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-b269e1800ffca0840d424c48690347d615019cd43798be671a4e5b30f632aa7b | At an international meeting with 1985 participants, in each subset of three participants, there are at least two persons who speak the same language. If each person speaks at most 5 languages, find the minimum number of participants who spoke the same language. | [
"200"
] | [
"At an international meeting with 1985 participants, in each subset of three participants, there are at least two persons who speak the same language. If each person speaks at most 5 languages, find the minimum number of participants who spoke the same language."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-a7232e64f06c094d49ed29b4dd41d5edb5874e0769f5da377633c7c3c65bebe5 | At an oil refinery, pump-1 and pump-2 are used to fill a holding tank with volume $V$. Pump-1 alone can fill the tank in four hours, and pump-2 alone can fill it in six hours. Initially, both pumps are used simultaneously for $a$ hours. Then, charging continues using only pump-1 for $b$ hours and continues again using ... | [
"\\( b = \\frac{3}{2} - a \\), \\( c = \\frac{15}{4} - a \\), \\( 0 \\le a \\le \\frac{3}{2} \\)"
] | [
"At an oil refinery, pump-1 and pump-2 are used to fill a holding tank with volume $V$. Pump-1 alone can fill the tank in four hours, and pump-2 alone can fill it in six hours. Initially, both pumps are used simultaneously for $a$ hours. Then, charging continues using only pump-1 for $b$ hours and continues again u... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-5ec5c30f45652664222ac9ddac74686667717cad4391abe623574ea94920833f | At any moment, John has two strips with letters formed by repeatedly transferring letters from one strip to the other. Initially, one strip has the letter $A$ and the other has the letter $B$. After 10 minutes, calculate the total number of distinct palindromic sequences possible on both strips combined. | [
"2"
] | [
"At any moment, John has two strips with letters formed by repeatedly transferring letters from one strip to the other. Initially, one strip has the letter $A$ and the other has the letter $B$. After 10 minutes, calculate the total number of distinct palindromic sequences possible on both strips combined."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-23411612a4306e9f8025624627007d98bbc51a7f4205025165384f118f1220d3 | At any point in time, the outer endpoints of the hour, minute, and second hands on a 12-hour analog clock can be connected with line segments to form a clock triangle. If the hour, minute, and second hands are all length 1, over the course of one day from 12:00:00 AM to 1:00:00 AM, at how many distinct times is the clo... | [
"0"
] | [
"At any point in time, the outer endpoints of the hour, minute, and second hands on a 12-hour analog clock can be connected with line segments to form a clock triangle. If the hour, minute, and second hands are all length 1, over the course of one day from 12:00:00 AM to 1:00:00 AM, at how many distinct times is th... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-3e42e493af4d2b014577bc7faa6213fb6aa53079c8f3d499857e672aec39bab4 | At breakfast, lunch, and dinner, Joe randomly chooses with equal probabilities either an apple, an orange, or a banana to eat. On a given day, what is the probability that Joe will eat at least two different kinds of fruit? | [
"\\frac{8}{9}"
] | [
"At breakfast, lunch, and dinner, Joe randomly chooses with equal probabilities either an apple, an orange, or a banana to eat. On a given day, what is the probability that Joe will eat at least two different kinds of fruit?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7e766d45b05dc03a0b41296879a19c3cabc29c09c6bdb1c6dc566592d83d4634 | At constant temperature, the pressure of a sample of gas is inversely proportional to its volume. I have some oxygen in a 2.28 liter container with a pressure of 5 kPa. If I move all of it to a 5.7 liter container at the same temperature, what will the new pressure be in kPa? | [
"2"
] | [
"At constant temperature, the pressure of a sample of gas is inversely proportional to its volume. I have some oxygen in a 2.28 liter container with a pressure of 5 kPa. If I move all of it to a 5.7 liter container at the same temperature, what will the new pressure be in kPa?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-446a4f9191c8bef15832621560d319141f026f07bc85b0997dca1ef9978dd208 | At department tea, there are several cups of milk-contaminated tea, each filled to the same amount. The cup I'm holding has a sixth of all the milk and a quarter of all the tea. How many cups are there? | [
"5"
] | [
"At department tea, there are several cups of milk-contaminated tea, each filled to the same amount. The cup I'm holding has a sixth of all the milk and a quarter of all the tea. How many cups are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-209c5a584f261f8027f0a473b4caf6b17bfdb1e0de9f22d6e9569ac58bc5cd3e | At different times during the day, Ms. Segura will drop a letter to be typed into her secretary’s tray. Ms. Segura numbers these letters consecutively from 1 to 6 as she writes them. When there is time, her secretary takes the top letter out of the tray and types it. Which sequence below could not be the order in which... | [
"c) 4,2,5,6,3,1"
] | [
"At different times during the day, Ms. Segura will drop a letter to be typed into her secretary’s tray. Ms. Segura numbers these letters consecutively from 1 to 6 as she writes them. When there is time, her secretary takes the top letter out of the tray and types it. Which sequence below could not be the order in ... | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a53e4ec621e0d9e381ea3a60fb767a3231c0b917293f31db0b0911f380bb5871 | At each corner of a cube, an integer is written. A legal transition of the cube consists in picking any corner of the cube and adding the value written at that corner to the value written at some adjacent corner. Given an initial configuration where the integers at the corners are $1, 2, 3, 4, 5, 6, 7, 8$, find the min... | [
"12"
] | [
"At each corner of a cube, an integer is written. A legal transition of the cube consists in picking any corner of the cube and adding the value written at that corner to the value written at some adjacent corner. Given an initial configuration where the integers at the corners are $1, 2, 3, 4, 5, 6, 7, 8$, find th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-948453ff2f073ca024cfd843fcbf2fe14c4b44d8e1f7c9d8655cc5c20ba34778 | At each of $2n$ points around a table, we place a disk with one white side and one black side. We may perform the following move: select a black disk, and flip over its two neighbors. Find all initial configurations from which some sequence of such moves leads to a position where all disks but one are white. | [
"\\text{All configurations with an odd number of black disks}"
] | [
"At each of $2n$ points around a table, we place a disk with one white side and one black side. We may perform the following move: select a black disk, and flip over its two neighbors. Find all initial configurations from which some sequence of such moves leads to a position where all disks but one are white."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b1c21e19967c66f28210b44e842517a15c4dbee733ba0966df63d3b02b1246e | At each of the sixteen circles in a network, stands a student. A total of 3360 coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of coins ... | [
"280"
] | [
