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-8dbfb1ff49186671af6420bd600e4245b6c05b00dfdc734fa6c141c91d6ded81 | A screen covering the front of a fireplace has wires $1$ mm in diameter, spaced in a mesh that has $5$ mm square openings between the wires. A spark $2$ mm in diameter pops out of the fire and heads directly to the screen. What is the probability that the spark misses the wires? | [
"\\dfrac{1}{4}"
] | [
"A screen covering the front of a fireplace has wires $1$ mm in diameter, spaced in a mesh that has $5$ mm square openings between the wires. A spark $2$ mm in diameter pops out of the fire and heads directly to the screen. What is the probability that the spark misses the wires?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1cec0d921963c55e7f3a4f6e890332bd1436df79ccde4eea1ab3c5ea65ee9975 | A screen covering the front of a fireplace has wires 1mm in diameter, spaced in a mesh that has 5mm square openings between the wires. A spark 2mm in diameter pops out of the fire and heads directly toward the screen. What is the probability that the spark misses the wires? | [
"\\(\\frac{4}{9}\\)"
] | [
"A screen covering the front of a fireplace has wires 1mm in diameter, spaced in a mesh that has 5mm square openings between the wires. A spark 2mm in diameter pops out of the fire and heads directly toward the screen. What is the probability that the spark misses the wires?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-688b8563ea0f9621774ff953a7fcd9d0426c8342e3e0da8a33cae8f2b6c69080 | A sculptor cuts a cuboid into two pieces along a plane parallel to one of its faces, creating two new cuboids. Let \( A_1 \) be the surface area of the black cuboid and \( A_2 \) be the surface area of the white cuboid. If the original cuboid has dimensions \( x \times y \times z \) and is cut into two pieces with dime... | [
"\\dfrac{xy + xz + yz}{2xy}"
] | [
"A sculptor cuts a cuboid into two pieces along a plane parallel to one of its faces, creating two new cuboids. Let \\( A_1 \\) be the surface area of the black cuboid and \\( A_2 \\) be the surface area of the white cuboid. If the original cuboid has dimensions \\( x \\times y \\times z \\) and is cut into two pie... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-62d3dae0b1de20ac9c218974a6492e932d05bfe935fc81cece1fad17e9d7ecf1 | A sealed cylinder drum of radius \( r \) is 90% filled with paint. If the drum is tilted to rest on its side, what fraction of its curved surface area (not counting the flat sides) will be under the paint? | [
"\\(\\frac{1}{12}\\) to \\(\\frac{1}{6}\\)"
] | [
"A sealed cylinder drum of radius \\( r \\) is 90% filled with paint. If the drum is tilted to rest on its side, what fraction of its curved surface area (not counting the flat sides) will be under the paint?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-397e4ae8875a5926be0b87b54324062bd9224830f168188faa77de51ba2aefe8 | A search plane has a cruising speed of 250 mph and can fly for at most 5 hours. If there is a wind averaging 30 mph, and the plane travels with the wind one way and against it the other, how far can the search plane travel? | [
"616 miles"
] | [
"A search plane has a cruising speed of 250 mph and can fly for at most 5 hours. If there is a wind averaging 30 mph, and the plane travels with the wind one way and against it the other, how far can the search plane travel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-e6e88b21d5a7ac8f05c81cb20019d0e51793586c74818aeece7dd9a854459289 | A searchlight that is 10 feet away from a wall rotates at $\frac{\pi}{18}$ rad/sec. At $t=0$, the light shines on the part of the wall closest to it. How fast, in feet per second, is the light moving along the wall at $t=6$? | [
"\\(\\frac{20\\pi}{9}\\) feet/sec"
] | [
"A searchlight that is 10 feet away from a wall rotates at $\\frac{\\pi}{18}$ rad/sec. At $t=0$, the light shines on the part of the wall closest to it. How fast, in feet per second, is the light moving along the wall at $t=6$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-5d09d13d05564e02cd64765e70897672ad0bf99415a42c52162db348b0d12cae | A secant $OM$ and a tangent $MC$ are drawn from the point $M$ to a circle of radius of 3 cm with center at a point $O$, the tangent touching the circle at a point $C$. Find the radius of the circle which touches the given circle and the straight lines $MC$ and $OM$ and lies in the interior of the triangle $OMC$ if $| O... | [
"$\\frac{2}{3}\\left(3-\\sqrt 5\\right)$"
] | [
"A secant $OM$ and a tangent $MC$ are drawn from the point $M$ to a circle of radius of 3 cm with center at a point $O$, the tangent touching the circle at a point $C$. Find the radius of the circle which touches the given circle and the straight lines $MC$ and $OM$ and lies in the interior of the triangle $OMC$ if... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4401bfa7d74f6fb9cf4a0484b32114f3023363ee5ab6a3e4dd7ec09f0494591a | A secant intersects a unit circle at points \(A\) and \(B\) and is perpendicular to the radius it bisects. If the perpendicular to the secant at \(A\) and the tangent to the circle at \(B\) meet at \(C\), compute the area of triangle \(ABC\). | [
"\\(\\frac{3\\sqrt{3}}{2}\\)"
] | [
"A secant intersects a unit circle at points \\(A\\) and \\(B\\) and is perpendicular to the radius it bisects. If the perpendicular to the secant at \\(A\\) and the tangent to the circle at \\(B\\) meet at \\(C\\), compute the area of triangle \\(ABC\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-bbb0229ed68123680e0b0b62cbbdf420ba69bfb71da38bae01ea7ab7bdeb49a5 | A secant line intersects a circle at points \( A \) and \( C \) and the circle's diameter at point \( B \) with a \( 45^\circ \) angle. If \( AB = 1 \) and \( BC = 7 \), find the circle's radius. | [
"5"
] | [
"A secant line intersects a circle at points \\( A \\) and \\( C \\) and the circle's diameter at point \\( B \\) with a \\( 45^\\circ \\) angle. If \\( AB = 1 \\) and \\( BC = 7 \\), find the circle's radius."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-585d64dc8717c235b04f5ad30334b210c989590e472a8a5ee2d7d98f086dca5d | A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares have two corners on the secant line and two on the circumference. The ratio of the squares' side lengths is $5:9$. Compute the ratio of the secant line to the circle radius. | [
"\\dfrac{3\\sqrt{10}}{5}"
] | [
"A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares have two corners on the secant line and two on the circumference. The ratio of the squares' side lengths is $5:9$. Compute the ratio of the secant line to the circle radius."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-4991b4ac086aec79297158ea85f7078ba7d61cf2a1f1b193ff193006feaf7579 | A secant line splits a circle into two segments. Inside those segments, two squares are drawn such that each square has two corners on the secant line and two on the circumference. The ratio of the side lengths of the squares is $5:9$. Compute the ratio of the secant line to the circle radius. | [
"\\( \\frac{3}{5}\\sqrt{10} \\)"
] | [
"A secant line splits a circle into two segments. Inside those segments, two squares are drawn such that each square has two corners on the secant line and two on the circumference. The ratio of the side lengths of the squares is $5:9$. Compute the ratio of the secant line to the circle radius."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 19
} | 67 | 0 | 0 |
omr-ws-8ee4ed52044f2c3eade20d3c39cf7f3c20f2dd832073fe2384c2cc8629ba95a5 | A secant meets one circle at points $A_1$, $B_1$ and a second circle at points $A_2$, $B_2$. Another secant meets the first circle at points $C_1$, $D_1$ and the second circle at points $C_2$, $D_2$. Let $P = A_1C_1 \cap B_2D_2$, $Q = A_1C_1 \cap A_2C_2$, $R = A_2C_2 \cap B_1D_1$, and $S = B_2D_2 \cap B_1D_1$. If the t... | [
"\\dfrac{r_1 r_2}{d}"
] | [
"A secant meets one circle at points $A_1$, $B_1$ and a second circle at points $A_2$, $B_2$. Another secant meets the first circle at points $C_1$, $D_1$ and the second circle at points $C_2$, $D_2$. Let $P = A_1C_1 \\cap B_2D_2$, $Q = A_1C_1 \\cap A_2C_2$, $R = A_2C_2 \\cap B_1D_1$, and $S = B_2D_2 \\cap B_1D_1$.... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-e59f11efd445b6fe3ff299254cdf198eab2d4e8a7e034d3d21f0fe6bcd827d0d | A secretary put \( N \) letters into \( N \) envelopes in a random way. Let \( X \) be the number of letters that went to the correct envelopes. Calculate the expected value and variance of \( X \). | [
"\\( E(X) = 1 \\), \\( \\text{Var}(X) = 1 \\)"
] | [
"A secretary put \\( N \\) letters into \\( N \\) envelopes in a random way. Let \\( X \\) be the number of letters that went to the correct envelopes. Calculate the expected value and variance of \\( X \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-c8efb2c1f5975f003644ec4b6edf80f42480617925b180d41c73d8bddebbf1b4 | A secretary types 3 letters and addresses 3 envelopes (to 3 different locations). If she picks them up and inserts each letter randomly into an envelope, what is the probability that each letter will be inserted into its correct envelope? | [
"\\(\\frac{1}{6}\\)"
] | [
"A secretary types 3 letters and addresses 3 envelopes (to 3 different locations). If she picks them up and inserts each letter randomly into an envelope, what is the probability that each letter will be inserted into its correct envelope?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-bf0d934584836478a3784e9099cb18ba075a95551852617a3b03813dd46c4ef4 | A secretary works in a building located 9 blocks east and 8 blocks north of his home. Every day he walks 17 blocks to work. How many different routes are possible if the street in the easterly direction, which begins 4 blocks east and 3 blocks north of his home, is under water (and he can't swim)? | [
"19900"
] | [
"A secretary works in a building located 9 blocks east and 8 blocks north of his home. Every day he walks 17 blocks to work. How many different routes are possible if the street in the easterly direction, which begins 4 blocks east and 3 blocks north of his home, is under water (and he can't swim)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-eeb90ce1c3b3cc74abce645c99ff8198c04e9762788c0600425bbe065ee39259 | A secretary writes letters to 8 different people and addresses 8 envelopes with the people's addresses. He randomly puts the letters in the envelopes. What is the probability that he gets exactly 6 letters in the correct envelopes? | [
"\\(\\frac{1}{1440}\\)",
"\\(\\frac{1}{1440}\\) \n\nThe relevant discussion is in Post 2, where the probability of exactly 6 letters being in the correct envelopes is calculated as \\(\\frac{C(8,6)}{8!} = \\frac{1}{1440}\\)."
