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-e614f84616a9e5c5163bba4d2e2b6693482e84c279e0960050bb4ee742614636 | A frog starts at the origin and alternately jumps over two lines intersecting at an angle $\theta = 72^{\circ}$. If the frog's position after $n$ jumps is represented by the complex number $z_n$, find the smallest positive integer $n$ such that $z_n = z_0$. | [
"5"
] | [
"A frog starts at the origin and alternately jumps over two lines intersecting at an angle $\\theta = 72^{\\circ}$. If the frog's position after $n$ jumps is represented by the complex number $z_n$, find the smallest positive integer $n$ such that $z_n = z_0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-f9fa7faeb0c81124171e4564d4e06ad50817b8ec4b953bfcbcb5be5f35f3465c | A frog starts at the origin and alternately jumps over two lines intersecting at an angle $\theta = 90^{\circ}$. After each jump, the frog's position is recorded. If the frog makes 2023 jumps, find the coordinates of the frog's final position. | [
"(1012, 1011)"
] | [
"A frog starts at the origin and alternately jumps over two lines intersecting at an angle $\\theta = 90^{\\circ}$. After each jump, the frog's position is recorded. If the frog makes 2023 jumps, find the coordinates of the frog's final position."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-17cb734a2fd3ba1828d444cda7f9f20df691aab0e4c7644cea72a35931c09161 | A frog starts at the origin of a Cartesian coordinate system. Every second, the frog jumps to one of the four adjacent points with integer coordinates. How many different points can the frog be at after 2015 seconds? | [
"4064256",
"\\(2016^2\\)"
] | [
"A frog starts at the origin of a Cartesian coordinate system. Every second, the frog jumps to one of the four adjacent points with integer coordinates. How many different points can the frog be at after 2015 seconds?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 4,
"tir": 19
} | 52 | 0 | 0 |
omr-ws-251ab055ec24006ea0a9ac86dfc8feca5b22f90664c33175a0d37f02926bcbed | A frog starts at the point $(0,0)$ at time $t=0$. Every unit of time, starting at $t=1$, the frog randomly makes a move up or right by one lattice point (i.e. from $(0,0)$, it can move to either $(1,0)$ or $(0,1)$). At $t=10$, the frog reaches $(5,5)$, and stops. The frog's expected average distance from the line $x=y$... | [
"97"
] | [
"A frog starts at the point $(0,0)$ at time $t=0$. Every unit of time, starting at $t=1$, the frog randomly makes a move up or right by one lattice point (i.e. from $(0,0)$, it can move to either $(1,0)$ or $(0,1)$). At $t=10$, the frog reaches $(5,5)$, and stops. The frog's expected average distance from the line ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b3cf725f6dd36dbad4d6294dd256a7710efafe5b2546f4b455d1549e13036e2 | A frog starts at the point $(0,2)$ and can jump one unit at a time to any adjacent lattice points. How many different paths of length $n$ can the frog take such that it ends up on the y-axis? | [
"\\dbinom{2n}{n}"
] | [
"A frog starts at the point $(0,2)$ and can jump one unit at a time to any adjacent lattice points. How many different paths of length $n$ can the frog take such that it ends up on the y-axis?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-894ceb54555735c8b236a0aa68cf9bd782a2563b73c89a0ac52f11ce65d5587c | A frog starts at vertex \( A \) of an octagon and jumps to one of the two adjacent vertices at each step, except when it reaches the opposite vertex \( E \), where it stops. Let \( a_n \) be the number of distinct paths of exactly \( n \) jumps ending at \( E \). Calculate the value of \( a_{10} \). | [
"96"
] | [
"A frog starts at vertex \\( A \\) of an octagon and jumps to one of the two adjacent vertices at each step, except when it reaches the opposite vertex \\( E \\), where it stops. Let \\( a_n \\) be the number of distinct paths of exactly \\( n \\) jumps ending at \\( E \\). Calculate the value of \\( a_{10} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-ea8e2f4eb92b83bb354bf92d88b5fa00c360c1ca23b32656ba243dd1acff2708 | A frog starts in the center of a $3 \times 3$ square table and jumps exactly one square in a random direction (up, down, left, right) upon hearing a call. At each subsequent call, the frog randomly chooses a direction from the four given directions but cannot jump back to where it just came from. The frog can jump outs... | [
"\\(\\frac{36}{11}\\)"
] | [
"A frog starts in the center of a $3 \\times 3$ square table and jumps exactly one square in a random direction (up, down, left, right) upon hearing a call. At each subsequent call, the frog randomly chooses a direction from the four given directions but cannot jump back to where it just came from. The frog can jum... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 6,
"tir": 2
} | 27 | 0 | 0 |
omr-ws-f64639c1e682aec652018a9e422e841a5ccea9bc6f31af8759ef88674cf7816d | A frog starts on lily pad 1 and can jump to the next lily pad (1 square forward) or skip one lily pad (2 squares forward) with equal probability of $\frac{1}{2}$. What is the probability that the frog lands on lily pad 7? Express the probability as $\frac{p}{q}$ where $p$ and $q$ are coprime, and find $p+q$. | [
"107"
] | [
"A frog starts on lily pad 1 and can jump to the next lily pad (1 square forward) or skip one lily pad (2 squares forward) with equal probability of $\\frac{1}{2}$. What is the probability that the frog lands on lily pad 7? Express the probability as $\\frac{p}{q}$ where $p$ and $q$ are coprime, and find $p+q$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-c44c8a786928d07e59be09d443f9d184b9bf55ad31fee4175044a229559c97ed | A frog starts on one of $n (>1)$ lotus leaves arranged in a circle and jumps according to the following rule: it always moves clockwise, first skipping one leaf, then two leaves, and so on. If the frog visits every leaf at least once, determine the remainder when the smallest possible value of $n$ that is even is divid... | [
"2"
] | [
"A frog starts on one of $n (>1)$ lotus leaves arranged in a circle and jumps according to the following rule: it always moves clockwise, first skipping one leaf, then two leaves, and so on. If the frog visits every leaf at least once, determine the remainder when the smallest possible value of $n$ that is even is ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-a72db36419bcf17c21385e73542a36e932e1543f2741cf0c79727747304b2096 | A frog starts on pad $1$ in a group of $11$ lily pads labeled $0$ to $10$. If the frog is on pad $n$, it will jump to pad $n-1$ with probability $\frac{n}{10}$ and to pad $n+1$ with probability $1-\frac{n}{10}$. What is the probability that the frog arrives on pad $10$ and leaves, never having arrived on pad $0$ before... | [
"\\dfrac{63}{146}"
] | [
"A frog starts on pad $1$ in a group of $11$ lily pads labeled $0$ to $10$. If the frog is on pad $n$, it will jump to pad $n-1$ with probability $\\frac{n}{10}$ and to pad $n+1$ with probability $1-\\frac{n}{10}$. What is the probability that the frog arrives on pad $10$ and leaves, never having arrived on pad $0$... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-195a5caabe918da0ce034125959c98f3684a2a7ea0d54f2a0640ed5dc3a80222 | A frog starts on pad 1 and can jump to either pad $k+1$ or pad $k+2$ from any pad $k$, each with probability $\tfrac{1}{2}$. What is the probability that the frog visits pad 7? Express the probability as $\tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, and find $p+q$. | [
"107"
] | [
"A frog starts on pad 1 and can jump to either pad $k+1$ or pad $k+2$ from any pad $k$, each with probability $\\tfrac{1}{2}$. What is the probability that the frog visits pad 7? Express the probability as $\\tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, and find $p+q$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-9ee6995b2afc8f3bc65d334bda6272b2200f1949a86cf856c4dfef947fd1469f | A frog starts on pad 1 and can jump to either pad \( k + 1 \) or pad \( k + 2 \) with equal probability of \(\frac{1}{2}\) from any pad \( k \). The probability that the frog visits pad 7 is \(\frac{m}{n}\), where \( \gcd(m, n) = 1 \). Evaluate \( n - m \). | [
"51"
] | [
"A frog starts on pad 1 and can jump to either pad \\( k + 1 \\) or pad \\( k + 2 \\) with equal probability of \\(\\frac{1}{2}\\) from any pad \\( k \\). The probability that the frog visits pad 7 is \\(\\frac{m}{n}\\), where \\( \\gcd(m, n) = 1 \\). Evaluate \\( n - m \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc3069eba0ddf2ccb1832d00cb5f0fdcbc1eb8ebdc691ae83089e86d567242b8 | A frog starts on the top left square of a $4 \times 4$ grid. Every minute, the frog travels to one of the adjacent unit squares at random. The frog cannot move outside of the $4 \times 4$ grid. The frog stops when it has passed through all $16$ of the unit squares. What is the probability that exactly $20$ minutes have... | [
"0"
] | [
"A frog starts on the top left square of a $4 \\times 4$ grid. Every minute, the frog travels to one of the adjacent unit squares at random. The frog cannot move outside of the $4 \\times 4$ grid. The frog stops when it has passed through all $16$ of the unit squares. What is the probability that exactly $20$ minut... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4da42556f761f3b1c1512c509d8fd0cd571e8ec9d59dd3aa38fe00cecfbebfdd | A frog takes jumps of length 1 meter in a random direction. After 3 jumps, what is the probability that the frog is within 1 meter of its starting point? | [
"\\dfrac{1}{4}"
] | [
"A frog takes jumps of length 1 meter in a random direction. After 3 jumps, what is the probability that the frog is within 1 meter of its starting point?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-c8fddb67653ae04dae64ffb6c843d1ca36e23a507736a7b9cb4fcfe2bcdbe2a5 | A frog takes two random hops, each to some random point one meter away from where it was when it started the hop. Its first hop thus carries it to some point on a circle about its initial position, and its second hop carries it to some point on a circle about where it first landed. What is the probability that the frog... | [
"\\dfrac{1}{3}"
] | [