"At each of the sixteen circles in a network, stands a student. A total of 3360 coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of c... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-2d554c57c793468b179260e266291408d6ea451546a6f7439f811bb7c4b5d896 | At each step, we are changing the places of exactly two numbers from the sequence $1, 2, 3, 4, 5, 6, 7$. How many different arrangements can be formed after two steps? | [
"176"
] | [
"At each step, we are changing the places of exactly two numbers from the sequence $1, 2, 3, 4, 5, 6, 7$. How many different arrangements can be formed after two steps?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 1
} | 48 | 0 | 0 |
omr-ws-4dd4f46ae1431a8e235bb87825465bc91fd8312382520184616827cfd3c8d134 | At each time period \( t = 1, 2, 3, \ldots \), a red coin and a blue coin are flipped simultaneously. The red coin comes up heads with probability \( p_r \) and the blue coin comes up heads with probability \( p_b \). Calculate the probability that at least 3 red coins come up heads before the first blue coin comes up ... | [
"\\(\\frac{(p_r)^3 (1 - p_b)^3}{1 - p_r (1 - p_b)}\\)"
] | [
"At each time period \\( t = 1, 2, 3, \\ldots \\), a red coin and a blue coin are flipped simultaneously. The red coin comes up heads with probability \\( p_r \\) and the blue coin comes up heads with probability \\( p_b \\). Calculate the probability that at least 3 red coins come up heads before the first blue co... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eebe03528fa472a28830530217450d5abf8084a4588e677964aa58c66834ca7e | At each vertex of a $13$-sided polygon, we write one of the numbers $1, 2, 3, \ldots, 12, 13$, without repeating. Then, on each side of the polygon, we write the absolute difference of the numbers of the vertices of its ends. If it is possible to number the vertices of the polygon so that only the numbers $3, 4$, and $... | [
"10"
] | [
"At each vertex of a $13$-sided polygon, we write one of the numbers $1, 2, 3, \\ldots, 12, 13$, without repeating. Then, on each side of the polygon, we write the absolute difference of the numbers of the vertices of its ends. If it is possible to number the vertices of the polygon so that only the numbers $3, 4$,... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-782c1aa4b41f0e3a0ed304f095eabfeb5d55b9de2792f2da905f20a145ddc4f2 | At each vertex of a $4 \times 5$ rectangle, there is a house. Find the path of the minimum length connecting all these houses. | [
"18"
] | [
"At each vertex of a $4 \\times 5$ rectangle, there is a house. Find the path of the minimum length connecting all these houses."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9562d26eb57812de0b3f7e6a7b872b475d000951fc720b4ada12037cd8e93191 | At each vertex of a convex polygon with \( n \) sides, there is a light bulb. Initially, exactly \( k \) lights are on. Each time, we choose an edge or a diagonal and change the state of one of the two bulbs at its ends. For what value of \( k \) is it possible to make all the lamps turn on after switching states on ea... | [
"\\( k \\) is even."
] | [
"At each vertex of a convex polygon with \\( n \\) sides, there is a light bulb. Initially, exactly \\( k \\) lights are on. Each time, we choose an edge or a diagonal and change the state of one of the two bulbs at its ends. For what value of \\( k \\) is it possible to make all the lamps turn on after switching s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 2
} | 14 | 0 | 0 |
omr-ws-48d5d0521ed9aea8a8fde6bfdcd972cee94a95669ed8c744130d0a3ea198fed0 | At each vertex of a cube, a tetrahedron is cut off, resulting in a solid with 6 regular octagons and 8 equilateral triangles as its faces. What fraction of the cube was cut off? | [
"\\dfrac{10 - 7\\sqrt{2}}{3}"
] | [
"At each vertex of a cube, a tetrahedron is cut off, resulting in a solid with 6 regular octagons and 8 equilateral triangles as its faces. What fraction of the cube was cut off?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-473a0e496a31a553124e41c4723289c849a5230d178f81f9cde20e22a4b6828a | At each vertex of a non-convex polygon, Vasya measured the angle between the rays whose sides lie on the edges of the polygon. The sum of all these angles turned out to be $2008^\circ$. What is the minimum number of sides the polygon could have for this to happen? | [
"14"
] | [
"At each vertex of a non-convex polygon, Vasya measured the angle between the rays whose sides lie on the edges of the polygon. The sum of all these angles turned out to be $2008^\\circ$. What is the minimum number of sides the polygon could have for this to happen?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-609566816da7acd4bf9bff0d14aee77fa7e6a280ec643cc218c100d6bd80b9ef | At each vertex of a regular hexagon, the numbers 1, 0, 1, 0, 0, 0 are written in clockwise order. At each step, we can increase or decrease any 3 consecutive numbers by an arbitrary real number. What is the minimum number of steps required to make all six numbers equal? | [
"2"
] | [
"At each vertex of a regular hexagon, the numbers 1, 0, 1, 0, 0, 0 are written in clockwise order. At each step, we can increase or decrease any 3 consecutive numbers by an arbitrary real number. What is the minimum number of steps required to make all six numbers equal?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4775d4ae757a2223e86de48e0da776387ddc42eb2be4062a7f3cbc4d38c314f5 | At each vertex of a unit square, we draw the smallest circle centered at that vertex which contains the entire square. If points \( A \) and \( B \) both lie on or inside all four circles, what is the maximum possible distance \( AB \)? | [
"\\(\\sqrt{2}\\)"
] | [
"At each vertex of a unit square, we draw the smallest circle centered at that vertex which contains the entire square. If points \\( A \\) and \\( B \\) both lie on or inside all four circles, what is the maximum possible distance \\( AB \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 7
} | 9 | 0 | 0 |
omr-ws-1a3798649d85df6c294d3ea092cc1f58d3962f5dbc83ef927f6aa0108a2829a1 | At every step, a random number between 1 and 8 is chosen. The process continues until every number from 1 to 8 has been chosen at least once. What is the mathematical expectation of the number of steps required? | [
"\\( 8H_8 \\)"
] | [
"At every step, a random number between 1 and 8 is chosen. The process continues until every number from 1 to 8 has been chosen at least once. What is the mathematical expectation of the number of steps required?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-023a66cc4bdcb042839d86fb117db8593a7b79053c38d4d37826c4801453162e | At every vertex $A_k$ (where $1 \le k \le n$) of a regular $n$-gon, $k$ coins are placed. In each step, one can choose two coins and move them to their adjacent vertices, one clockwise and one anticlockwise. Find all positive integers $n$ such that after a finite number of operations, we can reach the configuration whe... | [
"1"
] | [
"At every vertex $A_k$ (where $1 \\le k \\le n$) of a regular $n$-gon, $k$ coins are placed. In each step, one can choose two coins and move them to their adjacent vertices, one clockwise and one anticlockwise. Find all positive integers $n$ such that after a finite number of operations, we can reach the configurat... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 26,
"genselect": 7
} | 33 | 0 | 0 |
omr-ws-1d0cc5f6cab9f195ba3f16afabee974ac8c99aaae08c880390c4448e5c939fda | At first, on a board, the number $1$ is written $100$ times. Every minute, we pick a number $a$ from the board, erase it, and write $a/3$ thrice instead. We say that a positive integer $n$ is persistent if after any amount of time, regardless of the numbers we pick, we can find at least $n$ equal numbers on the board. ... | [
"67"
] | [