] | [
"A secretary writes letters to 8 different people and addresses 8 envelopes with the people's addresses. He randomly puts the letters in the envelopes. What is the probability that he gets exactly 6 letters in the correct envelopes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 56
} | 87 | 0 | 0 |
omr-ws-603dacf2a08fba0649c981b2eb7ed3ed7dc2e35a243a6bfe34e6e03e0e276917 | A section has 199 seats in each row and is reserved for students from South Carolina high schools, with each high school allowed to send up to 39 students. What is the least number of rows needed to guarantee that if 2006 students show up, all students from the same high school can be seated in a row? | [
"12"
] | [
"A section has 199 seats in each row and is reserved for students from South Carolina high schools, with each high school allowed to send up to 39 students. What is the least number of rows needed to guarantee that if 2006 students show up, all students from the same high school can be seated in a row?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 7
} | 26 | 0 | 0 |
omr-ws-ef6d36510d8ed2efcae522eb7a346668fd4dd8c708197375a680fe3f5734668a | A section of a regular tetragonal pyramid is a regular pentagon. Find the ratio of its side to the side of the base of the pyramid. | [
"\\dfrac{\\sqrt{5} - 1}{2}"
] | [
"A section of a regular tetragonal pyramid is a regular pentagon. Find the ratio of its side to the side of the base of the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3daaf98333e8ac0f36115c15617adc18cb39f43e9c9a5c7097159d0eb15cd0b3 | A section of a regular tetragonal pyramid is a regular pentagon. Find the ratio of the side of this pentagon to the side of the base of the pyramid. | [
"\\dfrac{\\sqrt{5} - 1}{2}"
] | [
"A section of a regular tetragonal pyramid is a regular pentagon. Find the ratio of the side of this pentagon to the side of the base of the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-5927a86be519a14b61b80667bec19238a9e7b9c942ae0a93de924788b3089a23 | A section which is drawn through the side $a$ of the base of a regular triangular prism at an angle $\alpha $ to it divides a lateral edge into parts in the ratio $m : n$, reckoning from the upper base. Find the volume of the resulting parts and the area of the section. | [
"$\\frac{(3m+2n)a^3\\tan\\alpha }{8n}:\\frac{a^3\\tan\\alpha }{8}$, $S=\\frac{a^2\\sqrt 3}{4\\cos \\alpha }$"
] | [
"A section which is drawn through the side $a$ of the base of a regular triangular prism at an angle $\\alpha $ to it divides a lateral edge into parts in the ratio $m : n$, reckoning from the upper base. Find the volume of the resulting parts and the area of the section."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aca741d7599bc5ec05b45546d80c76c8f7587a313af77c5cdefdb50cb5ea0009 | A sector has a perimeter of $20 \text{ cm}$ and its central angle is less than $360^\circ$. Find the range of the radius of the sector such that the area, $A$, satisfies $16 \text{ cm}^2 \le A \le 24 \text{ cm}^2$. | [
"\\(1.6 \\leq r \\leq 2.4\\)"
] | [
"A sector has a perimeter of $20 \\text{ cm}$ and its central angle is less than $360^\\circ$. Find the range of the radius of the sector such that the area, $A$, satisfies $16 \\text{ cm}^2 \\le A \\le 24 \\text{ cm}^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0654c1043f8265df2c946d8e9ea51f8f8d666e7d4f1857932eed0852cab953c5 | A sector has a perimeter of $20$ cm and its central angle is less than $360^\circ$. Find the range of the radius of the sector such that the area $A$ of the sector satisfies $16 \text{ cm}^2 \le A \le 24 \text{ cm}^2$. | [
"\\( 2 \\le r \\le 4 \\quad \\text{or} \\quad 6 \\le r \\le 8 \\)"
] | [
"A sector has a perimeter of $20$ cm and its central angle is less than $360^\\circ$. Find the range of the radius of the sector such that the area $A$ of the sector satisfies $16 \\text{ cm}^2 \\le A \\le 24 \\text{ cm}^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 25
} | 44 | 0 | 0 |
omr-ws-0b7646c06a55e03c4cc76beb5bee3468f47550f3cd544804dbc18adab1255d93 | A sector is cut out from a circular sheet of paper, and the two edges of the cut are joined together to form a cone. The cone formed has a radius of 6 and a volume of 96π. What is the number of degrees of the sector that was cut off? | [
"144"
] | [
"A sector is cut out from a circular sheet of paper, and the two edges of the cut are joined together to form a cone. The cone formed has a radius of 6 and a volume of 96π. What is the number of degrees of the sector that was cut off?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 3
} | 26 | 0 | 0 |
omr-ws-8ae4433774f478362e5c9e9b662275b1142d57a81ddf5f089feea5c79d5313c3 | A sector of a circle has an arc length of 1 and a sector angle of 60 degrees. A circle is inscribed in this sector, touching both radii and the arc. Compute the radius of the inscribed circle. | [
"$\\frac{1}{\\pi}$"
] | [
"A sector of a circle has an arc length of 1 and a sector angle of 60 degrees. A circle is inscribed in this sector, touching both radii and the arc. Compute the radius of the inscribed circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-263cddbd271edbcf9a97e586e37d883f13ce073dfa9a29221924697824d178d7 | A sector of radius \( R \) and central angle \( \theta \) is folded to form a cone. Find the volume and lateral surface area of the cone. | [
"Lateral Surface Area: \\( \\frac{1}{2} R^2 \\theta \\), Volume: \\( \\frac{1}{3} \\pi \\left(\\frac{R \\theta}{2\\pi}\\right)^2 \\sqrt{R^2 - \\left(\\frac{R \\theta}{2\\pi}\\right)^2} \\)",
"Lateral Surface Area: \\( \\frac{\\theta R^2}{2} \\), Volume: \\( \\frac{1}{3} \\pi \\left(\\frac{\\theta R}{2\\pi}\\right... | [
"A sector of radius \\( R \\) and central angle \\( \\theta \\) is folded to form a cone. Find the volume and lateral surface area of the cone."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-6086e1b960cda9b38f3776188771860a2e34dee2f3fdf0f037e73b97655f1aa1 | A sector of radius \( R \) and central angle \( a \) radians is folded to form a cone. Find the volume and lateral surface area of the cone. | [
"Lateral surface area = \\(\\frac{1}{2}R^2a\\), Volume = \\(\\frac{\\pi}{3} \\frac{R^3a^2}{4\\pi^2} \\sqrt{1-\\frac{a^2}{4\\pi^2}}\\)"
] | [
"A sector of radius \\( R \\) and central angle \\( a \\) radians is folded to form a cone. Find the volume and lateral surface area of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-52868533d2a7df38dfeeefa550fd4e0a4871aeeeafcbf62c04b5c54e041ba6dd | A sector with acute central angle \( \theta \) is cut from a circle of radius 6. What is the radius of the circle circumscribed about the sector? | [
"\\( 3 \\sec\\left(\\frac{1}{2}\\theta\\right) \\)"
] | [