"A frog takes two random hops, each to some random point one meter away from where it was when it started the hop. Its first hop thus carries it to some point on a circle about its initial position, and its second hop carries it to some point on a circle about where it first landed. What is the probability that the... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8dbdb90cccd52791ccb06227e1779178dc6820bc1f6889aed7d9a4f1869ebf13 | A frog trainer places one frog at each vertex of an equilateral triangle \(ABC\) with unit side length. The trainer can make one frog jump over another along the line joining the two, so that the total length of the jump is an even multiple of the distance between the two frogs just before the jump. Let \(M\) and \(N\)... | [
"\\(\\frac{\\lfloor\\frac{l+2}{2}\\rfloor\\lfloor\\frac{l+4}{2}\\rfloor(\\lfloor\\frac{l+1}{2}\\rfloor\\lfloor\\frac{l+3}{2}\\rfloor)^2}{8}\\)"
] | [
"A frog trainer places one frog at each vertex of an equilateral triangle \\(ABC\\) with unit side length. The trainer can make one frog jump over another along the line joining the two, so that the total length of the jump is an even multiple of the distance between the two frogs just before the jump. Let \\(M\\) ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-60e774f0d282c953f8939a2afe26dda54801e8d89203549eca50582075d17eba | A frustum has regular hexagonal bases with side lengths of $5$ and $6$ and a slant height of $7$. What is the height of the frustum? | [
"$\\sqrt{193}/2$"
] | [
"A frustum has regular hexagonal bases with side lengths of $5$ and $6$ and a slant height of $7$. What is the height of the frustum?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-c95a8c254807cb6316b182db7660a4278c9626fddbd840feafe6ea01fdc7cb2c | A frustum is formed from a square right pyramid, with the tip being cut off, forming 2 parallel square bases. The side lengths of the square bases are 9 and 16, and a sphere is inscribed in the frustum that is tangent to all faces. If each leg has length \( g \), and the radius of the sphere has length \( r \), what is... | [
"75"
] | [
"A frustum is formed from a square right pyramid, with the tip being cut off, forming 2 parallel square bases. The side lengths of the square bases are 9 and 16, and a sphere is inscribed in the frustum that is tangent to all faces. If each leg has length \\( g \\), and the radius of the sphere has length \\( r \\)... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-42f453d9ae40c691496cf5679f0e42474d0395ef887044fc34d451acd4993665 | A frustum is obtained by slicing a right-circular cone perpendicular to its axis and removing the small cone above the slice. The original cone has a base radius of 6 cm and a height of 8 cm. The curved surface area of the frustum is equal to the area of the two circular faces. What is the height of the frustum? | [
"4"
] | [
"A frustum is obtained by slicing a right-circular cone perpendicular to its axis and removing the small cone above the slice. The original cone has a base radius of 6 cm and a height of 8 cm. The curved surface area of the frustum is equal to the area of the two circular faces. What is the height of the frustum?"
... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3e6d29f9b817e01357572f28149a1fdae0f9ff03e7f65f2857024bd5b6497126 | A frustum of a cone is inscribed in a sphere of radius $R$. Then bases of the frustum cut off the sphere two segments with ares in the axial section equal to $\alpha $ and $\beta $, respectively. Find the curved surface of the frustum. | [
"$S_{curved}=2\\pi R^2\\sin\\frac{\\alpha +\\beta }{2}\\cos\\frac{\\alpha -\\beta }{4}$"
] | [
"A frustum of a cone is inscribed in a sphere of radius $R$. Then bases of the frustum cut off the sphere two segments with ares in the axial section equal to $\\alpha $ and $\\beta $, respectively. Find the curved surface of the frustum."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-289fc5057dd4281adb668671bb4c5655b4807a297e4d73e9a917b244ad942db3 | A frustum of a right circular cone has a height of 5 cm, an upper base radius of 1 cm, and a lower base radius of 4 cm. A plane cuts the frustum obliquely such that it cuts all the elements and is tangent to both the upper and lower bases. Find the ratio of the volume of the resulting upper part to the volume of the re... | [
"\\(\\frac{1}{8}\\)"
] | [
"A frustum of a right circular cone has a height of 5 cm, an upper base radius of 1 cm, and a lower base radius of 4 cm. A plane cuts the frustum obliquely such that it cuts all the elements and is tangent to both the upper and lower bases. Find the ratio of the volume of the resulting upper part to the volume of t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-a44071bcf53908f55f36e969888ae8546b0f2db516950731741c0eeab8bb747a | A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If a particular frustum has a lower base radius of 6 inches, an upper base radius of 3 inches, and a height of 4 inches, what is its lateral surface area? (The lateral surface area of a cone or frustum is the curved su... | [
"45\\pi"
] | [
"A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If a particular frustum has a lower base radius of 6 inches, an upper base radius of 3 inches, and a height of 4 inches, what is its lateral surface area? (The lateral surface area of a cone or frustum is the cur... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ffe821115cdd2bcd72c6440544b5ead13e3c1a54e99663c4895824359f8d2eaf | A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If a particular frustum has an altitude of $24$ centimeters, the area of its lower base is $225\pi$ sq cm and the area of its upper base is $25\pi$ sq cm, what is the altitude of the small cone that was cut off? [asy]s... | [
"12"
] | [
"A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If a particular frustum has an altitude of $24$ centimeters, the area of its lower base is $225\\pi$ sq cm and the area of its upper base is $25\\pi$ sq cm, what is the altitude of the small cone that was cut off?... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2d9e4d0b7fdd8e1c5741bd5fe4ebbc6f78b5334b07379990dc71268c29504ffe | A frustum of a solid right circular cone has a bottom base whose area is twice the area of its upper base. A hemisphere whose flat portion is the upper base of the frustum is carved into the frustum and has its nearest point to the bottom base located 1 centimeter from the center of the bottom base. The frustum has the... | [
"29.63"
] | [
"A frustum of a solid right circular cone has a bottom base whose area is twice the area of its upper base. A hemisphere whose flat portion is the upper base of the frustum is carved into the frustum and has its nearest point to the bottom base located 1 centimeter from the center of the bottom base. The frustum ha... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-235120dbcd81f90bfaf9accb34e4cfc9f66addc9a1b201810fe63c9231905c41 | A fudgeflake is a planar fractal figure with a $120^\circ$ rotational symmetry such that three identical fudgeflakes in the same orientation fit together without gaps to form a larger fudgeflake with its orientation $30^\circ$ clockwise of the smaller fudgeflakes' orientation. If the distance between the centers of the... | [
"\\dfrac{\\sqrt{3}}{12}"
] | [
"A fudgeflake is a planar fractal figure with a $120^\\circ$ rotational symmetry such that three identical fudgeflakes in the same orientation fit together without gaps to form a larger fudgeflake with its orientation $30^\\circ$ clockwise of the smaller fudgeflakes' orientation. If the distance between the centers... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-c7af987d72b4a7622ee8742a62357289bc8a749efab2b0fe660e8fc8ccfc2183 | A fuel tank for a rocket engine can be mapped out in the 3D coordinate space through the regions below the plane equation \(z = 8 - x^2 - y^2\), above the plane equation \(z = -\sqrt{4x^2 + 4y^2}\), and in the interior of the plane equation \(x^2 + y^2 = 4\). What is the volume of this tank? Express your answer in the ... | [
"\\(\\frac{104\\pi}{3}\\)"
] | [
"A fuel tank for a rocket engine can be mapped out in the 3D coordinate space through the regions below the plane equation \\(z = 8 - x^2 - y^2\\), above the plane equation \\(z = -\\sqrt{4x^2 + 4y^2}\\), and in the interior of the plane equation \\(x^2 + y^2 = 4\\). What is the volume of this tank? Express your an... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-16704e2a18c71a8a2ec8472839cf7f8256d498b8cd8d756801f47a3bbd84cfc4 | A fuel tank is in the shape of a right frustum (truncated cone). Its parallel circular bases have areas of \( A_1 \) and \( A_2 \) square feet, respectively. When the fuel tank is half filled (up to the axis of symmetry), the area of the surface of the fuel (cross section perpendicular to the bases, including the axis ... | [
"648"
] | [
"A fuel tank is in the shape of a right frustum (truncated cone). Its parallel circular bases have areas of \\( A_1 \\) and \\( A_2 \\) square feet, respectively. When the fuel tank is half filled (up to the axis of symmetry), the area of the surface of the fuel (cross section perpendicular to the bases, including ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-e8a2a8ace4200bc8c8059cf34a6590cd540cfc7ff82a71f9b7e572f04e78f19a | A fuel tank receives a continuous, steady flow of 2000 liters per hour. The tank experiences a steady rate of fuel usage within each of the 6 consecutive 4-hour periods every day. The usage during these periods is, respectively, 6000, 13500, 7300, 10000, 8000, and 3200 liters. What is the capacity, in liters, of the sm... | [
"7000"
] | [
"A fuel tank receives a continuous, steady flow of 2000 liters per hour. The tank experiences a steady rate of fuel usage within each of the 6 consecutive 4-hour periods every day. The usage during these periods is, respectively, 6000, 13500, 7300, 10000, 8000, and 3200 liters. What is the capacity, in liters, of t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 11
} | 32 | 0 | 0 |
omr-ws-1767d3cfe1b8ea5bff4b9a28fbabd08e596d1d1441b2cb60d7514f488c4180e6 | A fulcrum is placed under the center of a board. For two objects to balance on the board, the distance (in feet) of each object from the center of the board must vary inversely with its weight (in pounds). If object A weighs 7.5 pounds and object B weighs 12.5 pounds, and the distance between the two objects is 5 feet,... | [
"\\( A \\) is \\( 3.125 \\) ft away from the fulcrum, and \\( B \\) is \\( 1.875 \\) ft away."