"At first, on a board, the number $1$ is written $100$ times. Every minute, we pick a number $a$ from the board, erase it, and write $a/3$ thrice instead. We say that a positive integer $n$ is persistent if after any amount of time, regardless of the numbers we pick, we can find at least $n$ equal numbers on the bo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 41,
"genselect": 7,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-749bfbacb82668e130b2dfa78e9dd7d8bf2d1acfff3b985f4023d080647eaef6 | At high tide, the water level of a certain pier is 10 feet below the pier. At low tide, the water level is 26 feet below the pier. Assuming sinusoidal behavior, give a formula for \( y = f(t) \), where \( y \) is the water level relative to the pier at a given time \( t \) (in hours) if at \( t = 0 \) the water level i... | [
"\\( f(t) = 18 - 8 \\sin\\left(\\frac{\\pi}{3} t - \\frac{\\pi}{2}\\right) \\)"
] | [
"At high tide, the water level of a certain pier is 10 feet below the pier. At low tide, the water level is 26 feet below the pier. Assuming sinusoidal behavior, give a formula for \\( y = f(t) \\), where \\( y \\) is the water level relative to the pier at a given time \\( t \\) (in hours) if at \\( t = 0 \\) the ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 9
} | 14 | 0 | 0 |
omr-ws-2960d1c37e9f7bb45c3e554d860cc848b55b9fdf81e5fce9c1fbec03eecb2149 | At his usual rowing rate, a man rows 15 miles downstream in five hours less than it takes him to return. If he doubles his usual rowing rate, the time downstream is only one hour less than the time upstream. What is the rate of the stream’s current? Given equations: 1. $(r+c)t = 15$ 2. $(r-c)(t+5) = 15$ 3. $(2r+c)b = 1... | [
"2"
] | [
"At his usual rowing rate, a man rows 15 miles downstream in five hours less than it takes him to return. If he doubles his usual rowing rate, the time downstream is only one hour less than the time upstream. What is the rate of the stream’s current?\n\nGiven equations:\n1. $(r+c)t = 15$\n2. $(r-c)(t+5) = 15$\n3. $... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-80701b002ea01a465343d064bd4be4f9d731a8d4252ea5cd1382d44632883082 | At his usual rowing rate, a man rows 15 miles downstream in five hours less time than it takes him to return. If he doubles his usual rowing rate, the time downstream is only one hour less than the time upstream. Find the rate of the stream's current in miles per hour. | [
"2"
] | [
"At his usual rowing rate, a man rows 15 miles downstream in five hours less time than it takes him to return. If he doubles his usual rowing rate, the time downstream is only one hour less than the time upstream. Find the rate of the stream's current in miles per hour."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ef4e0bb42b3878077a19f83d2095ce158ddfef9590eb571ff2b23dbe5f2ede09 | At how many minutes after $8$ o'clock does the minute hand lie on top of the hour hand? | [
"\\(\\frac{480}{11}\\)"
] | [
"At how many minutes after $8$ o'clock does the minute hand lie on top of the hour hand?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-53e16cedf5f67796c2764c24477674d78a5426265b74d431144a25c97189ba59 | At how many minutes past 8:00 will the minute and hour hands form equal angles with the vertical for the first time? | [
"\\dfrac{240}{13}"
] | [
"At how many minutes past 8:00 will the minute and hour hands form equal angles with the vertical for the first time?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bbc56f628e7db32a3915da0b3bc59cdfda2da92575ce4c17653eb32c45de3462 | At how many points do the curves \( y = x^{12} \) and \( y = 2^x \) intersect? | [
"3"
] | [
"At how many points do the curves \\( y = x^{12} \\) and \\( y = 2^x \\) intersect?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 2
} | 22 | 0 | 0 |
omr-ws-faa203d7eac328a1bdf3bc3c7c670074c7ca0d2f759c70b1ad851807bd650144 | At how many points in the $xy$-plane do the graphs of $y=x^{12}$ and $y=2^x$ intersect? | [
"3"
] | [
"At how many points in the $xy$-plane do the graphs of $y=x^{12}$ and $y=2^x$ intersect?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-99b83d0c34a6f70c10870276d53288e7920b99b9d3077c302f394a90abaa0576 | At how many points in the Cartesian plane do the functions $f(x)=e^x-1$ and $g(x)=\ln{x}+1$ intersect? | [
"0"
] | [
"At how many points in the Cartesian plane do the functions $f(x)=e^x-1$ and $g(x)=\\ln{x}+1$ intersect?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f9251f0abf6e16498e719fe6e380d6a7c784e9c78176ad163860a6a0d3c3a11f | At how many points in the xy-plane do the graphs of \( y = x^{12} \) and \( y = 2^x \) intersect? | [
"3"
] | [
"At how many points in the xy-plane do the graphs of \\( y = x^{12} \\) and \\( y = 2^x \\) intersect?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 2
} | 25 | 0 | 0 |
omr-ws-61a1beb27f1ed394eec701d01cd973f9e6439080098217388c28e5e84bbcd4b9 | At how many points, excluding the endpoints, will an ant intersect the grid lines if it travels in a straight line from (0,0) to (240,300) on a large grid? | [
"479"
] | [
"At how many points, excluding the endpoints, will an ant intersect the grid lines if it travels in a straight line from (0,0) to (240,300) on a large grid?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-a3fe6d61d24b2abc4456a34fe0221bb048f65885d618c63c33b5b468b4fde042 | At how many times of the 24-hour day is the distance between the ends of the hour and the minute hand a positive multiple of 6, given that both hands are 6 units long and the clock has a radius of 6? | [
"66"
] | [
"At how many times of the 24-hour day is the distance between the ends of the hour and the minute hand a positive multiple of 6, given that both hands are 6 units long and the clock has a radius of 6?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f63c565359b2af565b7a19557d9031b71a586d23f344a1c7e8f7ce0f1d194023 | At how many values of $x$ does the graph of the function $f(x) = xe^{-x^2-x^{-2}}$ for $x \neq 0$ and $f(0) = 0$ have a horizontal tangent line? | [
"3"
] | [
"At how many values of $x$ does the graph of the function $f(x) = xe^{-x^2-x^{-2}}$ for $x \\neq 0$ and $f(0) = 0$ have a horizontal tangent line?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1073fd7d731f3d51dbe04a2ac3061884f050129b91a3b6cae1719bcb14ee3342 | At least how many $Z$ shapes are necessary to cover an $8 \times 8$ table, where each $Z$ shape covers the cells $(i,j)$, $(i,j+1)$, $(i+1,j+1)$, $(i+2,j+1)$, and $(i+2,j+2)$, and $Z$ shapes can overlap and rotate? | [
"13"
] | [
"At least how many $Z$ shapes are necessary to cover an $8 \\times 8$ table, where each $Z$ shape covers the cells $(i,j)$, $(i,j+1)$, $(i+1,j+1)$, $(i+2,j+1)$, and $(i+2,j+2)$, and $Z$ shapes can overlap and rotate?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-b8df8f9b9afd1d10389392891d0a1b4c2c9a070fd283c85b910b45faa785cad8 | At least how many 3-digit composite numbers should be chosen to ensure that at least two of the chosen numbers are not relatively prime? | [
"12"
] | [
"At least how many 3-digit composite numbers should be chosen to ensure that at least two of the chosen numbers are not relatively prime?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 56,
"genselect": 11,
"tir": 4
} | 71 | 0 | 0 |
omr-ws-240e2c452b2bb56f373a6e7d33513332843223f711fff8264d4f114996bbfae6 | At least how many 8's are needed after 1 to make a number which can be expressed as \( x^{64} \), where \( x \) is an integer? | [
"0"
] | [
"At least how many 8's are needed after 1 to make a number which can be expressed as \\( x^{64} \\), where \\( x \\) is an integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-81ef0ecb848e59520e6907d01f72ddc24848d75446d5bd2ad3c0112695ee9854 | At least how many days are necessary if each of 25 students must participate in a meeting at least once with all the other students, given that each day there are exactly three classes consisting of 3, 4, and 5 students respectively, and no student can be in more than one class on any day? | [