"A sector with acute central angle \\( \\theta \\) is cut from a circle of radius 6. What is the radius of the circle circumscribed about the sector?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-40dc263b52f0c7606234ee70a593a7c0823ab5020c1a72db6e954643ed17ccda | A sector with an acute central angle \( \theta \) is cut from a circle of radius 6. What is the radius of the circle circumscribed about the sector? | [
"\\( 3 \\sec \\frac12 \\theta \\)"
] | [
"A sector with an acute central angle \\( \\theta \\) is cut from a circle of radius 6. What is the radius of the circle circumscribed about the sector?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-8329a55d6269ac38f4ecf27bed1797ad8e353adac173db2f0f44d5b7be4467a3 | A sector with an interior angle of 135 degrees is cut from a circle of radius 8 and discarded. The remainder of the circle is formed into a cone by taping together the straight edges. What is the radius of the base of the cone? | [
"5"
] | [
"A sector with an interior angle of 135 degrees is cut from a circle of radius 8 and discarded. The remainder of the circle is formed into a cone by taping together the straight edges. What is the radius of the base of the cone?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c165a11971cc438264664e562060f9d870c6ca88497b9112e3802561ac9e2ea8 | A security guard enters a building at some time between 1 am and 3 am and stays for 10 minutes. A burglar breaks in at some time between 1 am and 2 am and stays for 15 minutes. The probability that the security guard and the burglar are in the building at the same time can be expressed as \( \frac{m}{n} \) where \( m \... | [
"173"
] | [
"A security guard enters a building at some time between 1 am and 3 am and stays for 10 minutes. A burglar breaks in at some time between 1 am and 2 am and stays for 15 minutes. The probability that the security guard and the burglar are in the building at the same time can be expressed as \\( \\frac{m}{n} \\) wher... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 4
} | 27 | 0 | 0 |
omr-ws-2177394e93d311b0b663288cc20738618a86c2979a4518b6b886a6cbd2037833 | A segment $AB$ is fixed on the plane. Consider all acute-angled triangles with side $AB$. Find the locus of the vertices of their greatest angles. | [
"\\text{The two arcs of the circles with radius } AB \\text{ centered at } A \\text{ and } B \\text{ that lie outside the circle with diameter } AB"
] | [
"A segment $AB$ is fixed on the plane. Consider all acute-angled triangles with side $AB$. Find the locus of the vertices of their greatest angles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-dbad3bb305b6f7cb09c3b382d0d8750640fdfb2c52e6d9b1a978360db3b4543b | A segment $AB$ is fixed on the plane. Consider all acute-angled triangles with side $AB$. Find the locus of their incenters. | [
"\\text{The interior of the circle with diameter } AB"
] | [
"A segment $AB$ is fixed on the plane. Consider all acute-angled triangles with side $AB$. Find the locus of their incenters."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-e93707a7d30344daacb3df653ad5905f5785712f1d029b5aea89136426a4893f | A segment $AB$ is marked on a line $t$. The segment is moved on the plane so that it remains parallel to $t$ and that the traces of points $A$ and $B$ do not intersect. The segment finally returns onto $t$. How far can point $A$ now be from its initial position? | [
"AB",
"arbitrarily large"
] | [
"A segment $AB$ is marked on a line $t$. The segment is moved on the plane so that it remains parallel to $t$ and that the traces of points $A$ and $B$ do not intersect. The segment finally returns onto $t$. How far can point $A$ now be from its initial position?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 2
} | 24 | 0 | 0 |
omr-ws-0818b0c604ffcec5a0f48a8fa18ba20925c75f3416d2486d58a495626e11cfb0 | A segment $BC$ of constant length slides along the sides of the angle $\angle PAQ$ ($B$ on $AP$, $C$ on $AQ$). Let $N$ be the Nagel point of triangle $ABC$. If the length of $BC$ is $2$, and the angle $\angle PAQ$ is $60^\circ$, find the radius of the circle traced by the Nagel point $N$ as $B$ and $C$ move along $AP$ ... | [
"\\dfrac{2\\sqrt{3}}{3}"
] | [
"A segment $BC$ of constant length slides along the sides of the angle $\\angle PAQ$ ($B$ on $AP$, $C$ on $AQ$). Let $N$ be the Nagel point of triangle $ABC$. If the length of $BC$ is $2$, and the angle $\\angle PAQ$ is $60^\\circ$, find the radius of the circle traced by the Nagel point $N$ as $B$ and $C$ move alo... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-89e1f0f5dadf710a34b3e6c6175f2a5dff9c4ffc0fef04bbbebf386bd597c04c | A segment $\overline{AD}$ is divided by two points $B$ and $C$. Semicircular arcs are constructed over $\overline{AD}$, $\overline{AB}$, $\overline{BC}$, and $\overline{CD}$. Points $P$ and $Q$ are on the semicircles above $\overline{AD}$ and $\overline{BC}$ respectively, such that $PQ$ is perpendicular to $\overline{A... | [
"4\\pi"
] | [
"A segment $\\overline{AD}$ is divided by two points $B$ and $C$. Semicircular arcs are constructed over $\\overline{AD}$, $\\overline{AB}$, $\\overline{BC}$, and $\\overline{CD}$. Points $P$ and $Q$ are on the semicircles above $\\overline{AD}$ and $\\overline{BC}$ respectively, such that $PQ$ is perpendicular to ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-384ea79641efab24673b75c2c9b5768ea840affda48d4a2e083c6f7ef52f0716 | A segment is divided into a finite number of smaller segments, some of which are colored blue and the rest red (the ends of the segments are not colored). It is known that the set of all red segments can be superimposed on the set of all blue segments (without shifting one-color segments relative to each other) so that... | [
"\\dfrac{1}{4}"
] | [
"A segment is divided into a finite number of smaller segments, some of which are colored blue and the rest red (the ends of the segments are not colored). It is known that the set of all red segments can be superimposed on the set of all blue segments (without shifting one-color segments relative to each other) so... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-2687c80330d26872afb471e08f3a6d2ddab1b92176054f553b26ab4bc3beb60c | A segment is drawn from $(362,194)$ to $(670,579)$. A regular hexagon of side length $\frac{1}{2}$ is centered at each lattice point in the plane. Given that hexagons may be rotated about their centers, what is the greatest number of hexagons that could possibly have two adjacent sides intersecting the segment? | [
"386"
] | [
"A segment is drawn from $(362,194)$ to $(670,579)$. A regular hexagon of side length $\\frac{1}{2}$ is centered at each lattice point in the plane. Given that hexagons may be rotated about their centers, what is the greatest number of hexagons that could possibly have two adjacent sides intersecting the segment?"
... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-be77411fbab86f578c07d352e3b07d73f50c6e98ee70ff5d069f233c93f42dac | A segment of a line with its endpoints on sides AB and AC of triangle ABC bisects the triangle into two regions of equal area. Find the length of the shortest such segment. | [
"\\(\\sqrt{bc(1-\\cos A)}\\)"
] | [
"A segment of a line with its endpoints on sides AB and AC of triangle ABC bisects the triangle into two regions of equal area. Find the length of the shortest such segment."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 3,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-cce92529b3af968226a65e5787183f7855d5c76eb111bea27cf7fe3ee4430cfa | A segment of length $1$ is completely covered by a finite number of smaller segments lying on it. Find the maximum possible value of the sum of the lengths of a selection of these segments such that no two segments in the selection intersect. Let this maximum sum be $S$. Calculate $S$. | [
"1"
] | [
"A segment of length $1$ is completely covered by a finite number of smaller segments lying on it. Find the maximum possible value of the sum of the lengths of a selection of these segments such that no two segments in the selection intersect. Let this maximum sum be $S$. Calculate $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-c0658690f1a809f9d6cfef2af841d2887904a2a8d976bf70913591c6085a9319 | A segment of length $2$ is divided into $n$, $n \ge 2$, subintervals. A square is constructed on each subinterval. Assume that the sum of the areas of all such squares is greater than $1$. Let $S$ be the set of all possible total lengths of two chosen subintervals under this assumption. Find the minimum value of $S$. | [
"0"
] | [
"A segment of length $2$ is divided into $n$, $n \\ge 2$, subintervals. A square is constructed on each subinterval. Assume that the sum of the areas of all such squares is greater than $1$. Let $S$ be the set of all possible total lengths of two chosen subintervals under this assumption. Find the minimum value of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-3a4410aef8bc3b5def36ece8f6c68359e98d2449888a43ced4768350eba3fb25 | A segment of length $2$ is divided into $n$, $n \geq 2$, subintervals. A square is constructed on each subinterval. Suppose the sum of the areas of all such squares is exactly $1.5$. Let $L$ be the maximum possible total length of any two chosen subintervals. Find the minimum possible value of $L$. | [
"\\dfrac{\\sqrt{6}}{2}"
] | [
"A segment of length $2$ is divided into $n$, $n \\geq 2$, subintervals. A square is constructed on each subinterval. Suppose the sum of the areas of all such squares is exactly $1.5$. Let $L$ be the maximum possible total length of any two chosen subintervals. Find the minimum possible value of $L$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2102eb211e019c98c33711d9f0f0ae1adfc48b4722de760a6d12fd257513db1f | A segment of length 2 is divided into \( n \) intervals (\( n \geq 2 \)). On each subinterval, a square is drawn with the subinterval as its side. If the sum of the areas of all these squares is greater than 1, find the minimum value of \( n \) such that it is always possible to choose two subintervals whose total leng... | [
"4"
] | [
"A segment of length 2 is divided into \\( n \\) intervals (\\( n \\geq 2 \\)). On each subinterval, a square is drawn with the subinterval as its side. If the sum of the areas of all these squares is greater than 1, find the minimum value of \\( n \\) such that it is always possible to choose two subintervals whos... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-c9b91dcf37e8e4bdba3772801fa3396aa06913d3bc99e7cec66ff4d3e689c5d4 | A segment of unit length is cut into eleven smaller segments, each with length no more than $a$. For what values of $a$ can one guarantee that any three segments can form a triangle? | [
"\\(\\frac{1}{11} \\leq a < \\frac{1}{10}\\)"
] | [
"A segment of unit length is cut into eleven smaller segments, each with length no more than $a$. For what values of $a$ can one guarantee that any three segments can form a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-aa9639be6cd03af93c9dad51b91c8f453f34129f5a5b55bcd62f58003e67e9eb | A segment with endpoints at $A(2, -2)$ and $B(14, 4)$ is extended through $B$ to point $C$. If $BC = \frac{1}{3} \cdot AB$, what are the coordinates for point $C$? Express your answer as an ordered pair. | [
"(18,6)"
] | [
"A segment with endpoints at $A(2, -2)$ and $B(14, 4)$ is extended through $B$ to point $C$. If $BC = \\frac{1}{3} \\cdot AB$, what are the coordinates for point $C$? Express your answer as an ordered pair."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-378b025de19b2bfc3b3709449ed1a5a1898cd347248e476bb5f7b2fa17d1e7bf | A segment with the endpoints $A (3, -2)$ and $B (6, 4)$ is divided into three equal parts. Find the coordinates of the points of division. | [
"$(4,0),\\ (5,2)$"
] | [
"A segment with the endpoints $A (3, -2)$ and $B (6, 4)$ is divided into three equal parts. Find the coordinates of the points of division."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-968f0d80d27cafaf2f782a5044c5ed18c6d82ea67ec18dc1677230241f0c8739 | A self-avoiding rook walk on a chessboard (a rectangular grid of unit squares) is a path traced by a sequence of moves parallel to an edge of the board from one unit square to another, such that each move begins where the previous move ended and such that no move ever crosses a square that has previously been crossed. ... | [
"\\(2^{n-1} + 2^{n-2}\\)"
] | [
"A self-avoiding rook walk on a chessboard (a rectangular grid of unit squares) is a path traced by a sequence of moves parallel to an edge of the board from one unit square to another, such that each move begins where the previous move ended and such that no move ever crosses a square that has previously been cros... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-878bf80f5455ef02684972c462fb2f7f122804ca2a643dfc3cca51abd2cdcd40 | A semi-circle $ABCD$ has diameter $AD$ of length $4$. Chord $AB$ has length $1$ and chord $BC$ has length $1$. Find the length of chord $CD$. | [
"\\(\\frac{7}{2}\\)"
] | [
"A semi-circle $ABCD$ has diameter $AD$ of length $4$. Chord $AB$ has length $1$ and chord $BC$ has length $1$. Find the length of chord $CD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 25
} | 73 | 0 | 0 |
omr-ws-a4f88e275118445fe0efdb1c899722eb178197e984b8d23a1c6827f12c1a3b7e | A semi-underground maths bunker has a $1$ metre square window with the bottom side exactly at ground level and parallel to the ground. A disc with a diameter of $5$ metres rolls past just outside the window at a velocity of $1$ m/s. For how long is at least part of the window obscured? | [
"6"
] | [
"A semi-underground maths bunker has a $1$ metre square window with the bottom side exactly at ground level and parallel to the ground. A disc with a diameter of $5$ metres rolls past just outside the window at a velocity of $1$ m/s. For how long is at least part of the window obscured?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-270cb21ec022953fc8a1f64f88b9eb4553318fd809bcd544fc9ffe05c55c6e52 | A semicircle $(O)$ is drawn with center $O$ on a line $\ell$. Points $C$ and $D$ lie on the semicircle, and the tangent lines at $C$ and $D$ intersect $\ell$ at points $B$ and $A$, respectively, with $O$ between $B$ and $A$. Let $E$ be the intersection point of $AC$ and $BD$, and let $F$ be the point on $\ell$ such tha... | [
"\\dfrac{r}{2}"
] | [
"A semicircle $(O)$ is drawn with center $O$ on a line $\\ell$. Points $C$ and $D$ lie on the semicircle, and the tangent lines at $C$ and $D$ intersect $\\ell$ at points $B$ and $A$, respectively, with $O$ between $B$ and $A$. Let $E$ be the intersection point of $AC$ and $BD$, and let $F$ be the point on $\\ell$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-ab1004aa6b7282dfdc7aaaa442775fa77e483e093ed5d3e477eba6d920a249bc | A semicircle $\Gamma$ is drawn with diameter on line $l$. Points $C$ and $D$ are on $\Gamma$ but not on $l$. The tangents to $\Gamma$ at $C$ and $D$ intersect $l$ at points $B$ and $A$, respectively. Let $E$ be the intersection of $AC$ and $BD$, and let $F$ be the point on $l$ such that $EF$ is perpendicular to $l$. If... | [
"1"
] | [
"A semicircle $\\Gamma$ is drawn with diameter on line $l$. Points $C$ and $D$ are on $\\Gamma$ but not on $l$. The tangents to $\\Gamma$ at $C$ and $D$ intersect $l$ at points $B$ and $A$, respectively. Let $E$ be the intersection of $AC$ and $BD$, and let $F$ be the point on $l$ such that $EF$ is perpendicular to... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-5080f084eaf501353e59558ff8e27cb4240a4422c5acff602ef6953cd1896733 | A semicircle $\omega$ has a diameter $\overline{XY}$ with length $8$. A point $I$ is variable along semicircle $\omega$ and $J$ is a point lying on ray $\overrightarrow{IX}$ such that $\overline{IJ} = \overline{IY}$. The possible locations of $J$ trace a curve. Find the length of this curve traced by $J$. | [
"\\(4\\pi\\sqrt{2}\\)"
] | [
"A semicircle $\\omega$ has a diameter $\\overline{XY}$ with length $8$. A point $I$ is variable along semicircle $\\omega$ and $J$ is a point lying on ray $\\overrightarrow{IX}$ such that $\\overline{IJ} = \\overline{IY}$. The possible locations of $J$ trace a curve. Find the length of this curve traced by $J$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-8ee359bf264d563e35adb97f2b555ab418c39f71ea5236066c19d88654717983 | A semicircle $\omega$ has radius $1$ and diameter $\overline{AB}$. Two distinct circles $\omega_1$ and $\omega_2$ of radius $r$ are constructed internally tangent to $\omega$ and tangent to the segment $\overline{AB}$. Circle $\mathcal{A}$ is constructed externally tangent to $\omega_1$ and $\omega_2$ as well as $\over... | [
"\\( \\frac{\\sqrt{2}}{4} \\)"
] | [
"A semicircle $\\omega$ has radius $1$ and diameter $\\overline{AB}$. Two distinct circles $\\omega_1$ and $\\omega_2$ of radius $r$ are constructed internally tangent to $\\omega$ and tangent to the segment $\\overline{AB}$. Circle $\\mathcal{A}$ is constructed externally tangent to $\\omega_1$ and $\\omega_2$ as ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-f06a8f8bd79e511d4eedff58342cee69e5947ff3c52a602cd71f7a9ba239e729 | A semicircle has \( O \) as its diameter midpoint and radius \( 1 \). Two points \( P_1 \) and \( P_2 \) are randomly and independently picked on the perimeter of the semicircle, including its diameter. Let \( \triangle \) denote the area of triangle \( OP_1P_2 \). What is the expected value of \( \triangle \)? | [
"\\dfrac{1}{\\pi + 2}"
] | [
"A semicircle has \\( O \\) as its diameter midpoint and radius \\( 1 \\). Two points \\( P_1 \\) and \\( P_2 \\) are randomly and independently picked on the perimeter of the semicircle, including its diameter. Let \\( \\triangle \\) denote the area of triangle \\( OP_1P_2 \\). What is the expected value of \\( \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-41b61b1b0f4bca6078e411d1be627cd5c5e6413bb706f3d8f3e8f87c36baecfb | A semicircle has center $O$ and diameter $AB$. Chord $CD$ is parallel to the diameter $AB$ and $\angle AOC = \angle DOB = \frac{7}{16}$ radians. Determine which of the two parts created by chord $CD$ has a larger area. | [
"The part containing the angle \\( \\angle COD \\) has a larger area."