] | [
"A fulcrum is placed under the center of a board. For two objects to balance on the board, the distance (in feet) of each object from the center of the board must vary inversely with its weight (in pounds). If object A weighs 7.5 pounds and object B weighs 12.5 pounds, and the distance between the two objects is 5 ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 29
} | 40 | 0 | 0 |
omr-ws-8b647f9abb221a9ec9c966f0c4871dd9642cfb3eb2fcab1ab6ba10904e8ceb67 | A function $S(m, n)$ satisfies the initial conditions $S(1, n) = n$, $S(m, 1) = 1$, and the recurrence $S(m, n) = S(m - 1, n)S(m, n - 1)$ for $m \geq 2, n \geq 2$. Find the largest integer $k$ such that $2^k$ divides $S(7, 7)$. | [
"803"
] | [
"A function $S(m, n)$ satisfies the initial conditions $S(1, n) = n$, $S(m, 1) = 1$, and the recurrence $S(m, n) = S(m - 1, n)S(m, n - 1)$ for $m \\geq 2, n \\geq 2$. Find the largest integer $k$ such that $2^k$ divides $S(7, 7)$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-34533f2ad10bf9bd8db6f7dee5200e7dc7ce0731a5d4d99cd2a648953612a0a8 | A function $f \colon \mathbb N \to \mathbb N$ is called [i]Canadian[/i] if it satisfies \[ \gcd\left(f(f(x)), f(x+y)\right)=\gcd(x, y) \] for all pairs of positive integers $x$ and $y$. Find all positive integers $m$ such that $f(m)=m$ for all Canadian functions $f$. | [
"1"
] | [
"A function $f \\colon \\mathbb N \\to \\mathbb N$ is called [i]Canadian[/i] if it satisfies \\[ \\gcd\\left(f(f(x)), f(x+y)\\right)=\\gcd(x, y) \\]\nfor all pairs of positive integers $x$ and $y$. Find all positive integers $m$ such that $f(m)=m$ for all Canadian functions $f$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-f1a1eb68e8fa2e2f403c8adaa0f8bec3b2da0a963aed28433642e23731b2906c | A function $f$ from the integers to the integers is defined as follows: \[f(n) = \left\{ \begin{array}{cl} n + 3 & \text{if $n$ is odd}, \\ n/2 & \text{if $n$ is even}. \end{array} \right.\]Suppose $k$ is odd and $f(f(f(k))) = 27.$ Find $k.$ | [
"105"
] | [
"A function $f$ from the integers to the integers is defined as follows:\n\\[f(n) = \\left\\{\n\\begin{array}{cl}\nn + 3 & \\text{if $n$ is odd}, \\\\\nn/2 & \\text{if $n$ is even}.\n\\end{array}\n\\right.\\]Suppose $k$ is odd and $f(f(f(k))) = 27.$ Find $k.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-96c9b9f5d34d8de1225de17ec1144671439c29074a5d7d16ca77bd65065b665f | A function $f$ from the positive integers to the positive integers is called Canadian if it satisfies \[ \gcd\left(f(f(x)), f(x+y)\right)=\gcd(x, y) \] for all pairs of positive integers $x$ and $y$. Find all positive integers $m$ such that $f(m)=m$ for all Canadian functions $f$. | [
"1"
] | [
"A function $f$ from the positive integers to the positive integers is called Canadian if it satisfies\n\\[ \\gcd\\left(f(f(x)), f(x+y)\\right)=\\gcd(x, y) \\]\nfor all pairs of positive integers $x$ and $y$. Find all positive integers $m$ such that $f(m)=m$ for all Canadian functions $f$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-6c2a7ba9387e20951e88b39b61af8a7c5fd7f756aa836ce3e53317568b0b0175 | A function $f$ has domain $[0,2]$ and range $[0,1]$. (The notation $[a,b]$ denotes $\{x:a \le x \le b \}$.) Let \[g(x) = 1 - f(x + 1).\]Then the domain of $g(x)$ is $[a,b],$ and the range of $g(x)$ is $[c,d].$ Enter the ordered quadruple $(a,b,c,d).$ | [
"(-1,1,0,1)"
] | [
"A function $f$ has domain $[0,2]$ and range $[0,1]$. (The notation $[a,b]$ denotes $\\{x:a \\le x \\le b \\}$.) Let\n\\[g(x) = 1 - f(x + 1).\\]Then the domain of $g(x)$ is $[a,b],$ and the range of $g(x)$ is $[c,d].$ Enter the ordered quadruple $(a,b,c,d).$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9393a34af10cb0040e022d26139c5834ecaac0c7d54ee1de353f8612c7171d37 | A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$? | [
"7"
] | [
"A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ebd17a226971464b00c8690c37cc6fea99b284d6f4b147381a5fcce1ab97c4a4 | A function $f$ is defined as in the following table: \[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline f(x) & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 1 \\ \hline \end{array} \] Based on the definition of the function $f$, a sequence is defined by $U_1 = 2$ and $U_{n+1} =... | [
"U_n = (n \\bmod 10) + 1"
] | [
"A function $f$ is defined as in the following table:\n\\[\n\\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\nx & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline\nf(x) & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 1 \\\\\n\\hline\n\\end{array}\n\\]\nBased on the definition of the function $f$, a sequence is defined b... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2db9c528c67243538a231a8627b08d4b762bd3dcef950783a325589bfbec9c1d | A function $f$ is defined by $f(x) = 1 + 2x + x^2 + 2x^3 + x^4 + \cdots$. Find the interval of convergence of the series and find an explicit formula for $f(x)$. | [
"\\(\\frac{1+2x}{1-x^2}\\), interval of convergence = (-1,1)"
] | [
"A function $f$ is defined by $f(x) = 1 + 2x + x^2 + 2x^3 + x^4 + \\cdots$. Find the interval of convergence of the series and find an explicit formula for $f(x)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 27
} | 34 | 0 | 0 |
omr-ws-443e46f0c0dc1676680443e16fe2c3ac8784bac76fd043823ec58eddc7ac4492 | A function $f$ is defined by $f(x)=\int_{0}^{x}\frac{dt}{1+t^2}$. The normal line to $y=f(x)$ at $x=1$ has $x$-intercept $X$ and $y$-intercept $Y$. Calculate the value of $Y-2X$. | [
"0"
] | [
"A function $f$ is defined by $f(x)=\\int_{0}^{x}\\frac{dt}{1+t^2}$. The normal line to $y=f(x)$ at $x=1$ has $x$-intercept $X$ and $y$-intercept $Y$. Calculate the value of $Y-2X$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b57aecf6d84dbf67c7d4bfa2ac02bb0f9501f81f9254dd4ae550bc486f0feb96 | A function $f$ is defined by $f(z) = (4 + i) z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $| \alpha | + |\gamma |$? | [
"\\sqrt{2}"
] | [
"A function $f$ is defined by $f(z) = (4 + i) z^2 + \\alpha z + \\gamma$ for all complex numbers $z$, where $\\alpha$ and $\\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $| \\alpha | + |\\gamma |$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6513aa67b10ead1db742bd25d776487d7e52dae69d5f2f78655c7a2d7a59b6d8 | A function $f$ is defined by $f(z) = (4 + i) z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers and $i^2 = -1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $|\alpha| + |\gamma|$? | [
"\\(\\sqrt{2}\\)"
] | [
"A function $f$ is defined by $f(z) = (4 + i) z^2 + \\alpha z + \\gamma$ for all complex numbers $z$, where $\\alpha$ and $\\gamma$ are complex numbers and $i^2 = -1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $|\\alpha| + |\\gamma|$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-9a1a1d2263867219172d032bfe36bebeff252306450f28f2c6c055535da2eed9 | A function $f$ is defined by $f(z) = i\overline{z}$, where $i^2 = -1$ and $\overline{z}$ is the complex conjugate of $z$. How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$? | [
"2"
] | [
"A function $f$ is defined by $f(z) = i\\overline{z}$, where $i^2 = -1$ and $\\overline{z}$ is the complex conjugate of $z$. How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-73f863aa29e85d8f674f99ca286b4ec0356efe71aa969221485dbf7d9382383b | A function $f$ is defined for all real numbers and has the property that for any $a$, $f(x+a) - f(x) = 3x^2a - 9a^2$. Find $f'(-4)$. | [
