), and \( XP, YP, ZP \) intersect \( w \) at \( A, B, C \) respectively. \( D, E, F \) are the incenters of triangles \( PBC, PCA, PAB \). Let \( G \) be the point of concurrency of \( AD, BE, CF \). I... | [
"0"
] | [
"Triangle \\( XYZ \\) is equilateral, and circle \\( w \\) is its circumcircle. Point \\( P \\) is in the interior of \\( XYZ \\), and \\( XP, YP, ZP \\) intersect \\( w \\) at \\( A, B, C \\) respectively. \\( D, E, F \\) are the incenters of triangles \\( PBC, PCA, PAB \\). Let \\( G \\) be the point of concurren... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-e47857b03a8cc79d21465da28034089ca1098df006ddee6df8c26827ee6fed9a | Triangle \( \text{ABC} \) has \( \text{AC} = 4 \), \( \cos \angle \text{B} = \frac{1}{3} \), and \( \frac{\text{AB}}{\text{BC}} = 2 \). The perpendicular bisector of side \( \overline{\text{AB}} \) meets the perpendicular trisector of side \( \overline{\text{AC}} \) closer to \( \text{A} \) at \( \text{P} \). What is \... | [
"\\dfrac{16}{3}"
] | [
"Triangle \\( \\text{ABC} \\) has \\( \\text{AC} = 4 \\), \\( \\cos \\angle \\text{B} = \\frac{1}{3} \\), and \\( \\frac{\\text{AB}}{\\text{BC}} = 2 \\). The perpendicular bisector of side \\( \\overline{\\text{AB}} \\) meets the perpendicular trisector of side \\( \\overline{\\text{AC}} \\) closer to \\( \\text{A}... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f491684d9e6721d1a363403d3d13db559312f50448c92dc1eab91a5afa334eb7 | Triangle \( \triangle ABC \) has \( \angle B = 120^\circ \) and \( AB = 1 \). Find the largest real number \( x \) such that \( CA - CB > x \) for all possible triangles \( \triangle ABC \). | [
"\\(\\frac{1}{2}\\)"
] | [
"Triangle \\( \\triangle ABC \\) has \\( \\angle B = 120^\\circ \\) and \\( AB = 1 \\). Find the largest real number \\( x \\) such that \\( CA - CB > x \\) for all possible triangles \\( \\triangle ABC \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-65528f39ef1c4a00be860fc4c3e37a37ba6902fcf83d62dc6c6669267b63dc20 | Triangle \( \triangle ABC \) has side lengths \( AB = 9 \) and \( AC = 14 \). Point \( X \) is on \( AB \) such that \( AX = 7 \). Point \( Y \) is on \( AC \) such that \( AY = 6 \). If the area of triangle \( \triangle ABC \) is 51, what is the area of \( \triangle AXY \)? | [
"17"
] | [
"Triangle \\( \\triangle ABC \\) has side lengths \\( AB = 9 \\) and \\( AC = 14 \\). Point \\( X \\) is on \\( AB \\) such that \\( AX = 7 \\). Point \\( Y \\) is on \\( AC \\) such that \\( AY = 6 \\). If the area of triangle \\( \\triangle ABC \\) is 51, what is the area of \\( \\triangle AXY \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
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