"16"
] | [
"At least how many days are necessary if each of 25 students must participate in a meeting at least once with all the other students, given that each day there are exactly three classes consisting of 3, 4, and 5 students respectively, and no student can be in more than one class on any day?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-7e72bc68a81d525eb00adf9cc665826bacbbf681809e7b5de58defefb7a61a32 | At least how many elements does the set which contains all of the midpoints of segments connecting 2003 different points in a plane have? | [
"None of the preceeding"
] | [
"At least how many elements does the set which contains all of the midpoints of segments connecting 2003 different points in a plane have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f80f9c47a784c208896b7becf7d22c7e64f2a15a1e610a1fb30df68ef0112ce1 | At least how many elements must be selected from the group $(\mathbb{Z}_{15},+)$ such that among the selected elements there exist three (not necessarily distinct) with sum $0$? | [
"7"
] | [
"At least how many elements must be selected from the group $(\\mathbb{Z}_{15},+)$ such that among the selected elements there exist three (not necessarily distinct) with sum $0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-0ba59c60b30d8e04f1be795f55931b59489293d0c4885cf1253336973f106e06 | At least how many elements must be selected from the group $(\mathbb{Z}_{2k},+)$ such that among the selected elements there exist three (not necessarily distinct) with sum $0$? | [
"\\( k+1 \\)"
] | [
"At least how many elements must be selected from the group $(\\mathbb{Z}_{2k},+)$ such that among the selected elements there exist three (not necessarily distinct) with sum $0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 1
} | 19 | 0 | 0 |
omr-ws-f7980fa9657998d2b74e1bb3e6ba5c5b05b8cadcaef76e1a67fa754048a42af8 | At least how many games does the champion team win in a competition where 3 countries $A, B, C$ each have 9 representatives, and the competition rules involve a knockout system with representatives from two countries competing each round? | [
"9"
] | [
"At least how many games does the champion team win in a competition where 3 countries $A, B, C$ each have 9 representatives, and the competition rules involve a knockout system with representatives from two countries competing each round?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-5234f47c74ff62599af05ffd5ec3b434bbc84aa1defc7adee2aa7ae47b071399 | At least how many non-zero real numbers do we have to select such that every one of them can be written as a sum of 2019 other selected numbers, where the selected numbers are not necessarily different? | [
"2020"
] | [
"At least how many non-zero real numbers do we have to select such that every one of them can be written as a sum of 2019 other selected numbers, where the selected numbers are not necessarily different?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-d54cf2c9204ec8af695be02f0d6b9fa880629052faee6f19a725321060170403 | At least how many non-zero real numbers do we have to select such that every one of them can be written as a sum of 2019 other selected numbers, where the selected numbers are pairwise different? | [
"4039"
] | [
"At least how many non-zero real numbers do we have to select such that every one of them can be written as a sum of 2019 other selected numbers, where the selected numbers are pairwise different?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-9606ada1f72679bd511632510a8542507725a76f2ad6c642b03da8afd893c82d | At least how many numbers are to be taken from the sequence $2, 3, 8, 13, 18, 23, \ldots, 118$ such that the sum of any two of the taken numbers is $126$? | [
"2"
] | [
"At least how many numbers are to be taken from the sequence $2, 3, 8, 13, 18, 23, \\ldots, 118$ such that the sum of any two of the taken numbers is $126$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-6463a7c9257bbaf859acddfd7b5917fff6c181fac055fb0e693d89c577ade7a1 | At least how many numbers must be chosen from the set \( S = \{3, 8, 13, \ldots, 118\} \) to ensure that the sum of any two chosen numbers is 126? | [
"14"
] | [
"At least how many numbers must be chosen from the set \\( S = \\{3, 8, 13, \\ldots, 118\\} \\) to ensure that the sum of any two chosen numbers is 126?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 25
} | 48 | 0 | 0 |
omr-ws-916f7a316c0788cb9dee650c0a50aa8f88ac1898bf387387fe3d20dd1c15fca4 | At least how many numbers should be selected from the set \(\{1, 5, 9, 13, \ldots, 125\}\) to be assured that two of the numbers selected have a sum of 238? | [
"31"
] | [
"At least how many numbers should be selected from the set \\(\\{1, 5, 9, 13, \\ldots, 125\\}\\) to be assured that two of the numbers selected have a sum of 238?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-985fe36cfb080b418b7d5ede7bb1d7bd994fe5cb0db35c1df44d86c6d89051c7 | At least how many people are needed to ensure that at least 2 of them were born in the month of January? | [
"336"
] | [
"At least how many people are needed to ensure that at least 2 of them were born in the month of January?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f3bda69f4ad14be3ca52d6dc3770ffcedbb152f3123b97c8dbe2afed0231fc1e | At least how many weighings of a balanced scale are needed to order four stones with distinct weights from the lightest to the heaviest? | [
"5"
] | [
"At least how many weighings of a balanced scale are needed to order four stones with distinct weights from the lightest to the heaviest?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-e263191a319032d695d52aa2f7b14d3e5205e149bfb79b1622e4596492bd0c01 | At lunch, $60\%$ of the students selected soda while $20\%$ selected milk. If 72 students selected soda, how many students selected milk? | [
"24"
] | [
"At lunch, $60\\%$ of the students selected soda while $20\\%$ selected milk. If 72 students selected soda, how many students selected milk?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bc247aea86fe97acce2009bfcec756494c697b14cca64751ac40114353428986 | At most how many 2x2 squares can be placed inside a 5x5 square without overlapping? | [
"4"
] | [
"At most how many 2x2 squares can be placed inside a 5x5 square without overlapping?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-3d18cb6b816fea2096a3dd145a1c628662913cfb6404b11418ace98b6463d14f | At most how many cells in a $2021 \times 2021$ grid can be painted black such that every $2 \times 2$ square contains at most two black cells? | [
"2043231"
] | [
"At most how many cells in a $2021 \\times 2021$ grid can be painted black such that every $2 \\times 2$ square contains at most two black cells?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 39,
"genselect": 6,
"tir": 6
} | 51 | 0 | 0 |
omr-ws-7e6d586ac0199762dd789cab2bbdc58e5c6c1fb21226bd740b5ecec79549210d | At most how many cells in a $2021 \times 2021$ grid can be painted black such that every $2 \times 2$ subgrid contains at most two black cells? | [
"2042220"
] | [
"At most how many cells in a $2021 \\times 2021$ grid can be painted black such that every $2 \\times 2$ subgrid contains at most two black cells?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d64e29555f4ba6c01cefb75c082fe1d6a4b03802ae9a40e25d6563d6b1354de | At most how many cells in an $n \times n$ grid can be painted black such that every $2 \times 2$ subgrid contains at most two black cells? | [
"\\left\\lceil \\dfrac{n^2}{2} \\right\\rceil"
] | [