"48"
] | [
"A function $f$ is defined for all real numbers and has the property that for any $a$, $f(x+a) - f(x) = 3x^2a - 9a^2$. Find $f'(-4)$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-100481179aa9ee44876241e2186f339387b84f4b734fc4e2eb56193409dceaac | A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$. If $x=0$ is a root for $f(x)=0$, what is the least number of roots $f(x)=0$ must have in the interval $-1000 \leq x \leq 1000$? | [
"401"
] | [
"A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$. If $x=0$ is a root for $f(x)=0$, what is the least number of roots $f(x)=0$ must have in the interval $-1000 \\leq x \\leq 1000$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-28b79a1cd2b4d4037bb49a37a041115c5bbf3e03d628e331e85efc6cdbe401c8 | A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$. If $x=0$ is a root for $f(x)=0$, what is the least number of roots $f(x)=0$ must have in the interval $-1000\leq x \leq 1000$? | [
"401"
] | [
"A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$. If $x=0$ is a root for $f(x)=0$, what is the least number of roots $f(x)=0$ must have in the interval $-1000\\leq x \\leq 1000$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0663db8326c884571ddf2d6fbf807d96f8bdfebc69640747c237cc6df8af2e93 | A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x.$ If $f(0) = 0,$ what is the least number of roots $f(x)=0$ must have in the interval $-1000\leq x \leq 1000$? | [
"401"
] | [
"A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x.$ If $f(0) = 0,$ what is the least number of roots $f(x)=0$ must have in the interval $-1000\\leq x \\leq 1000$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-312089e3fa5f4784967de73d75bff435b9b4fa6368fd6c0472e56cbeef967245 | A function $f$ is defined on $[0,1]$ by $f(x) = x$ if $x \in \mathbb{Q}$ and $f(x) = x^2$ if $x \in \mathbb{R} \setminus \mathbb{Q}$. Find the lower integral $\underline{\int}_0^1 f(x) \, dx$ and the upper integral $\overline{\int}_0^1 f(x) \, dx$. | [
"\\(\\underline{\\int}_0^1 f(x) \\, dx = \\frac{1}{3}\\) and \\(\\overline{\\int}_0^1 f(x) \\, dx = \\frac{1}{2}\\)"
] | [
"A function $f$ is defined on $[0,1]$ by $f(x) = x$ if $x \\in \\mathbb{Q}$ and $f(x) = x^2$ if $x \\in \\mathbb{R} \\setminus \\mathbb{Q}$. Find the lower integral $\\underline{\\int}_0^1 f(x) \\, dx$ and the upper integral $\\overline{\\int}_0^1 f(x) \\, dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 25
} | 26 | 0 | 0 |
omr-ws-c1505ea1fa7677ad08b17ecff527be56c4b0f0452793262e88f4a2e7833d876f | A function $f$ is defined on the complex numbers by $f(z) = (a + bi)z$, where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given $|a + bi| = 8$, find the value of $b^2$. | [
"63.75"
] | [
"A function $f$ is defined on the complex numbers by $f(z) = (a + bi)z$, where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given $|a + bi| = 8$, find the value of $b^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-59000e368411ee52c187ebcd73b7fccf39e7a89b993db1c1c1ca8a5505ed5c0c | A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z$, where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $\left | a+bi \right |=8$, find the value of $b^2$. | [
"\\(\\frac{255}{4}\\)"
] | [
"A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z$, where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $\\left | a+bi \\right |=8$, find the value of $b^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-a7ec120c4cc90ea3f8b1bf12447666f84e772f8bb1549e45bf986cc4cbc4592a | A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that for each complex number $z$, $f(z)$ is equidistant from both $z$ and the origin. Given that $|a+bi|=8$, find $b^2.$ | [
"\\frac{255}{4}"
] | [
"A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that for each complex number $z$, $f(z)$ is equidistant from both $z$ and the origin. Given that $|a+bi|=8$, find $b^2.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d82d4abeee64eb83f8f9009cdeac688ad800ddfcfded2c7fd613d29aaae6d959 | A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=\frac{m}{n},$ where $m$ and $n$ are relatively p... | [
"259"
] | [
"A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=\\frac{m}{n},$ where $m$ and $n$ are relati... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9347e913880b3025b8ddde0cc3194ef3145454a505c5d76cdfbdd009fef0cfcf | A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a_{}$ and $b_{}$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m_{}$ and $n_{}$ are relative... | [
"259"
] | [
"A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a_{}$ and $b_{}$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m_{}$ and $n_{}$ ar... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b239b68541e3d047a5b3ccccf7dace0d818fcf9627d7ce4fcdabae5ffea634d0 | A function $f$ is defined on the natural numbers $\mathbb{N}$ with the following properties: - $f(1)=1$; - $f(2n)=f(n)$ and $f(2n+1)=f(2n)+1$ for all $n\in \mathbb{N}$. Calculate the maximum value $m$ of the set $\{f(n):n\in \mathbb{N}, 1\le n\le 1989\}$, and determine the number of natural numbers $n$, with $1\le n\le... | [
"5"
] | [
"A function $f$ is defined on the natural numbers $\\mathbb{N}$ with the following properties:\n- $f(1)=1$;\n- $f(2n)=f(n)$ and $f(2n+1)=f(2n)+1$ for all $n\\in \\mathbb{N}$.\n\nCalculate the maximum value $m$ of the set $\\{f(n):n\\in \\mathbb{N}, 1\\le n\\le 1989\\}$, and determine the number of natural numbers $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff4b14b644742574986ba56ed6e96dc2a72a014475c7d88005d648d403a981a9 | A function $f$ is defined recursively by $f(1)=f(2)=1$ and \[f(n)=f(n-1)-f(n-2)+n\]for all integers $n \geq 3$. What is $f(2018)$? | [
"2017"
] | [
"A function $f$ is defined recursively by $f(1)=f(2)=1$ and \\[f(n)=f(n-1)-f(n-2)+n\\]for all integers $n \\geq 3$. What is $f(2018)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c42c08c5d05945d1bf8b267005a4c1b07810fc296088e55a64c89a58db11605e | A function $f(a, b)$ satisfies the properties $f(a, f(b, c)) = f(a, b) \cdot c$ and $f(a, a) = 1$ for all nonzero real numbers $a, b,$ and $c$. Given the equation $f(2016, f(84, x)) = 1008$, find the value of $x$. | [
"42"
] | [
"A function $f(a, b)$ satisfies the properties $f(a, f(b, c)) = f(a, b) \\cdot c$ and $f(a, a) = 1$ for all nonzero real numbers $a, b,$ and $c$. Given the equation $f(2016, f(84, x)) = 1008$, find the value of $x$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-8af07752e1ca22a9898c81015e4abc9956cae52e7013c6c22f3be10921109f96 | A function $f(x)$ has 2 steps: i) Multiplies by 1337. ii) Takes the 42nd root. After a large number of repeating this function on a number $n$, the number approaches $\sqrt[a]{b}$. Find $a+b$. | [
"1378"
] | [
"A function $f(x)$ has 2 steps:\ni) Multiplies by 1337.\nii) Takes the 42nd root.\nAfter a large number of repeating this function on a number $n$, the number approaches $\\sqrt[a]{b}$. Find $a+b$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7411612ee46e8517771a0257474ee972bc1a730e0458a810212e1bc0b3820e31 | A function $f(x)$ is defined as convex if for any two points $x_1$ and $x_2$ in its domain and any $t$ in the interval $[0,1]$, the following inequality holds: $f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2)$. Using this concept, find the minimum value of the expression $3x^2 + 4y^2 + 5z^2$ subject to the constraint $x ... | [
