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-c69cbad9be73e59bf69ae1777d141cc6a2e24080f08a44546cdc27f2f7fe97ff | A person leaves the house between 4:00 and 5:00 P.M. and returns between 7:00 and 8:00 P.M. He notices that the positions of the hour hand and the minute hand have exactly interchanged. What time did he leave his house? | [
"4:\\!36\\!\\frac{12}{13}"
] | [
"A person leaves the house between 4:00 and 5:00 P.M. and returns between 7:00 and 8:00 P.M. He notices that the positions of the hour hand and the minute hand have exactly interchanged. What time did he leave his house?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6422b97712a562958789970beb316ce0fca563b5804d557615ecebe98db12b96 | A person left home between 4 p.m. and 5 p.m. and returned between 5 p.m. and 6 p.m., finding that the positions of the hour and minute hands of his watch had exactly exchanged places. At what time did he leave home? | [
"4:26:51.18"
] | [
"A person left home between 4 p.m. and 5 p.m. and returned between 5 p.m. and 6 p.m., finding that the positions of the hour and minute hands of his watch had exactly exchanged places. At what time did he leave home?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-4c00c6830fdb7aebbab998f537205a0d112b352bfdc3bce68d81af36f2975a4b | A person lives in a town $B$ connected by a railway with towns $A$ and $C$. The trains running between towns $A$ and $C$ stop at town $B$. A train leaves town $B$ in each direction every hour. A person arrives at the station at a random moment and takes the first train to arrive at the station. How must the time-table ... | [
"The trains going in the direction $AC$ must arrive 10 minutes after the departure of the train going in the direction $CA$"
] | [
"A person lives in a town $B$ connected by a railway with towns $A$ and $C$. The trains running between towns $A$ and $C$ stop at town $B$. A train leaves town $B$ in each direction every hour. A person arrives at the station at a random moment and takes the first train to arrive at the station. How must the time-t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d1511589fcd636f21fb4e8b8748f162dba5ee6b8f2f942ddacaf420616e45b7 | A person maintains an average of 36 mph driving from $P$ to $Q$. On the return trip over the same route, he averages $k$ mph, where $k$ is an integer and $1 \le k \le 100$. Determine the sum of all integers $k$ for which his overall average for the entire trip is an integer. | [
"198"
] | [
"A person maintains an average of 36 mph driving from $P$ to $Q$. On the return trip over the same route, he averages $k$ mph, where $k$ is an integer and $1 \\le k \\le 100$. Determine the sum of all integers $k$ for which his overall average for the entire trip is an integer."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3bc7679f53ad75f12dbd8d433f27f2ef6bee5f11d07bde2b44f61a848b026eee | A person makes a $2.50 profit on every small painting and a $4.00 profit on every large painting. He generally sells 5 large paintings for every 2 small paintings. How many of each painting does he need to sell to make a total profit of $10,000? | [
"800 small paintings, 2000 large paintings"
] | [
"A person makes a $2.50 profit on every small painting and a $4.00 profit on every large painting. He generally sells 5 large paintings for every 2 small paintings. How many of each painting does he need to sell to make a total profit of $10,000?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-d03cce2213f18e1c517795ff2cf599024c1633c51a6fedac58860b0a1e82f453 | A person moves in the $x-y$ plane along points with integer coordinates $x$ and $y$. When at a point $(x,y)$, she moves according to the following rules: - If $x+y$ is even, she moves to either $(x+1,y)$ or $(x+1,y+1)$. - If $x+y$ is odd, she moves to either $(x,y+1)$ or $(x+1,y+1)$. How many distinct paths can she tak... | [
"\\(\\binom{11}{5}\\)"
] | [
"A person moves in the $x-y$ plane along points with integer coordinates $x$ and $y$. When at a point $(x,y)$, she moves according to the following rules:\n- If $x+y$ is even, she moves to either $(x+1,y)$ or $(x+1,y+1)$.\n- If $x+y$ is odd, she moves to either $(x,y+1)$ or $(x+1,y+1)$.\nHow many distinct paths can... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-73ad7ba88c6b872cf39a16a52992c27e25c528ecfb0b60d8b1e0a9afb5390a14 | A person moves on the real number line starting at \( x = 0 \). Each coin toss results in a move one unit right for heads or one unit left for tails. After 15 coin tosses, what is the probability that the person is at \( x = 9 \) at some point during the sequence of tosses? | [
"\\(\\frac{697}{32768}\\)"
] | [
"A person moves on the real number line starting at \\( x = 0 \\). Each coin toss results in a move one unit right for heads or one unit left for tails. After 15 coin tosses, what is the probability that the person is at \\( x = 9 \\) at some point during the sequence of tosses?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 15
} | 38 | 0 | 0 |
omr-ws-56824385bc0892d96791d828a6e20e1d0d8e6ecef12638e4b03a2acc5a33d5ad | A person moves on the real number line starting at \( x = 0 \). Each coin toss results in a move one unit right for heads or one unit left for tails. After 30 coin tosses, what is the probability that the person is at \( x = 19 \) at some point during the sequence of tosses? Is there a recursive relation or method to s... | [