"At most how many cells in an $n \\times n$ grid can be painted black such that every $2 \\times 2$ subgrid contains at most two black cells?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-e6532c1ec94a33f540e940e213db309727c1b228e196b96e3fe76046ae167e45 | At most how many consecutive positive integers can be added to get 2020? | [
"5"
] | [
"At most how many consecutive positive integers can be added to get 2020?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4ac80df0c5da3b49c4754c61b35337a86d8b4c91d9f6bb5e360b09d3f1023c9b | At most how many consecutive singular numbers are there, where a singular number is defined as a positive number whose prime factorization contains no even powers other than $0$s? | [
"7"
] | [
"At most how many consecutive singular numbers are there, where a singular number is defined as a positive number whose prime factorization contains no even powers other than $0$s?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-7cfe118999268d7bbcf47d4596ac3d90bfe6e37e702cf51f358840bc5d3de0bb | At most how many intersection points can a 6-gon polygon have if the intersections can be outside of the polygon? | [
"15"
] | [
"At most how many intersection points can a 6-gon polygon have if the intersections can be outside of the polygon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-a920471ee415ac0495c2e11b708921254e61b47321ec8b4ebacc2806c34236e9 | At most how many numbers within a set of 10 consecutive integers are not divisible by 2, 3, 5, and 7? | [
"4"
] | [
"At most how many numbers within a set of 10 consecutive integers are not divisible by 2, 3, 5, and 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-304a44f2f489696025530921b26c8651f9b446f1e6c7d862bab0455d8289e497 | At most how many of the first 100 positive integers can be chosen if no two of the chosen numbers have a sum divisible by 5? | [
"41"
] | [
"At most how many of the first 100 positive integers can be chosen if no two of the chosen numbers have a sum divisible by 5?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 10
} | 32 | 0 | 0 |
omr-ws-7ed431037e0ce159c1607ebb43b9aa5b63f784be3212b21d2f13087a761a7f99 | At most how many of the following assertions can mutually be true? $a$) If $b$ is true, then this assertion is false. $b$) If the number of true assertions is greater than two, then one of them is $c$. $c$) At least one of $a$ and $d$ is false. $d$) $b$ and $c$ are both true or both false. $e$) $b$ is true or false. | [
"3"
] | [
"At most how many of the following assertions can mutually be true?\n$a$) If $b$ is true, then this assertion is false.\n$b$) If the number of true assertions is greater than two, then one of them is $c$.\n$c$) At least one of $a$ and $d$ is false.\n$d$) $b$ and $c$ are both true or both false.\n$e$) $b$ is true or... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 24,
"genselect": 2
} | 26 | 0 | 0 |
omr-ws-5e235802e0b7fa60061167025d9e5b43011302024cd68553ba972279bdb8bd30 | At most how many points with integer coordinates are there over a circle with center $(\sqrt{20}, \sqrt{10})$ in the $xy$-plane? | [
"1"
] | [
"At most how many points with integer coordinates are there over a circle with center $(\\sqrt{20}, \\sqrt{10})$ in the $xy$-plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 46
} | 46 | 0 | 0 |
omr-ws-abeb8fc59ac33e106df24d8ebbe586985fb0fd5fc2e99131c4eaba5818f8c1aa | At most how many points with integer coordinates are there over a circle with center of \( (\sqrt{20}, \sqrt{10}) \) in the \( xy \)-plane? | [
"1"
] | [
"At most how many points with integer coordinates are there over a circle with center of \\( (\\sqrt{20}, \\sqrt{10}) \\) in the \\( xy \\)-plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 45
} | 45 | 0 | 0 |
omr-ws-a06288f5bbc75bf6bf96a7472ece1b8e55602608c9e61ce57a4619c95281a837 | At most how many positive integers less than $51$ are there such that no one is triple of another one? | [
"38"
] | [
"At most how many positive integers less than $51$ are there such that no one is triple of another one?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 21
} | 44 | 0 | 0 |
omr-ws-1c48124c5c3dd237962cc550c5c9c5576e781260ec4b2a5a7cf017e6eec78c22 | At most how many primes can $n$ consecutive integers include? | [
"\\pi(n+1)"
] | [
"At most how many primes can $n$ consecutive integers include?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-24689e15800a52afcfbe1e86684d3ae1dff4ca9e5e1bc21624bcfb8721fbf24c | At most how many right triangles can be formed by using five lines on the plane? | [
"6"
] | [
"At most how many right triangles can be formed by using five lines on the plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-8b8dfb66fa78ef048587c629f6855662c43abcbbad4e127272fbf910fc481c9f | At most how many right triangles with smaller sides parallel to the axes can be formed using $k^2$ lattice points in the Cartesian plane? | [
"k^2(k-1)^2"
] | [
"At most how many right triangles with smaller sides parallel to the axes can be formed using $k^2$ lattice points in the Cartesian plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-77bac3142c5042511f267d62c2ac2b776adad75d147bdc7c22596c190f14a310 | At most how many subsets of size 1012 can be chosen from the set \(\{1, 2, \ldots, 2024\}\) such that the intersection of any three subsets has no more than 1 element? | [
"4"
] | [
"At most how many subsets of size 1012 can be chosen from the set \\(\\{1, 2, \\ldots, 2024\\}\\) such that the intersection of any three subsets has no more than 1 element?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-b79d3931e1eafaed1ee3461de16544a3d9cacf67daf79ae50e35e20850894567 | At most how much money does Beth need to spend to deduce Angela's polynomial with positive integer coefficients, given that Beth can pay Angela a dollar to know the value of \( P(n) \) for a positive integer \( n \) of Beth's choice? | [
"2"
] | [
"At most how much money does Beth need to spend to deduce Angela's polynomial with positive integer coefficients, given that Beth can pay Angela a dollar to know the value of \\( P(n) \\) for a positive integer \\( n \\) of Beth's choice?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b3684fa7b5c6e32dfbc58195dad475fbe551bc96a2fb3b1d69ab16c5b592b5f8 | At most, how many regions do 10 lines and 10 circles divide the plane? | [
"346"
] | [
"At most, how many regions do 10 lines and 10 circles divide the plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-dd4c83e69145e46fd039b8306487cc7e33db75b3b63fb969a23a24e826616f6e | At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values... | [
"60"
] | [
"At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible v... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f81955f168ecce4b9adbb840bb4613b645be94a43dbbc14ead50f920c4572cf1 | At noon sharp a pedestrian and a cyclist left point $A$ for point $B$ and a horseman left point $B$ for point $A$. Two hours later the cyclist and the horseman met 3 km from the half-way point between $A$ and $B$, and another 48 min later the pedestrian and the horseman met. Find the speed of each of them and the dista... | [
"The speeds of the pedestrian, the cyclist and the horseman are 6 km/h, 9 km/h and 12 km/h, respectively. The distance is 42 km"
] | [
"At noon sharp a pedestrian and a cyclist left point $A$ for point $B$ and a horseman left point $B$ for point $A$. Two hours later the cyclist and the horseman met 3 km from the half-way point between $A$ and $B$, and another 48 min later the pedestrian and the horseman met. Find the speed of each of them and th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6f60d036c886067c33e5adc1a0930ae46721c844b34fdd5edd97c00bf10a569a | At noon, ship A is 100 km west of ship B. Ship B is sailing south at 35 km/h and ship A is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 PM? | [
"\\(\\frac{720}{13}\\) or approximately \\(55.4\\) kilometers per hour."