"\\dfrac{60}{47}"
] | [
"A function $f(x)$ is defined as convex if for any two points $x_1$ and $x_2$ in its domain and any $t$ in the interval $[0,1]$, the following inequality holds: $f(tx_1 + (1-t)x_2) \\leq tf(x_1) + (1-t)f(x_2)$. Using this concept, find the minimum value of the expression $3x^2 + 4y^2 + 5z^2$ subject to the constrai... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-166cff6f938a8e48c4fdf2825d967364931c794c91b858cbcea15cbd4f7b58f9 | A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have \[2f\left(x\right) + f\left(\frac{1}{x}\right) = 5x + 4\] Let $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$. | [
"601"
] | [
"A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-97982f445109ebeb9e9d79571ec542aaf879bdbda7f94996f1fb74d3239ec090 | A function $f(x)$ is made such that the value of $f(x)$ is $x$ plus the number formed by reversing the digits of $x$. For example, $f(15) = 15 + 51 = 66$ and $f(20) = 20 + 02 = 22$. Let $$S = f(1) + f(2) + f(3) + \ldots + f(98) + f(99).$$ What is the sum of the digits of $S$? | [
"27"
] | [
"A function $f(x)$ is made such that the value of $f(x)$ is $x$ plus the number formed by reversing the digits of $x$. For example, $f(15) = 15 + 51 = 66$ and $f(20) = 20 + 02 = 22$. Let $$S = f(1) + f(2) + f(3) + \\ldots + f(98) + f(99).$$ What is the sum of the digits of $S$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c42b117553e89bc7b17b625f75959ab3f2f33b9f70c5bcaaea4bc403b0dc2191 | A function $f(x)=ax^2+bx+c$ satisfies $|f(x)|\leq 1$ for $|x|\leq 1$. Determine all functions $f(x)$ such that $|f'(1)|=4$. | [
"\\( f(x) = -2x^2 + 1 \\) and \\( f(x) = 2x^2 - 1 \\)"
] | [
"A function $f(x)=ax^2+bx+c$ satisfies $|f(x)|\\leq 1$ for $|x|\\leq 1$. Determine all functions $f(x)$ such that $|f'(1)|=4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-7e3d67b174b899fb34bcff2d6e508f3381a8acde3fa80a1c2b6cae14219ab1df | A function $f(x)=ax^2+bx+c$ satisfies $|f(x)|\leq 1$ for $|x|\leq 1$. Find the maximum possible value of $|f'(1)|$. | [
"4"
] | [
"A function $f(x)=ax^2+bx+c$ satisfies $|f(x)|\\leq 1$ for $|x|\\leq 1$. Find the maximum possible value of $|f'(1)|$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a85cc1d5d5e83b011b61b75cec6045ff9d1f97ec73a331ba7a91674c8a2354e7 | A function $f: \mathbb{R}^* \rightarrow \mathbb{R}$ satisfies the equation $f(x+y-1) + 2f\left(\frac{y}{x}\right) = 2(x + f(y))$ for nonzero constants $x$ and $y$. Compute $f(2024)$. | [
"2"
] | [
"A function $f: \\mathbb{R}^* \\rightarrow \\mathbb{R}$ satisfies the equation $f(x+y-1) + 2f\\left(\\frac{y}{x}\\right) = 2(x + f(y))$ for nonzero constants $x$ and $y$. Compute $f(2024)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-ecf4f8f8631fe4747ff2d7d1deadf54953684b6f129b712b495958b35ed1f6e2 | A function $f:\mathbb{N} \rightarrow \mathbb{N}$ is called $\mathbb{N}$-good if $f(a^2+b)=f(b^2+a)$ for all $a, b \in \mathbb{N}$. What is the largest possible number of distinct values that can occur among $f(1), \ldots, f(2023)$, where $f$ is a $\mathbb{N}$-good function? | [
"1077"
] | [
"A function $f:\\mathbb{N} \\rightarrow \\mathbb{N}$ is called $\\mathbb{N}$-good if $f(a^2+b)=f(b^2+a)$ for all $a, b \\in \\mathbb{N}$. What is the largest possible number of distinct values that can occur among $f(1), \\ldots, f(2023)$, where $f$ is a $\\mathbb{N}$-good function?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 5,
"tir": 17
} | 25 | 0 | 0 |
omr-ws-23b315357c3f01a97b86e017ec9e41816b6ba4c9e7e78fe0d7ae7bf59cdc8b58 | A function $f:\mathbb{Z} \rightarrow \mathbb{Z}$ is called $\mathbb{Z}$-good if $f(a^2+b)=f(b^2+a)$ for all $a, b \in \mathbb{Z}$. What is the largest possible number of distinct values that can occur among $f(1), \ldots, f(2023)$, where $f$ is a $\mathbb{Z}$-good function? | [
"2"
] | [
"A function $f:\\mathbb{Z} \\rightarrow \\mathbb{Z}$ is called $\\mathbb{Z}$-good if $f(a^2+b)=f(b^2+a)$ for all $a, b \\in \\mathbb{Z}$. What is the largest possible number of distinct values that can occur among $f(1), \\ldots, f(2023)$, where $f$ is a $\\mathbb{Z}$-good function?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-0b1fd091aa1dc80b64845271645792785b025d9d892c5e043db6a3985646c6a4 | A function $f:\mathbb{Z} \to \mathbb{Z}$ satisfies \begin{align*} f(x+4)-f(x) &= 8x+20, \\ f(x^2-1) &= (f(x)-x)^2+x^2-2 \end{align*}for all integers $x.$ Enter the ordered pair $(f(0),f(1)).$ | [
"(-1,1)"
] | [
"A function $f:\\mathbb{Z} \\to \\mathbb{Z}$ satisfies\n\\begin{align*}\nf(x+4)-f(x) &= 8x+20, \\\\\nf(x^2-1) &= (f(x)-x)^2+x^2-2\n\\end{align*}for all integers $x.$ Enter the ordered pair $(f(0),f(1)).$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8d508da9bd944d145116de8f20c0d4b2d737ccebb97a1afa6ba0a0e219806acd | A function $g$ defined for all positive integers $n$ satisfies: - $g(1) = 1$ - For all $n \ge 1$, either $g(n+1) = g(n) + 1$ or $g(n+1) = g(n) - 1$ - For all $n \ge 1$, $g(3n) = g(n)$ Find the smallest positive integer $k$ such that $g(k) = 2001$. | [
"2 \\times 3^{1000} - 1"
] | [
"A function $g$ defined for all positive integers $n$ satisfies:\n- $g(1) = 1$\n- For all $n \\ge 1$, either $g(n+1) = g(n) + 1$ or $g(n+1) = g(n) - 1$\n- For all $n \\ge 1$, $g(3n) = g(n)$\nFind the smallest positive integer $k$ such that $g(k) = 2001$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-9c3bc7d7b32731059840fcac4229785bc68b03e0f0fd3be12abe7209c9422a79 | A function $g(x)$ has derivatives of all orders for all real numbers $x$. Assume $g(2) = 5$, $g'(2) = -6$, $g''(2) = 10$, $g'''(2) = -3$. Write the third-degree Taylor polynomial for $g(x)$ about $x=2$ and use it to approximate $g(1.8)$. If the fourth derivative satisfies the inequality $|g^{(4)}(x)| \le 5$ for all $x$... | [
"6.404"
] | [
"A function $g(x)$ has derivatives of all orders for all real numbers $x$. Assume $g(2) = 5$, $g'(2) = -6$, $g''(2) = 10$, $g'''(2) = -3$. Write the third-degree Taylor polynomial for $g(x)$ about $x=2$ and use it to approximate $g(1.8)$. If the fourth derivative satisfies the inequality $|g^{(4)}(x)| \\le 5$ for a... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-160c82d32d0efc6354a2b7d8029ce5a2326b07611aeba4889297902c2a5a95a6 | A function $g:\mathbb{Z}\to\mathbb{Z}$ is called adjective if $g(m)+g(n)>\max{(m^2,n^2)}$ for any pair of integers $m$ and $n$. Let $f$ be an adjective function such that the value of $f(1)+f(2)+f(3)+\cdots+f(30)$ is minimized. Find the smallest possible value of $f(25)$. | [
"313"
] | [
"A function $g:\\mathbb{Z}\\to\\mathbb{Z}$ is called adjective if $g(m)+g(n)>\\max{(m^2,n^2)}$ for any pair of integers $m$ and $n$. Let $f$ be an adjective function such that the value of $f(1)+f(2)+f(3)+\\cdots+f(30)$ is minimized. Find the smallest possible value of $f(25)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-3193da63d4411953cf9b40152cbc5e815b7c6fa1fb0a785c6c3053a16233b7bc | A function \( A(x, y) \) is defined by the following rules: - \( A(0, n) = n + 1 \) - \( A(m, 0) = A(m-1, 1) \) - \( A(m, n) = A(m-1, A(m, n-1)) \) Find \( A(2, 3) \). | [
"9"
] | [