"\\dfrac{174437}{536870912}"
] | [
"A person moves on the real number line starting at \\( x = 0 \\). Each coin toss results in a move one unit right for heads or one unit left for tails. After 30 coin tosses, what is the probability that the person is at \\( x = 19 \\) at some point during the sequence of tosses? Is there a recursive relation or me... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-50bb4f883488c89fb0684706a4f8e9d33af27f666d1c25bf97a9d5546da0e54d | A person needs to buy exactly 106 sweets. Sweets are sold in packs of 5 for $6 each or packs of 7 for $7 each. What is the lowest cost at which the person can buy the sweets? | [
"$109$"
] | [
"A person needs to buy exactly 106 sweets. Sweets are sold in packs of 5 for $6 each or packs of 7 for $7 each. What is the lowest cost at which the person can buy the sweets?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-0f29ae7e501275f93970ec5412c73f9d7a3ac476248db9a9cdccea139267ac39 | A person observes the angle of elevation of a peak from a point A on the ground to be $\alpha$. He goes up an incline of inclination $\beta$ (where $\beta < \alpha$) towards the top of the peak and observes that the angle of elevation of the peak is now $\gamma$. If B is the second place of observation and AB is $y$ me... | [
"\\(\\frac{y(\\sin\\beta-\\cos\\beta\\tan\\gamma)}{(1-\\cot\\alpha\\tan\\gamma)}\\)"
] | [
"A person observes the angle of elevation of a peak from a point A on the ground to be $\\alpha$. He goes up an incline of inclination $\\beta$ (where $\\beta < \\alpha$) towards the top of the peak and observes that the angle of elevation of the peak is now $\\gamma$. If B is the second place of observation and AB... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c803ad269742967c01f842940147646c987e66c17bd0dd4c5578e1df3095fda7 | A person of height 1.7 meters walks up a path with a constant gradient. After walking 20 meters, they walk an additional 5 meters and notice that the top of a tall billboard aligns horizontally with the top of a tall building. They then walk another 10 meters and notice that the top half of the building is visible abov... | [
"13.6"
] | [
"A person of height 1.7 meters walks up a path with a constant gradient. After walking 20 meters, they walk an additional 5 meters and notice that the top of a tall billboard aligns horizontally with the top of a tall building. They then walk another 10 meters and notice that the top half of the building is visible... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-6b301e6184ff6c0eac87d5df3ec42d20eb786cb74094184462628f3a2c613dc2 | A person picks \( k \) distinct numbers from the first \( n \) natural numbers. A second person picks another set of \( k \) distinct numbers from the remaining numbers. What is the minimum possible sum of the numbers the first person picked such that, whatever the numbers the second person picks, the sum of his set of... | [
"\\(\\sigma_A = \\frac{k(2n - k + 1)}{2} - q(n - k) - \\text{min}(r - k, \\lfloor r/2 \\rfloor)\\)"
] | [
"A person picks \\( k \\) distinct numbers from the first \\( n \\) natural numbers. A second person picks another set of \\( k \\) distinct numbers from the remaining numbers. What is the minimum possible sum of the numbers the first person picked such that, whatever the numbers the second person picks, the sum of... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7fdc5f1403b91cbc0bb065637e3ae72ac963d863aaa49c18b59c500e52b2a0e6 | A person picks \( n \) different prime numbers less than 150 and finds that they form an arithmetic sequence. What is the greatest possible value of \( n \)? | [
"5"
] | [
"A person picks \\( n \\) different prime numbers less than 150 and finds that they form an arithmetic sequence. What is the greatest possible value of \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3a85de9b2de31180ef29e4bd359fe1714559445c7fdcb5d93735a21678dbb563 | A person reads a book. On the first day, he reads \( n \) pages in \( t \) minutes, where \( n \) and \( t \) are positive integers. The next day he reads \( n + 1 \) pages in \( t + 1 \) minutes, and this pattern continues until he finishes the 374-page book in 319 minutes. Find \( n + t \). | [
"53"
] | [
"A person reads a book. On the first day, he reads \\( n \\) pages in \\( t \\) minutes, where \\( n \\) and \\( t \\) are positive integers. The next day he reads \\( n + 1 \\) pages in \\( t + 1 \\) minutes, and this pattern continues until he finishes the 374-page book in 319 minutes. Find \\( n + t \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-7b369dd148a7dc035185986f8cc3bcb3a69eb7f053bbc5c92a087d75ddbd3e2b | A person rolls a die, tosses a coin, and draws a card from an ordinary deck. He receives $3 for each point up on the die, $10 for a head and $0 for a tail, and $1 for each spot on the card (jack=11, queen=12, king=13). If we assume that the three random variables involved are independent and uniformly distributed, comp... | [
"65.25"
] | [
"A person rolls a die, tosses a coin, and draws a card from an ordinary deck. He receives $3 for each point up on the die, $10 for a head and $0 for a tail, and $1 for each spot on the card (jack=11, queen=12, king=13). If we assume that the three random variables involved are independent and uniformly distributed,... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 2
} | 11 | 0 | 0 |