] | [
"A semicircle has center $O$ and diameter $AB$. Chord $CD$ is parallel to the diameter $AB$ and $\\angle AOC = \\angle DOB = \\frac{7}{16}$ radians. Determine which of the two parts created by chord $CD$ has a larger area."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 25
} | 39 | 0 | 0 |
omr-ws-a293b5f668fa01b105770815294422fa833f67af42dbc0a99ecdcd95b9287bc5 | A semicircle has chords of lengths 3, 2, and 1. What is the radius of the semicircle? | [
"2.0565453"
] | [
"A semicircle has chords of lengths 3, 2, and 1. What is the radius of the semicircle?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-6854c7aa2b590757ceae2b5cc64a2ca1c56b437127fdf2b94830e6f7298c75ac | A semicircle has diameter $XY$. A square $PQRS$ with side length 12 is inscribed in the semicircle with $P$ and $S$ on the diameter. Another square $STUV$ has $T$ on $RS$, $U$ on the semicircle, and $V$ on $XY$. What is the area of square $STUV$? | [
"36"
] | [
"A semicircle has diameter $XY$. A square $PQRS$ with side length 12 is inscribed in the semicircle with $P$ and $S$ on the diameter. Another square $STUV$ has $T$ on $RS$, $U$ on the semicircle, and $V$ on $XY$. What is the area of square $STUV$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2e528d2e43d8480950c16a32c15eb33902be62529b3811d4aed0fb1c1ce359aa | A semicircle has diameter AC, and BD is perpendicular to AC at B. Given AB = 6 and BC = 24, find the length of BD. | [
"12"
] | [
"A semicircle has diameter AC, and BD is perpendicular to AC at B. Given AB = 6 and BC = 24, find the length of BD."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-f8c6e3ee80ecae737fd9b7df118e31926dd15759c1196d0e072391ad94b9b21d | A semicircle has diameter XY. A square PQRS with side length 12 is inscribed in the semicircle with P and S on the diameter. A square STUV has T on RS, U on the semicircle, and V on XY. What is the area of square STUV? | [
"36"
] | [
"A semicircle has diameter XY. A square PQRS with side length 12 is inscribed in the semicircle with P and S on the diameter. A square STUV has T on RS, U on the semicircle, and V on XY. What is the area of square STUV?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-135c1f2fb14c2327bb64ec5d22007f4c772a0c4c54f98a803fede03ea4906d68 | A semicircle has diameter \(AB\) and center \(S\), with a point \(M\) on the circumference. \(U\) and \(V\) are the incircles of sectors \(ASM\) and \(BSM\). Let \(d\) be the shortest distance between the centers of circles \(U\) and \(V\). If \(AB = 10\), find the value of \(d\) when \(M\) is at the midpoint of the ar... | [
"10(\\sqrt{2} - 1)"
] | [
"A semicircle has diameter \\(AB\\) and center \\(S\\), with a point \\(M\\) on the circumference. \\(U\\) and \\(V\\) are the incircles of sectors \\(ASM\\) and \\(BSM\\). Let \\(d\\) be the shortest distance between the centers of circles \\(U\\) and \\(V\\). If \\(AB = 10\\), find the value of \\(d\\) when \\(M\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-8f08a793f581bc7f0efe08cdee1cfb5e139c930bd431415fead68408539a325c | A semicircle is constructed on the base of an equilateral triangle as a diameter which bisects the triangle. The side of the triangle is $a$ long. Find the area of the part of the triangle which lies outside the semicircle. | [
"$\\frac{a^2\\left(3\\sqrt 3-\\pi \\right)}{24}$"
] | [
"A semicircle is constructed on the base of an equilateral triangle as a diameter which bisects the triangle. The side of the triangle is $a$ long. Find the area of the part of the triangle which lies outside the semicircle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1884b2b2213a5741cabfdfa725eef9ae29171b808511dd4060bda9784bcbbc74 | A semicircle is constructed with diameter $AD = 3$ cm. Points $B$ and $C$ are marked on the arc such that $AB = BC = 1$ cm. Find the length of the chord $CD$. | [
"\\( \\sqrt{7} \\)"
] | [
"A semicircle is constructed with diameter $AD = 3$ cm. Points $B$ and $C$ are marked on the arc such that $AB = BC = 1$ cm. Find the length of the chord $CD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-84b252044c4e6543857334d2f5fa319cc1ec3b167e0a31ea927393165e9035f4 | A semicircle is cut out of a piece of homogenous thin cardboard and hung freely by an endpoint of the diameter. Find the angle between the vertical and the diameter. | [
"\\(\\tan^{-1}\\left(\\frac{4}{3\\pi}\\right)\\)"
] | [
"A semicircle is cut out of a piece of homogenous thin cardboard and hung freely by an endpoint of the diameter. Find the angle between the vertical and the diameter."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 8,
"tir": 15
} | 50 | 0 | 0 |
omr-ws-c570810e66163d13c992a0d6e30ade916d65337cca3ff8e48ce843b949990a57 | A semicircle is drawn on $\overline{AB}$, a line segment of length $2$. Let $C_1$ be the circle of radius $\frac12$ that is tangent to $\overline{AB}$ and the semicircle. For $n\geq 2$, let $C_n$ be the circle that is tangent to $\overline{AB}$, the semicircle, and the circle $C_{n-1}$. (Circle $C_n$ is also smaller th... | [
"578"
] | [
"A semicircle is drawn on $\\overline{AB}$, a line segment of length $2$. Let $C_1$ be the circle of radius $\\frac12$ that is tangent to $\\overline{AB}$ and the semicircle. For $n\\geq 2$, let $C_n$ be the circle that is tangent to $\\overline{AB}$, the semicircle, and the circle $C_{n-1}$. (Circle $C_n$ is also ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-3c826d187edb37cb9338eb150f53d01e21730f5702152aa404b71c15034bee10 | A semicircle is erected over the segment \(AB\) with center \(M\). Let \(P\) be a point on the semicircle different from \(A\) and \(B\), and let \(Q\) be the midpoint of the arc \(AP\). The point of intersection of the line \(BP\) with the line parallel to \(PQ\) through \(M\) is \(S\). Calculate the ratio \(\frac{PS}... | [
"1"
] | [
"A semicircle is erected over the segment \\(AB\\) with center \\(M\\). Let \\(P\\) be a point on the semicircle different from \\(A\\) and \\(B\\), and let \\(Q\\) be the midpoint of the arc \\(AP\\). The point of intersection of the line \\(BP\\) with the line parallel to \\(PQ\\) through \\(M\\) is \\(S\\). Calc... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d502743c1fc529684d9fa5dca9810d205a2434eddec79a6d77b8367574de74b0 | A semicircle is given with diameter $AB$. An arbitrary point $C$ is chosen on the arc $AB$ (excluding points $A$ and $B$). Let $D$ be the orthogonal projection of point $C$ onto the diameter $AB$. A circle $\omega$ touches segments $AD$, $CD$, and the arc $AB$ at point $P$. Let $Q$ be the intersection point of the bise... | [
"no_valid_answer_found"
] | [
"A semicircle is given with diameter $AB$. An arbitrary point $C$ is chosen on the arc $AB$ (excluding points $A$ and $B$). Let $D$ be the orthogonal projection of point $C$ onto the diameter $AB$. A circle $\\omega$ touches segments $AD$, $CD$, and the arc $AB$ at point $P$. Let $Q$ be the intersection point of th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d7c0a42d1260eff99b8c56887dec88a76644a842dfa0d0101f014b26f5294dba | A semicircle is inscribed in a quadrilateral ABCD such that the midpoint of BC coincides with the center of the semicircle, and the diameter of the semicircle lies along a portion of BC. If AB = 4 and CD = 5, what is the length of BC? | [