"A function \\( A(x, y) \\) is defined by the following rules:\n- \\( A(0, n) = n + 1 \\)\n- \\( A(m, 0) = A(m-1, 1) \\)\n- \\( A(m, n) = A(m-1, A(m, n-1)) \\)\n\nFind \\( A(2, 3) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-b679ce309e04b1e9f66d4279b9ab5bcfd30ca87cb752d8ea68ca3200bd89e9aa | A function \( F(n) \) satisfies the recurrence \( F(n) \leq 4F\left(\left\lfloor \frac{n}{2} \right\rfloor\right) + n \) for all \( n \in \mathbb{N} \). Give an upper bound for \( F(n) \). | [
"O(n^2)"
] | [
"A function \\( F(n) \\) satisfies the recurrence \\( F(n) \\leq 4F\\left(\\left\\lfloor \\frac{n}{2} \\right\\rfloor\\right) + n \\) for all \\( n \\in \\mathbb{N} \\). Give an upper bound for \\( F(n) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c505a385979a0c0c034d4d2484fb05b6363a4cc0ef266883ff533299490d3d54 | A function \( P_n(x) \) is defined inductively by \[ P_0(x) = 0, \] \[ P_{n+1}(x) = \frac{1}{2}x^2 + P_n(x) - \frac{1}{2}P_n(x)^2 \quad (n = 0, 1, 2, \cdots). \] Determine the form or properties of \( P_n(x) \). | [
"P_n(x) \\text{ is a polynomial of degree } 2^n"
] | [
"A function \\( P_n(x) \\) is defined inductively by \n\\[ P_0(x) = 0, \\]\n\\[ P_{n+1}(x) = \\frac{1}{2}x^2 + P_n(x) - \\frac{1}{2}P_n(x)^2 \\quad (n = 0, 1, 2, \\cdots). \\]\nDetermine the form or properties of \\( P_n(x) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-ccd332f8cb640c25f98f48dff4d9dff638d5f2016fba4e3e34b31f12c0e626e0 | A function \( f : \mathbb{N} \rightarrow \mathbb{N} \) is defined such that \( f(1) = 1 \) and for all \( n \in \mathbb{N} \), \( f(2n) = f(n) \) and \( f(2n+1) = f(2n) + 1 \). Find a concise algorithm for \( f(n) \) in one line. | [
"\\( f(n) \\) equals the sum of the digits of \\( n \\) when written in binary."
] | [
"A function \\( f : \\mathbb{N} \\rightarrow \\mathbb{N} \\) is defined such that \\( f(1) = 1 \\) and for all \\( n \\in \\mathbb{N} \\), \\( f(2n) = f(n) \\) and \\( f(2n+1) = f(2n) + 1 \\). Find a concise algorithm for \\( f(n) \\) in one line."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-4d7f6683fa7b0cc2e3074941c70921a39bca9c566b41f38805e4346509f4daad | A function \( f : \mathbb{N} \to \mathbb{N} \) satisfies the condition \(\sum_{k = 1}^n \left(\frac{1}{f(k)f(k + 1)}\right) = \frac{f(f(n))}{f(n +1)}\) for all \( n \in \mathbb{N} \). Find \( f(2017) \). | [
"2017"
] | [
"A function \\( f : \\mathbb{N} \\to \\mathbb{N} \\) satisfies the condition \\(\\sum_{k = 1}^n \\left(\\frac{1}{f(k)f(k + 1)}\\right) = \\frac{f(f(n))}{f(n +1)}\\) for all \\( n \\in \\mathbb{N} \\). Find \\( f(2017) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-298f4a25df7d5fbf25ec6169b414fa3e2a184b4910628e545b35f2756e2c51f7 | A function \( f : \mathbb{N}^+ \rightarrow \mathbb{N}^+ \) satisfies \( f^n(n) = n \) for all positive integers \( n \). What are all possible values of \( f(1000) \)? | [
"All positive even numbers and all odd multiples of 5."
] | [
"A function \\( f : \\mathbb{N}^+ \\rightarrow \\mathbb{N}^+ \\) satisfies \\( f^n(n) = n \\) for all positive integers \\( n \\). What are all possible values of \\( f(1000) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 15
} | 32 | 0 | 0 |
omr-ws-e81845d82daebb4c262999cbb477b09996fbcf94ba42f42f4358bdfbe6a025fe | A function \( f : \mathbb{R} \rightarrow \mathbb{R} \) satisfies the property \( f(x^2) - f^2(x) \geq \frac{1}{4} \) for all \( x \). Verify if the function is one-to-one. | [
"The function is not one-to-one."
] | [
"A function \\( f : \\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfies the property \\( f(x^2) - f^2(x) \\geq \\frac{1}{4} \\) for all \\( x \\). Verify if the function is one-to-one."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-c5ebe0430c39491a08edbf98a881d115a41a948116330ed53463c7fe97e74998 | A function \( f : \mathbb{R} \to \mathbb{R} \) satisfies the equation \( f(x) - f(y) = (x - y) f'\left(\frac{x + y}{2}\right) \) for all \( x, y \in \mathbb{R} \). If \( f(0) = 1 \), \( f(1) = 3 \), and \( f(2) = 7 \), find the value of \( f(3) \). | [
"13"
] | [
"A function \\( f : \\mathbb{R} \\to \\mathbb{R} \\) satisfies the equation \\( f(x) - f(y) = (x - y) f'\\left(\\frac{x + y}{2}\\right) \\) for all \\( x, y \\in \\mathbb{R} \\). If \\( f(0) = 1 \\), \\( f(1) = 3 \\), and \\( f(2) = 7 \\), find the value of \\( f(3) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-559a4512568e591c24d0b59431b48c9ca76402c7785d27cd8135667192d70296 | A function \( f : \mathbb{Z} \to \mathbb{Q} \) has the property \( f(x+1) = \frac{1 + f(x)}{1 - f(x)} \) for every \( x \in \mathbb{Z} \). If \( f(2) = 2 \), then the value of the function \( f(2009) \) is ... | [
"\\dfrac{1}{3}"
] | [
"A function \\( f : \\mathbb{Z} \\to \\mathbb{Q} \\) has the property \\( f(x+1) = \\frac{1 + f(x)}{1 - f(x)} \\) for every \\( x \\in \\mathbb{Z} \\). If \\( f(2) = 2 \\), then the value of the function \\( f(2009) \\) is ..."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14865a9e1b7fe74d355136169ae8231f7e966ef202d6f811397076f8e487a13a | A function \( f : \mathbb{Z}_+ \to \mathbb{Z}_+ \), where \(\mathbb{Z}_+\) is the set of positive integers, is non-decreasing and satisfies \( f(mn) = f(m)f(n) \) for all relatively prime positive integers \( m \) and \( n \). Calculate the minimum possible value of \( \frac{f(8)f(13)}{(f(10))^2} \). | [
"1"
] | [
"A function \\( f : \\mathbb{Z}_+ \\to \\mathbb{Z}_+ \\), where \\(\\mathbb{Z}_+\\) is the set of positive integers, is non-decreasing and satisfies \\( f(mn) = f(m)f(n) \\) for all relatively prime positive integers \\( m \\) and \\( n \\). Calculate the minimum possible value of \\( \\frac{f(8)f(13)}{(f(10))^2} \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-e7f94077b6905bd13591619361f2e688256f4c7c4035a47bb0a5b60126987740 | A function \( f : \{1, 2, 3, \ldots, n\} \to \{1, 2, 3, \ldots, n\} \) satisfies that for all \( 1 \le m \le 25 \), there are exactly \( g(m) \) values \( 1 \le k \le n \) such that \( f^m(k) = k \), where \( g(m) \) is the sum of all distinct prime divisors of \( m \) and \( f^{g(m)}(k) \) is \( f \) iterated \( g(m) ... | [
"100"
] | [
"A function \\( f : \\{1, 2, 3, \\ldots, n\\} \\to \\{1, 2, 3, \\ldots, n\\} \\) satisfies that for all \\( 1 \\le m \\le 25 \\), there are exactly \\( g(m) \\) values \\( 1 \\le k \\le n \\) such that \\( f^m(k) = k \\), where \\( g(m) \\) is the sum of all distinct prime divisors of \\( m \\) and \\( f^{g(m)}(k) ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-7c86212ea79a4dbdec4d6fd62ea41b289ecace23917e8ae71e36904032786da1 | A function \( f : \{1, 2, \ldots, 8\} \rightarrow \{1, 2, \ldots, 8\} \) satisfies the following conditions: - For all \( x \) in its domain, at least one of \( x, f(x), f(f(x)) \) is equal to 8; - For exactly one \( y \) in its domain, more than one of \( y, f(y), f(f(y)) \) are equal to 8; - For exactly one ordered p... | [
"1"
] | [
"A function \\( f : \\{1, 2, \\ldots, 8\\} \\rightarrow \\{1, 2, \\ldots, 8\\} \\) satisfies the following conditions:\n- For all \\( x \\) in its domain, at least one of \\( x, f(x), f(f(x)) \\) is equal to 8;\n- For exactly one \\( y \\) in its domain, more than one of \\( y, f(y), f(f(y)) \\) are equal to 8;\n- ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-820862aee0d200cb11d1b24c053bce9b46d757d7702ba482e420180ec84c9fc8 | A function \( f : \{2, \ldots, N\} \to [0, \infty) \) satisfies the equation \( f(xy + 1) = f(x) + f(y) + 1 \) for all integers \( x, y \geq 2 \). Compute the largest possible value of \( N \). | [