] | [
"At noon, ship A is 100 km west of ship B. Ship B is sailing south at 35 km/h and ship A is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 PM?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-2c7e20b5122328db67ec9c9e2926e0bcda116916dadecc5cf70defa5e8a3f157 | At one moment, the end of the hour hand, the end of the minute hand, and one of the twelve numbers on a watch form an equilateral triangle. Suppose the ratio of the lengths of the minute hand (which is equal to the distance from the center of the watch plate to any of the twelve numbers) and the hour hand is \( k > 1 \... | [
"12"
] | [
"At one moment, the end of the hour hand, the end of the minute hand, and one of the twelve numbers on a watch form an equilateral triangle. Suppose the ratio of the lengths of the minute hand (which is equal to the distance from the center of the watch plate to any of the twelve numbers) and the hour hand is \\( k... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-78d586d6bcf2f4f23f4bf5bcb3f2a9cf6852f71e4df4067801d51fd2bfc8558c | At one moment, the end of the hour hand, the end of the minute hand, and one of the twelve numbers on a watch form an equilateral triangle. Suppose the ratio of the lengths of the minute hand (which is equal to the distance from the center of the watch to any of the twelve numbers) and the hour hand is \( k > 1 \). Fin... | [
"12"
] | [
"At one moment, the end of the hour hand, the end of the minute hand, and one of the twelve numbers on a watch form an equilateral triangle. Suppose the ratio of the lengths of the minute hand (which is equal to the distance from the center of the watch to any of the twelve numbers) and the hour hand is \\( k > 1 \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-33265dac8fa99f24a4d4942995dfaefd4404c94da0051fb89fd62ae7416314f5 | At one point as Lorena climbs a ladder, she finds that the number of steps above her is twice the number below her (not counting the step she is on). After climbing 5 more steps, she finds that the number of steps above and below her are equal. How many more steps must she climb to have the number below her be four tim... | [
"9"
] | [
"At one point as Lorena climbs a ladder, she finds that the number of steps above her is twice the number below her (not counting the step she is on). After climbing 5 more steps, she finds that the number of steps above and below her are equal. How many more steps must she climb to have the number below her be fou... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61e08afde3c1a38122d1404095a1b4497bc3605b6df7a8d08e9785faee51d850 | At precisely \( m \) minutes after 12:00 on a standard twelve-hour clock, the angle made by the hour hand and minute hand is exactly \( 1^\circ \). Determine all possible values of \( m \) where \( 1 \leq m \leq 720 \). | [
"\\( m = 262 \\) or \\( m = 458 \\)",
"\\( m = 65, 139, 213, 287, 361, 435, 509, 583, 657 \\)"
] | [
"At precisely \\( m \\) minutes after 12:00 on a standard twelve-hour clock, the angle made by the hour hand and minute hand is exactly \\( 1^\\circ \\). Determine all possible values of \\( m \\) where \\( 1 \\leq m \\leq 720 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 21
} | 28 | 0 | 0 |
omr-ws-8ba25b94ad467783826b33f8f0636b63b06f17932eca984d6ffe8d98c66c180d | At present, the sum of the ages of a father and his son is 43 years. In \( n \) years, the father will be four times the son's age. Determine the possible ages of the father and son. | [
"Father's age: 35, 38, 41; Son's age: 8, 5, 2"
] | [
"At present, the sum of the ages of a father and his son is 43 years. In \\( n \\) years, the father will be four times the son's age. Determine the possible ages of the father and son."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 29
} | 42 | 0 | 0 |
omr-ws-f55eab24c37b349aad00ef8ad67550aea036d27c9654a0a0e11e379435a708d9 | At some point inside a sphere, three mutually perpendicular planes were drawn, dividing the sphere's surface into $8$ curvilinear triangles. These triangles are painted in a checkerboard pattern with black and white. If the radius of the sphere is $r$, calculate the total area painted black on the sphere's surface. | [
"2\\pi r^2"
] | [
"At some point inside a sphere, three mutually perpendicular planes were drawn, dividing the sphere's surface into $8$ curvilinear triangles. These triangles are painted in a checkerboard pattern with black and white. If the radius of the sphere is $r$, calculate the total area painted black on the sphere's surface... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bb62e73bd251479871f8091e9c95d7be7b7c1347ae2eba0cf742b0ddd3dce4ea | At some point inside a sphere, three mutually perpendicular planes were drawn, dividing the sphere's surface into 8 curvilinear triangles. These triangles are painted in a checkerboard pattern with black and white colors. If the radius of the sphere is $r$, calculate the total area painted black on the sphere's surface... | [
"2\\pi r^2"
] | [
"At some point inside a sphere, three mutually perpendicular planes were drawn, dividing the sphere's surface into 8 curvilinear triangles. These triangles are painted in a checkerboard pattern with black and white colors. If the radius of the sphere is $r$, calculate the total area painted black on the sphere's su... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eb68a043e930888cb7d3400190cf10cb5398189ec31fabc207fc1e3a95fdfcfc | At the .05 significance level, can we conclude that college students watch fewer DVDs a month than high school students, given that high school students watched an average of 6.8 DVDs per month, and a random sample of 36 college students watched an average of 6.2 DVDs with a standard deviation of 0.5? | [
"Yes"
] | [
"At the .05 significance level, can we conclude that college students watch fewer DVDs a month than high school students, given that high school students watched an average of 6.8 DVDs per month, and a random sample of 36 college students watched an average of 6.2 DVDs with a standard deviation of 0.5?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-32dbfca77a194fae0b806c53727b8d268b2b5ed472e2e00fee2666d1c33ef5f8 | At the 2014 Mathcounts National Competition, the 224 competitors are seated randomly into 14 rows of 16 seats. What is the expected number of times two members of the New Jersey team are seated in adjacent seats? Express your answer as a common fraction. (Two seats are adjacent if they are consecutive seats in the same... | [
"\\(\\frac{627}{6244}\\)"
] | [