"\\( 4\\sqrt{5} \\)"
] | [
"A semicircle is inscribed in a quadrilateral ABCD such that the midpoint of BC coincides with the center of the semicircle, and the diameter of the semicircle lies along a portion of BC. If AB = 4 and CD = 5, what is the length of BC?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 10
} | 23 | 0 | 0 |
omr-ws-ea98d87c97cd4182417a12a02ca9ab1d28d67a237199c4a02819be07bc8b908a | A semicircle is inscribed in a quadrilateral \(ABCD\) such that the midpoint of \(BC\) coincides with the center of the semicircle, and the diameter of the semicircle lies along a portion of \(BC\). If \(AB = 4\) and \(CD = 5\), what is the length of \(BC\)? | [
"\\(4\\sqrt{5}\\)"
] | [
"A semicircle is inscribed in a quadrilateral \\(ABCD\\) such that the midpoint of \\(BC\\) coincides with the center of the semicircle, and the diameter of the semicircle lies along a portion of \\(BC\\). If \\(AB = 4\\) and \\(CD = 5\\), what is the length of \\(BC\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-0e3f1caa80a153c4fb83e536fa2d3e9ff7cf180e27fc545e09ed7cf332293549 | A semicircle is inscribed in a right triangle \(ABC\) with the right angle at \(B\). The semicircle has its diameter on \(AB\) and one end at point \(B\). Given that \(AB = 15\) and \(BC = 8\), determine the radius of the semicircle. | [
"\\(\\frac{24}{5}\\)"
] | [
"A semicircle is inscribed in a right triangle \\(ABC\\) with the right angle at \\(B\\). The semicircle has its diameter on \\(AB\\) and one end at point \\(B\\). Given that \\(AB = 15\\) and \\(BC = 8\\), determine the radius of the semicircle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-bb113eb82f1a30e06bf58152cc6565022a87ef75538551ecca625d030be5a400 | A semicircle is inscribed in a right triangle with sides 5, 12, and 13, such that the base of the semicircle is touching the side of the triangle that is 12 cm long, and the curving part of the semicircle is convex to the hypotenuse. What is the radius of the inscribed semicircle? | [
"\\(\\frac{10}{3}\\)",
"\\dfrac{10}{3}"
] | [
"A semicircle is inscribed in a right triangle with sides 5, 12, and 13, such that the base of the semicircle is touching the side of the triangle that is 12 cm long, and the curving part of the semicircle is convex to the hypotenuse. What is the radius of the inscribed semicircle?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 19
} | 50 | 0 | 0 |
omr-ws-de2a59e304e334e369a39c5822a1172cd58a46937aaa5ca4e69a1f001db3d534 | A semicircle is inscribed in a right triangle with sides 5, 12, and 13. The base of the semicircle is touching the side of the triangle that is 12 cm long, and the curving part of the semicircle is convex to the hypotenuse. What is the radius of the inscribed semicircle? | [
"\\(\\frac{10}{3}\\)"
] | [
"A semicircle is inscribed in a right triangle with sides 5, 12, and 13. The base of the semicircle is touching the side of the triangle that is 12 cm long, and the curving part of the semicircle is convex to the hypotenuse. What is the radius of the inscribed semicircle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-6a6762fa0dc74bfd5f9549e50ab7d680284b59c2009fff82f5395471018b7699 | A semicircle is inscribed in a triangle \(ABC\) such that the diameter \(\overline{EF}\) of the semicircle lies on the side \(\overline{AB}\). \(E\) is between \(F\) and \(A\), and the semicircle touches the sides of the triangle \(\overline{BC}\) and \(\overline{AC}\) at points \(G\) and \(H\) respectively. Let \(P\) ... | [
"d"
] | [
"A semicircle is inscribed in a triangle \\(ABC\\) such that the diameter \\(\\overline{EF}\\) of the semicircle lies on the side \\(\\overline{AB}\\). \\(E\\) is between \\(F\\) and \\(A\\), and the semicircle touches the sides of the triangle \\(\\overline{BC}\\) and \\(\\overline{AC}\\) at points \\(G\\) and \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-de8e7c153036bdf77eafdc924365a52e3ae42be08e96950a76b5cdb6de957a10 | A semicircle is inscribed in an equilateral triangle such that the diameter of the semicircle lies on the base of the triangle. Express the radius of the semicircle in terms of the side length \( S \) of the triangle. | [
"\\dfrac{\\sqrt{3}}{4} S"
] | [
"A semicircle is inscribed in an equilateral triangle such that the diameter of the semicircle lies on the base of the triangle. Express the radius of the semicircle in terms of the side length \\( S \\) of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-30958a3130dad265d709e380286a46fb11cd379838d95bba4e96378e7af16cdf | A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle. What is the radius of the semicircle? | [
"\\(\\frac{120}{17}\\)"
] | [
"A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle. What is the radius of the semicircle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 26
} | 74 | 0 | 0 |
omr-ws-9b6fdbf71acf359f1323c0bda5c10e40f66903b173f952e85280a27765359666 | A semicircle is inscribed in another semicircle such that the "apex" of the inscribed semicircle divides the diameter of the larger semicircle into two equal portions. What is the ratio of the radius of the smaller semicircle to the radius of the larger semicircle? | [
"\\(\\frac{\\sqrt{2}}{2}\\)"
] | [
"A semicircle is inscribed in another semicircle such that the \"apex\" of the inscribed semicircle divides the diameter of the larger semicircle into two equal portions. What is the ratio of the radius of the smaller semicircle to the radius of the larger semicircle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-d1ef686967d4b323ad93e88ffe8fea84a392b6fa8aac2c57147d67fc472c785b | A semicircle is inscribed in the acute triangle $ABC$ so that its diameter lies on the side $AB$ and the arc touches the sides $AC$ and $BC$. Find the radius of the circle which touches the arc of the semicircle and the sides $AC$ and $BC$ of the triangle if $| AC | = b$, $| BC | = a$, $\angle ACB = \alpha $. | [
"$\\frac{ab\\sin \\alpha }{a+b}\\cdot \\frac{1-\\sin \\frac{\\alpha }{2}} {1+\\sin \\frac{\\alpha }{2}}$"
] | [
"A semicircle is inscribed in the acute triangle $ABC$ so that its diameter lies on the side $AB$ and the arc touches the sides $AC$ and $BC$. Find the radius of the circle which touches the arc of the semicircle and the sides $AC$ and $BC$ of the triangle if $| AC | = b$, $| BC | = a$, $\\angle ACB = \\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc0baf1495fd52d4c0ec91384600b9650b193251f7f3cdfa80d2c77e74c4c7b1 | A semicircle is inscribed inside a quarter-circle with area $9\pi$. Find the maximum possible radius of the semicircle. | [
"\\(2\\sqrt{3}\\)"
] | [
"A semicircle is inscribed inside a quarter-circle with area $9\\pi$. Find the maximum possible radius of the semicircle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98eec89ce28eeabbbfe9770ec939c5919058fee3c07d792ce1ab7a30d0620212 | A semicircle of radius $2$ has a circle $\omega$ of radius $1$ inscribed within it, touching the semicircle at the midpoint $O$ of its diameter $AB$. Find the radius of the circle that is tangent to both $\omega$ and the semicircle. | [
"\\( 0.5 \\)"
] | [