"10"
] | [
"A function \\( f : \\{2, \\ldots, N\\} \\to [0, \\infty) \\) satisfies the equation \\( f(xy + 1) = f(x) + f(y) + 1 \\) for all integers \\( x, y \\geq 2 \\). Compute the largest possible value of \\( N \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6073713d020d153ba26dac78e74a79ccf4603ab7a250fd0517e39f0a7cefd489 | A function \( f \) defined on the interval \([-1, 1]\) satisfies \( f(0) = f(1) \) and the inequality \( |f(x_2) - f(x_1)| < |x_2 - x_1| \) for all \( x_1, x_2 \in [-1, 1] \). Let \( M \) be the maximum value of \( |f(x_2) - f(x_1)| \) over all \( x_1, x_2 \in [-1, 1] \). Calculate \( 2M \). | [
"2"
] | [
"A function \\( f \\) defined on the interval \\([-1, 1]\\) satisfies \\( f(0) = f(1) \\) and the inequality \\( |f(x_2) - f(x_1)| < |x_2 - x_1| \\) for all \\( x_1, x_2 \\in [-1, 1] \\). Let \\( M \\) be the maximum value of \\( |f(x_2) - f(x_1)| \\) over all \\( x_1, x_2 \\in [-1, 1] \\). Calculate \\( 2M \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-42fa6a09e72442436739d0d5f0b01c46f0536062f1613af2be9e0303fd4ab78f | A function \( f \) defined on the set of integers satisfies \[ f(x) + f(x+3) = x^2 \] for any integer \( x \). If \( f(19) = 94 \), find \( f(94) \). | [
"4335"
] | [
"A function \\( f \\) defined on the set of integers satisfies\n\n\\[ f(x) + f(x+3) = x^2 \\]\n\nfor any integer \\( x \\). If \\( f(19) = 94 \\), find \\( f(94) \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 16
} | 38 | 0 | 0 |
omr-ws-bf93e0e730260b283eb025834fa1c837dc02fe71a876bcfc084f745368347ace | A function \( f \) defined on the set of positive integers has \( f(1) = 2 \) and \( f(2) = 3 \). Additionally, \( f(f(f(n))) = n + 2 \) if \( n \) is even and \( f(f(f(n))) = n + 4 \) if \( n \) is odd. What is \( f(777) \)? | [
"390"
] | [
"A function \\( f \\) defined on the set of positive integers has \\( f(1) = 2 \\) and \\( f(2) = 3 \\). Additionally, \\( f(f(f(n))) = n + 2 \\) if \\( n \\) is even and \\( f(f(f(n))) = n + 4 \\) if \\( n \\) is odd. What is \\( f(777) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-d5fb154b14947f224c65257c20a3c9d012c44e7ee5081b4baaf0f53bc3a362db | A function \( f \) defined on the set of positive integers has the properties that, for any positive integer \( n \), \( f(f(n)) = 2n \) and \( f(4n + 1) = 4n + 3 \). What are the last three digits of \( f(2016) \)? | [
"904"
] | [
"A function \\( f \\) defined on the set of positive integers has the properties that, for any positive integer \\( n \\), \\( f(f(n)) = 2n \\) and \\( f(4n + 1) = 4n + 3 \\). What are the last three digits of \\( f(2016) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 4
} | 24 | 0 | 0 |
omr-ws-540d939b46624d703842e985911cafb15783721bfb409f4d73ebf1453832781f | A function \( f \) defined on the set of real numbers has the property \( f(a+b) = f(a)f(b) \). If \( f(4) = 625 \), what is the value of \( 3f(-2) \)? | [
"\\(\\frac{3}{25}\\)"
] | [
"A function \\( f \\) defined on the set of real numbers has the property \\( f(a+b) = f(a)f(b) \\). If \\( f(4) = 625 \\), what is the value of \\( 3f(-2) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-108af10ea6a6ec5e7228fe6adcb495a3adac5d93d14038685a2085629df35c2b | A function \( f \) from pairs of nonnegative integers to the real numbers satisfies the following conditions: \[ f(0,0) = 0, \] \[ f(2x,2y) = f(2x+1,2y+1) = f(x,y), \] \[ f(2x+1,2y) = f(2x,2y+1) = f(x,y) + 1 \] for all nonnegative integers \( x, y \). Let \( a, b \) and \( n \) be nonnegative integers such that \( f(a,... | [
"\\(2^n\\)"
] | [
"A function \\( f \\) from pairs of nonnegative integers to the real numbers satisfies the following conditions:\n\\[ f(0,0) = 0, \\]\n\\[ f(2x,2y) = f(2x+1,2y+1) = f(x,y), \\]\n\\[ f(2x+1,2y) = f(2x,2y+1) = f(x,y) + 1 \\]\nfor all nonnegative integers \\( x, y \\). Let \\( a, b \\) and \\( n \\) be nonnegative int... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-63d98f7e99142f9218744f406137b82470b37911da85d7f65d2ffa2ef55b5446 | A function \( f \) from the integers to the integers is defined as follows: \[ f(n) = \begin{cases} n + 3 & \text{if } n \text{ is odd} \\ n/2 & \text{if } n \text{ is even} \end{cases} \] Suppose \( k \) is odd and \( f(f(f(k))) = 27 \). What is the sum of the digits of \( k \)? | [
"6"
] | [
"A function \\( f \\) from the integers to the integers is defined as follows:\n\\[ f(n) = \\begin{cases} \nn + 3 & \\text{if } n \\text{ is odd} \\\\\nn/2 & \\text{if } n \\text{ is even} \n\\end{cases}\n\\]\nSuppose \\( k \\) is odd and \\( f(f(f(k))) = 27 \\). What is the sum of the digits of \\( k \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-89d7fde4e7702c32ac330887d49af6c7f4af5addbd9040b1b75dbd838fa6e163 | A function \( f \) from the real numbers to the real numbers satisfies \( f(5-x) = f(5+x) \). If \( f(x) = 0 \) has 5 distinct real roots, what is the sum of all of the distinct real roots? | [
"25"
] | [
"A function \\( f \\) from the real numbers to the real numbers satisfies \\( f(5-x) = f(5+x) \\). If \\( f(x) = 0 \\) has 5 distinct real roots, what is the sum of all of the distinct real roots?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9dd70ef401a0242d2f833146585d8f658a89f26e5ede458e82b473b55c27adfc | A function \( f \) from the set \(\{0, 1, \ldots, 2005\}\) into the nonnegative integers has the property that for all applicable integers \( x \), \[ f(2x+1) = f(2x), \] \[ f(3x+1) = f(3x), \] \[ f(5x+1) = f(5x). \] At most, how many distinct values can this function take? | [
"536"
] | [
"A function \\( f \\) from the set \\(\\{0, 1, \\ldots, 2005\\}\\) into the nonnegative integers has the property that for all applicable integers \\( x \\),\n\\[ f(2x+1) = f(2x), \\]\n\\[ f(3x+1) = f(3x), \\]\n\\[ f(5x+1) = f(5x). \\]\nAt most, how many distinct values can this function take?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-7f2ec00ce5a32ab862cdd9b09482634c0061661c51207f1fd650fd6bb0358f40 | A function \( f \) from the set of real numbers to itself satisfies \[ f \left(x^3 + y^3 \right) = (x + y) \left( f(x)^2 - f(x)f(y) + f(y)^2 \right) \] for arbitrary real numbers \( x \) and \( y \). If \( f(1) = 1 \), find the value of \( f(2014) \). | [
"2014"
] | [
"A function \\( f \\) from the set of real numbers to itself satisfies \n\\[ f \\left(x^3 + y^3 \\right) = (x + y) \\left( f(x)^2 - f(x)f(y) + f(y)^2 \\right) \\]\nfor arbitrary real numbers \\( x \\) and \\( y \\). If \\( f(1) = 1 \\), find the value of \\( f(2014) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-30d200211e5cb70e3ac6d97f83efd8cdcd27465ca06100c6c03ada911b07192f | A function \( f \) has a continuous third derivative everywhere and satisfies the relation \[ \lim_{x \rightarrow 0} \left( 1 + x + \frac{f(x)}{x} \right)^{\frac{1}{x}} = e^3. \] Compute \( f(0) \), \( f'(0) \), \( f''(0) \), and \( \lim_{x \rightarrow 0} \left( 1 + \frac{f(x)}{x} \right)^{\frac{1}{x}} \). | [
"\\( f(0) = 0, f'(0) = 2, f''(0) = 6, \\lim_{x \\rightarrow 0} \\left( 1 + \\frac{f(x)}{x} \\right)^{\\frac{1}{x}} = e^3 \\)."