"At the 2014 Mathcounts National Competition, the 224 competitors are seated randomly into 14 rows of 16 seats. What is the expected number of times two members of the New Jersey team are seated in adjacent seats? Express your answer as a common fraction. (Two seats are adjacent if they are consecutive seats in the... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-bb72045788acf7d3ea7fa031cd056cdf058bd8cf98ef9a026bf0c2aec212b366 | At the Amalgamated Institute of Math Excellence, there are 10 clubs and a number of students. For any two different students, there is some club such that exactly one of the two belongs to that club. For any three students, there is some club such that either one of the three belongs to that club, or all three belong t... | [
"1024"
] | [
"At the Amalgamated Institute of Math Excellence, there are 10 clubs and a number of students. For any two different students, there is some club such that exactly one of the two belongs to that club. For any three students, there is some club such that either one of the three belongs to that club, or all three bel... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5dd8c63386ab51664f8c4ad5c4a981b42f73904e6baee714c4f4e581623ebb2a | At the AoPS Carnival, there is a "Weighted Dice" game show featuring two 6-sided dice. For each integer $1 \leq i \leq 6$, Die A has a probability of $\frac{i}{21}$ of rolling the number $i$, while Die B has a probability of $\frac{7-i}{21}$ of rolling $i$. During one session, the host randomly chooses a die, rolls it ... | [
"156"
] | [
"At the AoPS Carnival, there is a \"Weighted Dice\" game show featuring two 6-sided dice. For each integer $1 \\leq i \\leq 6$, Die A has a probability of $\\frac{i}{21}$ of rolling the number $i$, while Die B has a probability of $\\frac{7-i}{21}$ of rolling $i$. During one session, the host randomly chooses a die... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-1287225da71fc34029aba2262d656b6f9e2e8a658c634088fbe9129a46b6cfd4 | At the Arsenal Soccer Center, 60% of the players use orange cleats as their main shoes. If 25 players are randomly surveyed, find the probability that exactly 10 do NOT use orange cleats as their main shoes. Round to 4 decimal places. | [
"0.1612"
] | [
"At the Arsenal Soccer Center, 60% of the players use orange cleats as their main shoes. If 25 players are randomly surveyed, find the probability that exactly 10 do NOT use orange cleats as their main shoes. Round to 4 decimal places."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-721be11e2367b1cd1b0158749fcb36b5dfca5c3faa8d08a470f070d6c325815d | At the Boba Math Tournament, the Blackberry Milk Team has answered $4$ out of the first $10$ questions on the Boba Round correctly. If they answer all $p$ remaining questions correctly, they will have answered exactly $\frac{9p}{5}\%$ of the questions correctly in total. How many questions are on the Boba Round? | [
"60"
] | [
"At the Boba Math Tournament, the Blackberry Milk Team has answered $4$ out of the first $10$ questions on the Boba Round correctly. If they answer all $p$ remaining questions correctly, they will have answered exactly $\\frac{9p}{5}\\%$ of the questions correctly in total. How many questions are on the Boba Round?... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-41757d9512097c3a8d8089299603baed2a9d042436d8f0124216e2433f6179cf | At the Gooddog Obedience School, dogs can learn to do three tricks: sit, stay, and roll over. Of the dogs at the school: \begin{tabular}{l@{\qquad}l} 50 dogs can sit & 17 dogs can sit and stay \\ 29 dogs can stay & 12 dogs can stay and roll over \\ 34 dogs can roll over & 18 dogs can sit and roll over \\ 9 dogs can do ... | [
"84"
] | [
"At the Gooddog Obedience School, dogs can learn to do three tricks: sit, stay, and roll over. Of the dogs at the school: \\begin{tabular}{l@{\\qquad}l}\n50 dogs can sit & 17 dogs can sit and stay \\\\\n29 dogs can stay & 12 dogs can stay and roll over \\\\\n34 dogs can roll over & 18 dogs can sit and roll over \\\... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-26fcfaec90bfcf61d21442bdd129178d4abe669da05247ef4e2d314d08667e16 | At the Hardey Fitness Center, the management did a survey of their membership. The average age of the female members was 40 years old. The average age of the male members was 25 years old. The average age of the entire membership was 30 years old. What is the ratio of the female to male members? Express your answer as ... | [
"\\frac{1}{2}"
] | [
"At the Hardey Fitness Center, the management did a survey of their membership. The average age of the female members was 40 years old. The average age of the male members was 25 years old. The average age of the entire membership was 30 years old. What is the ratio of the female to male members? Express your answe... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-68179ceaba70031a24ff015f338b8c69cc3c0174e29541da870d98a46d6469df | At the IMO, there are $n$ students, and $m$ pairs of these students are enemies. Suppose it is possible to divide the students into $k$ rooms so that there are at most $\frac{m}{k}$ pairs of enemies that are placed in the same room as each other. If $n=100$, $m=1000$, and $k=10$, find the maximum number of pairs of ene... | [
"100"
] | [
"At the IMO, there are $n$ students, and $m$ pairs of these students are enemies. Suppose it is possible to divide the students into $k$ rooms so that there are at most $\\frac{m}{k}$ pairs of enemies that are placed in the same room as each other. If $n=100$, $m=1000$, and $k=10$, find the maximum number of pairs ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-531e57469371812a20218be5a3a58fd593ca86fc257160c3448fb01a01288c28 | At the Ideias theater, the stage is a cube with each side measuring 6 meters. On each edge, there are rings every 2 meters. A triangular backdrop is made using fabric attached to these rings. Calculate the area of the fabric used for the triangular backdrop. | [
"\\( 14\\sqrt{3} \\)"
] | [
"At the Ideias theater, the stage is a cube with each side measuring 6 meters. On each edge, there are rings every 2 meters. A triangular backdrop is made using fabric attached to these rings. Calculate the area of the fabric used for the triangular backdrop."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d458956644cb54c46d714941030d3a0135e42655996b1dbf9f071aca2c8cd0e | At the Intergalactic Math Olympiad, there are 6 problems, each scored from 0 to 7. The contestant's score is the product of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks are equal. There are $8^6$ participants, and no two get the ... | [
"1"
] | [