"A semicircle of radius $2$ has a circle $\\omega$ of radius $1$ inscribed within it, touching the semicircle at the midpoint $O$ of its diameter $AB$. Find the radius of the circle that is tangent to both $\\omega$ and the semicircle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 4
} | 18 | 0 | 0 |
omr-ws-d60e13a8d57113d943be598c50b297dad94900f51c05a9318d8cfac3c9bc8572 | A semicircle of radius $5$ and a quarter of a circle of radius $8$ touch each other and are located inside a square. Find the length of the part of the common tangent that is enclosed within the square. | [
"13"
] | [
"A semicircle of radius $5$ and a quarter of a circle of radius $8$ touch each other and are located inside a square. Find the length of the part of the common tangent that is enclosed within the square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-fed76b224c36ccf788ef56ea8108ef8ba0efbd7d64ee52bf39a34fbeee02d0c2 | A semicircle of radius $r$ is divided into $n + 1$ equal parts. For each division point $k$, a triangle $A_k$ is formed with the ends of the semicircle. Calculate the limit, as $n$ tends to infinity, of the arithmetic mean of the areas of these triangles. | [
"\\dfrac{2r^2}{\\pi}"
] | [
"A semicircle of radius $r$ is divided into $n + 1$ equal parts. For each division point $k$, a triangle $A_k$ is formed with the ends of the semicircle. Calculate the limit, as $n$ tends to infinity, of the arithmetic mean of the areas of these triangles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-073d5328269e3bf38b7174fe2cf335589d13854c755537c0683c58faaca84881 | A semicircle rests on the negative \( x \)-axis and is tangent to the \( y \)-axis. A line intersects both axes and the semicircle, creating three segments of equal length. What is the slope of the line? | [
"\\(\\frac{\\sqrt{14}}{7}\\)"
] | [
"A semicircle rests on the negative \\( x \\)-axis and is tangent to the \\( y \\)-axis. A line intersects both axes and the semicircle, creating three segments of equal length. What is the slope of the line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-65c13b5954dec485a98e8daaf048f0207a9d23a5165ef434747e96b5b08bcb3d | A semicircle with center $O$ and diameter $AB$ is given. Point $M$ on the extension of $AB$ is taken so that $AM > BM$. A line through $M$ intersects the semicircle at $C$ and $D$ so that $CM < DM$. The circumcircles of triangles $AOD$ and $OBC$ meet again at point $K$. If $OK$ and $KM$ are perpendicular, calculate the... | [
"10\\sqrt{2}"
] | [
"A semicircle with center $O$ and diameter $AB$ is given. Point $M$ on the extension of $AB$ is taken so that $AM > BM$. A line through $M$ intersects the semicircle at $C$ and $D$ so that $CM < DM$. The circumcircles of triangles $AOD$ and $OBC$ meet again at point $K$. If $OK$ and $KM$ are perpendicular, calculat... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6775c28364cf894f02fe87cd7bd11c18f247736f649426be18c8831ab3bd763c | A semicircle with center \( O \) has diameter \( AB \). A line intersects \( AB \) at \( M \) and the semicircle at \( C \) and \( D \) such that \( AM > BM \) and \( CM > DM \). The circumcircles of triangles \( AOC \) and \( BOD \) intersect again at \( K \). Let \( P \) be the intersection point of \( AD \), \( BC \... | [
"no_valid_answer_found"
] | [
"A semicircle with center \\( O \\) has diameter \\( AB \\). A line intersects \\( AB \\) at \\( M \\) and the semicircle at \\( C \\) and \\( D \\) such that \\( AM > BM \\) and \\( CM > DM \\). The circumcircles of triangles \\( AOC \\) and \\( BOD \\) intersect again at \\( K \\). Let \\( P \\) be the intersecti... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-722cf5a549fad7e53fcab9057115f01c834bbd3e822280ab619cbfa2cb0df0a8 | A semicircle with center \(O\) is inscribed in quadrilateral \(ABCD\) such that \(O\) is the midpoint of \(AB\), and segments \(AD\), \(DC\), and \(CB\) are tangent to the semicircle. Given \(AD = 5\) and \(BC = 6\), find the length of \(AB\). | [
"\\(2\\sqrt{30}\\)"
] | [
"A semicircle with center \\(O\\) is inscribed in quadrilateral \\(ABCD\\) such that \\(O\\) is the midpoint of \\(AB\\), and segments \\(AD\\), \\(DC\\), and \\(CB\\) are tangent to the semicircle. Given \\(AD = 5\\) and \\(BC = 6\\), find the length of \\(AB\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 20
} | 42 | 0 | 0 |
omr-ws-161ac6e6853b3c5c2a3756dbbb07079266f6a0f4a3a307c72f97de6b5dd391e3 | A semicircle with diameter $AB$ and center $S$ is given. Points $C$ and $D$ are on the semicircle such that $\angle CSD = 90^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$, and $F$ be the intersection of lines $AD$ and $BC$. If $AB = 2r$, find the length of $EF$ in terms of $r$. | [
"2r"
] | [
"A semicircle with diameter $AB$ and center $S$ is given. Points $C$ and $D$ are on the semicircle such that $\\angle CSD = 90^\\circ$. Let $E$ be the intersection of lines $AC$ and $BD$, and $F$ be the intersection of lines $AD$ and $BC$. If $AB = 2r$, find the length of $EF$ in terms of $r$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c001b95faf0844e2c45e34c947488c061af833249299fe91f346f5250e3d37b7 | A semicircle with diameter $d$ is contained in a square whose sides have length $8$. Given the maximum value of $d$ is $m - \sqrt{n}$, find $m+n$. | [
"544"
] | [
"A semicircle with diameter $d$ is contained in a square whose sides have length $8$. Given the maximum value of $d$ is $m - \\sqrt{n}$, find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-b7bf466144ad6bab73fef4ea5f6c7a846981a2123994ff4b9ac33c5240767d11 | A semicircle with diameter $d$ is contained in a square whose sides have length 8. Given the maximum value of $d$ is $m - \sqrt{n},$ find $m+n.$ Contents 1 Problem 2 Solution 2.1 Solution 1 2.2 Solution 2 2.3 Solution 3 2.4 Solution 4 2.5 Solution 5 3 See also | [
"544"
] | [
"A semicircle with diameter $d$ is contained in a square whose sides have length 8. Given the maximum value of $d$ is $m - \\sqrt{n},$ find $m+n.$ \nContents 1 Problem 2 Solution 2.1 Solution 1 2.2 Solution 2 2.3 Solution 3 2.4 Solution 4 2.5 Solution 5 3 See... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 3
} | 21 | 0 | 0 |
omr-ws-f4841d11aa8e0871fd023b9abb47edae4229dd0a7ba55345a3b5fa3966e71fc6 | A semicircle with diameter \( d \) is contained in a square whose sides have length 8. Given the maximum value of \( d \) is \( m - \sqrt{n} \), find \( m + n \). | [
"544"
] | [
"A semicircle with diameter \\( d \\) is contained in a square whose sides have length 8. Given the maximum value of \\( d \\) is \\( m - \\sqrt{n} \\), find \\( m + n \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-f44991838e7130d48cf5f75517f3fcd274410ecf18b9a58c97b9616ee53c0a07 | A semicircle with diameter \(AB = 2r\) is divided into two sectors by an arbitrary radius. To each of the sectors, a circle is inscribed, and these two circles touch \(AB\) at points \(S\) and \(T\). If the distance between \(S\) and \(T\) is minimized, find the value of \(ST\) in terms of \(r\). | [
"2r(\\sqrt{2} - 1)"
] | [
"A semicircle with diameter \\(AB = 2r\\) is divided into two sectors by an arbitrary radius. To each of the sectors, a circle is inscribed, and these two circles touch \\(AB\\) at points \\(S\\) and \\(T\\). If the distance between \\(S\\) and \\(T\\) is minimized, find the value of \\(ST\\) in terms of \\(r\\)."
... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-8c53b694615c5a6c3e0aaef5a58aa57f488c46233ea7f7bf058b89d154eea559 | A semicircle with diameter length $16$ contains a circle of radius $3$ that is tangent to both the inside of the semicircle and its diameter. A second larger circle is tangent to the inside of the semicircle, the outside of the circle, and the diameter of the semicircle. The diameter of the second circle can be written... | [
"361"
] | [
"A semicircle with diameter length $16$ contains a circle of radius $3$ that is tangent to both the inside of the semicircle and its diameter. A second larger circle is tangent to the inside of the semicircle, the outside of the circle, and the diameter of the semicircle. The diameter of the second circle can be wr... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-d236f6afedf9e50cee2b09de433db0cc872b3fe073d4f10a5b9208c02711dce6 | A semicircle with radius $r$ is given. This semicircle is surrounded by right triangles $ABC$. Among all such triangles, find the one with the smallest area. | [
"$2r^2$"
] | [
"A semicircle with radius $r$ is given. This semicircle is surrounded by right triangles $ABC$. Among all such triangles, find the one with the smallest area."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-b8bef1791f412d02d7c88003e771daad935ed59440b8213807a806d2ed8ca77c | A semicircle with radius 1 and diameter AB lies in the plane. A line parallel to AB intersects the semicircle at points P and Q, dividing the semicircle into two regions with equal area. Find the length of PQ. | [
"\\( PQ \\doteq 1.829542 \\)"
] | [
"A semicircle with radius 1 and diameter AB lies in the plane. A line parallel to AB intersects the semicircle at points P and Q, dividing the semicircle into two regions with equal area. Find the length of PQ."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 28
} | 41 | 0 | 0 |
omr-ws-bddb11cad67373d6cf9d950595e05d32d71f8f25375373648af585e531a05dcf | A semicircle with radius 2 has a part of its arc folded so that it lands on the diameter, dividing the diameter in the ratio 3 : 1. Find the length of the fold. | [
"\\(\\frac{5}{4}\\)"
] | [
"A semicircle with radius 2 has a part of its arc folded so that it lands on the diameter, dividing the diameter in the ratio 3 : 1. Find the length of the fold."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4901cb9b316fe1dc2d872914b45363cb326b46f804d933725bbd61e1ed11c9a | A semicircle Γ is drawn on one side of a straight line l. Points C and D are on Γ. The tangents to Γ at C and D meet l at B and A respectively, with the center of the semicircle between them. Let E be the point of intersection of AC and BD, and F the point on l such that EF is perpendicular to l. If the radius of the s... | [
"10"
] | [
"A semicircle Γ is drawn on one side of a straight line l. Points C and D are on Γ. The tangents to Γ at C and D meet l at B and A respectively, with the center of the semicircle between them. Let E be the point of intersection of AC and BD, and F the point on l such that EF is perpendicular to l. If the radius of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-c53ec9cde323e5ad81e04742cb6dda5b9c694a09edc5d89a7c7ed9a98949f959 | A semicircular plate is centered at the origin with its straight side on the x-axis and radius $r$. Determine the centroid of the semicircular plate in the form $\frac{a}{b}r$. | [
"\\(\\frac{4}{3\\pi}r\\)"
] | [
"A semicircular plate is centered at the origin with its straight side on the x-axis and radius $r$. Determine the centroid of the semicircular plate in the form $\\frac{a}{b}r$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-c977fee1f23f37d523e533dd65081853c2e692e342d3a9f386c9b46831794c31 | A semielliptical arch for a tunnel under a river has a width of 100 feet and a height of 40 feet. Determine the height of the arch 5 feet from the edge of the tunnel. | [
"\\( 4 \\sqrt{19} \\)"
] | [
"A semielliptical arch for a tunnel under a river has a width of 100 feet and a height of 40 feet. Determine the height of the arch 5 feet from the edge of the tunnel."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 20
} | 40 | 0 | 0 |
omr-ws-e6db4cd1d44a8ee099af5e089bafa02ac24fb1ae201cb50b79c6f48ea2c41f07 | A sequence $ a_n $ has $ a_0 = 1 $, and $ a_{n + 1} = a_n + f_n $, where $ f_n $ is the largest prime factor of $ a_n $ (or 1). Thus, the first few terms of the sequence are $ 1, 2, 4, 6, 9, 12, \ldots $ Find a closed-form expression for $ a_n $. You may express your answer in terms of the $ n^{th} $ prime number $ p_n... | [
"p_k (n - p_k + 2)"
] | [
"A sequence $ a_n $ has $ a_0 = 1 $, and $ a_{n + 1} = a_n + f_n $, where $ f_n $ is the largest prime factor of $ a_n $ (or 1). Thus, the first few terms of the sequence are $ 1, 2, 4, 6, 9, 12, \\ldots $ Find a closed-form expression for $ a_n $. You may express your answer in terms of the $ n^{th} $ prime number... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6a5fa5dfae91c9d5c2f51c94203fa9245f4ed98ade2db9c1da58c9ac007f093d | A sequence $(S_{n})$ is defined by $S_{1}=1$ and $S_{n+1}=\frac{1}{3}S_{n}+\frac{2n-1}{(n+1)!.6^{n+1}}$ for $n=1,2,3,...$. Find $S_{n}$. | [
"no_valid_answer_found"
] | [
"A sequence $(S_{n})$ is defined by $S_{1}=1$ and $S_{n+1}=\\frac{1}{3}S_{n}+\\frac{2n-1}{(n+1)!.6^{n+1}}$ for $n=1,2,3,...$. Find $S_{n}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-594bab590f30395e51c47b522ed9f45ad69864fea15e4d33b7482543a8b99524 | A sequence $(a_1,b_1)$, $(a_2,b_2)$, $(a_3,b_3)$, $\ldots$ of points in the coordinate plane satisfies \[(a_{n + 1}, b_{n + 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n)\]for $n = 1,2,3,\ldots$. Suppose that $(a_{100},b_{100}) = (2,4)$. What is $a_1 + b_1$? Express your answer using exponential notation. | [
"\\frac{1}{2^{98}}"
] | [
"A sequence $(a_1,b_1)$, $(a_2,b_2)$, $(a_3,b_3)$, $\\ldots$ of points in the coordinate plane satisfies\n\\[(a_{n + 1}, b_{n + 1}) = (\\sqrt {3}a_n - b_n, \\sqrt {3}b_n + a_n)\\]for $n = 1,2,3,\\ldots$. Suppose that $(a_{100},b_{100}) = (2,4)$. What is $a_1 + b_1$? Express your answer using exponential notation.... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1bea0485a9a4ba24a137938da1571489eeac4e809635316b67ee956b885726ca | A sequence $(a_i)$ is defined by $a_1 = a_2 = 1$ and $a_n = \dfrac{n+1}{n-1}(a_1 + a_2 + \cdots + a_{n-1})$. Find $a_{2005}$. | [
"\\( 2006 \\cdot 2^{2002} \\)"
] | [
"A sequence $(a_i)$ is defined by $a_1 = a_2 = 1$ and $a_n = \\dfrac{n+1}{n-1}(a_1 + a_2 + \\cdots + a_{n-1})$. Find $a_{2005}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-4d6ccc318bed5d20156b8b9e5f3e041ce7969c27d209b63991d40deecfb0e3b3 | A sequence $(a_n)$ for $n \geq 1$ is defined by the following equations: - $a_1 = 1$, $a_2 = 2$, $a_3 = 1$, - $a_{2n-1}a_{2n} = a_2a_{2n-3} + (a_2a_{2n-3} + a_4a_{2n-5} + \ldots + a_{2n-2}a_1)$ for $n \geq 2$, - $na_{2n}a_{2n+1} = a_2a_{2n-2} + (a_2a_{2n-2} + a_4a_{2n-4} + \ldots + a_{2n-2}a_2)$ for $n \geq 2$. Find $a... | [
"\\(2^{1010}\\)"
] | [
"A sequence $(a_n)$ for $n \\geq 1$ is defined by the following equations:\n- $a_1 = 1$, $a_2 = 2$, $a_3 = 1$,\n- $a_{2n-1}a_{2n} = a_2a_{2n-3} + (a_2a_{2n-3} + a_4a_{2n-5} + \\ldots + a_{2n-2}a_1)$ for $n \\geq 2$,\n- $na_{2n}a_{2n+1} = a_2a_{2n-2} + (a_2a_{2n-2} + a_4a_{2n-4} + \\ldots + a_{2n-2}a_2)$ for $n \\ge... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 1
} | 48 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.