] | [
"A function \\( f \\) has a continuous third derivative everywhere and satisfies the relation\n\\[ \\lim_{x \\rightarrow 0} \\left( 1 + x + \\frac{f(x)}{x} \\right)^{\\frac{1}{x}} = e^3. \\]\nCompute \\( f(0) \\), \\( f'(0) \\), \\( f''(0) \\), and \\( \\lim_{x \\rightarrow 0} \\left( 1 + \\frac{f(x)}{x} \\right)^{... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d0db459110af79bbcc9af0cd5e857194a6f556f1944181f402df0eb7c97bfcbf | A function \( f \) has its domain defined on natural numbers. Given the conditions \( f(1) = 1 \), \( f(2n) = 2f(n) \), and \( n \cdot f(2n+1) = (2n+1) \cdot (f(n) + n) \), find the function \( f \). | [
"\\( f(n) = \\frac{n(n+1)}{2} \\)"
] | [
"A function \\( f \\) has its domain defined on natural numbers. Given the conditions \\( f(1) = 1 \\), \\( f(2n) = 2f(n) \\), and \\( n \\cdot f(2n+1) = (2n+1) \\cdot (f(n) + n) \\), find the function \\( f \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1c1dea094e037968d9d0b3dea79517a08b1bc063c343cbbf03d07344cacea773 | A function \( f \) is analytic in a domain \( D \). For each point \( z \in D \), there exists a non-negative integer \( n_z \) such that \( f^{(n_z)}(z) = 0 \). If \( D \) contains the closed disk \( \{ z : |z| \leq 2 \} \), find the maximum possible degree of the polynomial \( f(z) \) when restricted to this disk. | [
"2"
] | [
"A function \\( f \\) is analytic in a domain \\( D \\). For each point \\( z \\in D \\), there exists a non-negative integer \\( n_z \\) such that \\( f^{(n_z)}(z) = 0 \\). If \\( D \\) contains the closed disk \\( \\{ z : |z| \\leq 2 \\} \\), find the maximum possible degree of the polynomial \\( f(z) \\) when re... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e68ffdceb278a1dafa4ff650231714aa96bd37c747f9f89fa2527d8c4db51ed | A function \( f \) is analytic in the whole complex plane except for 4 poles. We know that \(-1\), \(2\), and \(1+5i\) are poles of \( f \) and that \( f \) takes only real values in the interval \((-1, 2)\). Let \( z_0 \) be the fourth pole of \( f \). Calculate the value of \( z_0 + \overline{z_0} \). | [
"2"
] | [
"A function \\( f \\) is analytic in the whole complex plane except for 4 poles. We know that \\(-1\\), \\(2\\), and \\(1+5i\\) are poles of \\( f \\) and that \\( f \\) takes only real values in the interval \\((-1, 2)\\). Let \\( z_0 \\) be the fourth pole of \\( f \\). Calculate the value of \\( z_0 + \\overline... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a9f3e93a9bb354859fd4fcd10bedcce367349965c75dd04003ca962a3d4230ce | A function \( f \) is called good if it assigns an integer value \( f(m,n) \) to every ordered pair of integers \( (m,n) \) such that for every pair of integers \( (m,n) \), the following conditions hold: \[ 2f(m,n) = f(m-n, n-m) + m + n \] \[ 2f(m,n) = f(m+1,n) + f(m,n+1) - 1 \] Find all good functions. | [
"\\( f(m,n) = m + n \\)"
] | [
"A function \\( f \\) is called good if it assigns an integer value \\( f(m,n) \\) to every ordered pair of integers \\( (m,n) \\) such that for every pair of integers \\( (m,n) \\), the following conditions hold:\n\\[ 2f(m,n) = f(m-n, n-m) + m + n \\]\n\\[ 2f(m,n) = f(m+1,n) + f(m,n+1) - 1 \\]\nFind all good funct... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4782ae52fb79a5c1b97fb2d5f103232b47aff89edaa51360b7e740044a539dff | A function \( f \) is continuous on \([0,1]\) and differentiable on \((0,1)\). Given \( f(0) = f(1) = 0 \), find the number of distinct values of \( x \in (0,1) \) such that \( f'(x) = f(x) \) if \( f(x) = x(1-x)e^{kx} \) for some constant \( k \). | [
"1"
] | [
"A function \\( f \\) is continuous on \\([0,1]\\) and differentiable on \\((0,1)\\). Given \\( f(0) = f(1) = 0 \\), find the number of distinct values of \\( x \\in (0,1) \\) such that \\( f'(x) = f(x) \\) if \\( f(x) = x(1-x)e^{kx} \\) for some constant \\( k \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a77a1fab8bddca25dec56867fadd24c5c94763c908b9f5f632bc53890e93c34d | A function \( f \) is defined as follows: \[ f(1) = 1 \] \[ f(2n) = f(n) \] \[ f(2n+1) = f(2n) + 1 \] Find the maximum value that the function \( f \) can attain for \( 1 \leq n \leq 2009 \). | [
"10"
] | [
"A function \\( f \\) is defined as follows:\n\\[ f(1) = 1 \\]\n\\[ f(2n) = f(n) \\]\n\\[ f(2n+1) = f(2n) + 1 \\]\n\nFind the maximum value that the function \\( f \\) can attain for \\( 1 \\leq n \\leq 2009 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-7170a56b7f14d150a0672e88bdfe98eb658449ba5ffd24a625c3084efe5b2b4f | A function \( f \) is defined by \( f(z) = (4 + i) z^2 + \alpha z + \gamma \) for all complex numbers \( z \), where \( \alpha \) and \( \gamma \) are complex numbers and \( i^2 = -1 \). Suppose that \( f(1) \) and \( f(i) \) are both real. What is the smallest possible value of \( | \alpha | + |\gamma | \)? | [
"\\(\\sqrt{2}\\)"
] | [
"A function \\( f \\) is defined by \\( f(z) = (4 + i) z^2 + \\alpha z + \\gamma \\) for all complex numbers \\( z \\), where \\( \\alpha \\) and \\( \\gamma \\) are complex numbers and \\( i^2 = -1 \\). Suppose that \\( f(1) \\) and \\( f(i) \\) are both real. What is the smallest possible value of \\( | \\alpha |... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f1170c1449161fe8a78fd12c1330f93376260e6e33a6a50bb28ef482369537f9 | A function \( f \) is defined by \( f(z) = (4 + i) z^2 + \alpha z + \gamma \) for all complex numbers \( z \), where \( \alpha \) and \( \gamma \) are complex numbers and \( i^2 = -1 \). Suppose that \( f(1) \) and \( f(i) \) are both real. What is the smallest possible value of \( |\alpha| + |\gamma| \)? | [
"\\(\\sqrt{2}\\)"
] | [
"A function \\( f \\) is defined by \\( f(z) = (4 + i) z^2 + \\alpha z + \\gamma \\) for all complex numbers \\( z \\), where \\( \\alpha \\) and \\( \\gamma \\) are complex numbers and \\( i^2 = -1 \\). Suppose that \\( f(1) \\) and \\( f(i) \\) are both real. What is the smallest possible value of \\( |\\alpha| +... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-52c233bbf2cd10423bfe515e3794d2d9b8e6d98ca01fcd9cb9f015c682d51258 | A function \( f \) is defined for all real numbers and satisfies \( f(2 + x) = f(2 - x) \) and \( f(7 + x) = f(7 - x) \) for all real \( x \). If \( x = 0 \) is a root of \( f(x) = 0 \), what is the least number of roots \( f(x) = 0 \) must have in the interval \( -1000 \le x \le 1000 \)? | [
"401"
] | [
"A function \\( f \\) is defined for all real numbers and satisfies \\( f(2 + x) = f(2 - x) \\) and \\( f(7 + x) = f(7 - x) \\) for all real \\( x \\). If \\( x = 0 \\) is a root of \\( f(x) = 0 \\), what is the least number of roots \\( f(x) = 0 \\) must have in the interval \\( -1000 \\le x \\le 1000 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 30,
"tir": 2
} | 32 | 0 | 0 |
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