"At the Intergalactic Math Olympiad, there are 6 problems, each scored from 0 to 7. The contestant's score is the product of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks are equal. There are $8^6$ participants, and no two get... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-67ab1e8723cb00b185043e3a8934c5d71bbc7e9a3167cd4a3502b12562e09bdb | At the MP Donut Hole Factory, Niraek, Theo, and Akshaj are coating spherical donut holes in powdered sugar. Niraek's donut holes have radius 6 mm, Theo's donut holes have radius 8 mm, and Akshaj's donut holes have radius 10 mm. All three workers coat the surface of the donut holes at the same rate and start at the same... | [
"400"
] | [
"At the MP Donut Hole Factory, Niraek, Theo, and Akshaj are coating spherical donut holes in powdered sugar. Niraek's donut holes have radius 6 mm, Theo's donut holes have radius 8 mm, and Akshaj's donut holes have radius 10 mm. All three workers coat the surface of the donut holes at the same rate and start at t... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-18b4276b973e7f92ba75cbd5a5802699e0faab33144dcc5be9060a3c4efcf8dd | At the Olympic Games, a number of medals are awarded each day as follows: - On the 1st day, one medal plus 1/13 of the remaining medals are awarded. - On the 2nd day, two medals plus 1/13 of the new remaining medals are awarded. - This pattern continues each day until all medals are awarded. Determine how many days the... | [
"12"
] | [
"At the Olympic Games, a number of medals are awarded each day as follows:\n- On the 1st day, one medal plus 1/13 of the remaining medals are awarded.\n- On the 2nd day, two medals plus 1/13 of the new remaining medals are awarded.\n- This pattern continues each day until all medals are awarded.\nDetermine how many... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-e1252913584abe78d01933c63738737647eaf8be124885a70eee2bf7028a6a76 | At the Outdian Institute of Technology, there are $511$ maths buildings, called Maths Building $1$, Maths Building $2$, etc. There exists pathways between these buildings such that for any positive integer $k \le 255$, there exists a pathway between Maths Building $k$ and Maths Building $2k$ and there exists a pathway ... | [
"510"
] | [
"At the Outdian Institute of Technology, there are $511$ maths buildings, called Maths Building $1$, Maths Building $2$, etc. There exists pathways between these buildings such that for any positive integer $k \\le 255$, there exists a pathway between Maths Building $k$ and Maths Building $2k$ and there exists a pa... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-036dfce016ff5ef199e7e394a3b904f3c6f4749767cd46f3eaa1bfb3f770773b | At the Polyglot Mathematics Olympiad, each participant speaks some of 4 languages. It is known that any two participants can communicate in a common language. If there are 100 participants, find the minimum number of participants who speak the most commonly spoken language. | [
"51"
] | [
"At the Polyglot Mathematics Olympiad, each participant speaks some of 4 languages. It is known that any two participants can communicate in a common language. If there are 100 participants, find the minimum number of participants who speak the most commonly spoken language."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-356813efae0f22feab806916107de9ef19637108eb11618adc0ae6796933a509 | At the Rice Mathematics Tournament, 80% of contestants wear blue jeans, 70% wear tennis shoes, and 80% of those who wear blue jeans also wear tennis shoes. What fraction of people wearing tennis shoes are wearing blue jeans? | [
"\\(\\frac{32}{35}\\)"
] | [
"At the Rice Mathematics Tournament, 80% of contestants wear blue jeans, 70% wear tennis shoes, and 80% of those who wear blue jeans also wear tennis shoes. What fraction of people wearing tennis shoes are wearing blue jeans?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-6a4f2681d10275cd8af80c85c3c92b02a7fd2dc5cac293fa0a9657d44f81813d | At the South Pole, ice grows in right triangular structures with an area of 60. If the legs of each right triangle $\overline{AB}$ and $\overline{AC}$ are integer lengths, how many distinct possible lengths are there for the hypotenuse $\overline{BC}$? | [
"1"
] | [
"At the South Pole, ice grows in right triangular structures with an area of 60. If the legs of each right triangle $\\overline{AB}$ and $\\overline{AC}$ are integer lengths, how many distinct possible lengths are there for the hypotenuse $\\overline{BC}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-87ade1473853b435e388dc362b88268076ca1dc3a342d756138c259c6789a62a | At the Winter Olympics, Belarus and the Czech Republic won $B$ and $C$ medals respectively, totaling 10 medals. When the number of medals the US won is divided by $B + C$, it leaves a remainder of 3, and when divided by $BC$, it leaves a remainder of 2. What is the least possible number of medals the U.S. could have wo... | [
"23"
] | [
"At the Winter Olympics, Belarus and the Czech Republic won $B$ and $C$ medals respectively, totaling 10 medals. When the number of medals the US won is divided by $B + C$, it leaves a remainder of 3, and when divided by $BC$, it leaves a remainder of 2. What is the least possible number of medals the U.S. could ha... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-5a728aa7ff53adc9b7b77b49bb9956e3931256d9501132f6444b917b0ecd4877 | At the base of a regular pyramid \(PABCD\) lies a square \(ABCD\) with side length 6. The section of the pyramid passes through vertex \(B\) and the midpoint of edge \(PD\) and is perpendicular to this edge. Find the cross-sectional area of the pyramid. | [
"\\(12\\sqrt{3}\\)"
] | [
"At the base of a regular pyramid \\(PABCD\\) lies a square \\(ABCD\\) with side length 6. The section of the pyramid passes through vertex \\(B\\) and the midpoint of edge \\(PD\\) and is perpendicular to this edge. Find the cross-sectional area of the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 7,
"tir": 17
} | 60 | 0 | 0 |
omr-ws-66d03fa148f46f5a269710dc75ea7f7f93d47489566ad8d9c2e66a838d76b571 | At the base of the prism \(ABCDA_1B_1C_1D_1\) lies the rectangle \(ABCD\). The acute angles \(D_1DA\) and \(D_1DC\) are equal to each other, the angle between the edge \(DD_1\) and the plane of the base of the prism is \(\arccos(\theta)\), and \(CD = 5\sqrt{6}\). All the faces of the prism are tangent to some sphere. F... | [
"\\(\\arccos \\frac{1}{5}\\)"
] | [
"At the base of the prism \\(ABCDA_1B_1C_1D_1\\) lies the rectangle \\(ABCD\\). The acute angles \\(D_1DA\\) and \\(D_1DC\\) are equal to each other, the angle between the edge \\(DD_1\\) and the plane of the base of the prism is \\(\\arccos(\\theta)\\), and \\(CD = 5\\sqrt{6}\\). All the faces of the prism are tan... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.