omr-ws-895648b3e66b492f886f88867c64e1b11280666050dae825f5596167d8598b7e | A person rolls a fair six-sided die four times. Determine the probability that none of the numbers drawn in the first two throws will match any of the numbers drawn in the last two throws. | [
"\\(\\frac{10}{27}\\)"
] | [
"A person rolls a fair six-sided die four times. Determine the probability that none of the numbers drawn in the first two throws will match any of the numbers drawn in the last two throws."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dcf17cbc03a113cea3c888251cb889ad04eb483f37ef472385c436cf9d5a27ee | A person shuffles 2000 cards and repeatedly places the first card on the table, the second card at the bottom of the stack, the third card on the table, the fourth card at the bottom of the stack, and so on, until all cards are on the table. If the cards on the table are in ascending order from left to right, how many ... | [
"1535"
] | [
"A person shuffles 2000 cards and repeatedly places the first card on the table, the second card at the bottom of the stack, the third card on the table, the fourth card at the bottom of the stack, and so on, until all cards are on the table. If the cards on the table are in ascending order from left to right, how ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-946bc8a7adc3d3a865e40be10e1a5dfce69cf40df8593e8513fd1639e78ff042 | A person standing at a point P on a flat plane starts walking. At each step, they walk exactly 1 foot in one of the directions: North, South, East, or West. After 6 steps, they return to the original position P. How many distinct paths can the person take? | [
"400"
] | [
"A person standing at a point P on a flat plane starts walking. At each step, they walk exactly 1 foot in one of the directions: North, South, East, or West. After 6 steps, they return to the original position P. How many distinct paths can the person take?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-090e1257a792ae1f118437306d06d091b6bec5831bb6261a75242abcd22022f0 | A person standing at a point \( P \) on a flat plane starts walking. At each step, they walk exactly 1 foot in one of the directions North, South, East, or West. After 6 steps, the person returns to the original position \( P \). How many distinct paths can the person take to return to \( P \)? | [
"400"
] | [
"A person standing at a point \\( P \\) on a flat plane starts walking. At each step, they walk exactly 1 foot in one of the directions North, South, East, or West. After 6 steps, the person returns to the original position \\( P \\). How many distinct paths can the person take to return to \\( P \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-9eb6976f4894a5a3125fa1cb69438c7f8bebcfeefb690904ce4c18f2423d4bda | A person standing at point A finds the angle of elevation of a nearby tower to be 60 degrees. From A, the person walks 100 feet to point B and then to point C such that angle ABC = 120 degrees. If the angles of elevation of the tower at both B and C are also 60 degrees, what is the height of the tower? | [
"\\( 100\\sqrt{3} \\) feet"
] | [
"A person standing at point A finds the angle of elevation of a nearby tower to be 60 degrees. From A, the person walks 100 feet to point B and then to point C such that angle ABC = 120 degrees. If the angles of elevation of the tower at both B and C are also 60 degrees, what is the height of the tower?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 2
} | 19 | 0 | 0 |
omr-ws-5cb1e6688ac5802d319f10bc924b61531b6d7727cf19393843755b62a025e36a | A person standing on point P on a flat plane starts walking. At each step, they walk exactly 1 foot in one of the directions North, South, East, or West. After 6 steps, they return to the original position P. How many distinct paths can the person take? | [
"400"
] | [
"A person standing on point P on a flat plane starts walking. At each step, they walk exactly 1 foot in one of the directions North, South, East, or West. After 6 steps, they return to the original position P. How many distinct paths can the person take?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-1871cee1ba577aa49d5ad05692a45f2b1979894d3fbc384b7d11185782d580f8 | A person stands at 0 on the number line. At each second, the person can move left or right with equal probability, unless the person is at 0, in which case the person must move right. Find the expected number of seconds needed for the person to reach the number 5. | [
"25"
] | [
"A person stands at 0 on the number line. At each second, the person can move left or right with equal probability, unless the person is at 0, in which case the person must move right. Find the expected number of seconds needed for the person to reach the number 5."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-9810bca4280690b8235e2df7db1c6d14bd3ff0ceb82c64acddad508670c6d8fa | A person starts 5 meters west of point $A$ on a north-south line. Each coin toss determines their movement: heads moves them 1 meter east, and tails moves them 1 meter north. They stop when they reach the line. Let $p_{n}$ be the probability that the person stops at a point $n$ meters north of $A$. Find the value of $n... | [
"\\( n = 3 \\) or \\( n = 4 \\)"
] | [
"A person starts 5 meters west of point $A$ on a north-south line. Each coin toss determines their movement: heads moves them 1 meter east, and tails moves them 1 meter north. They stop when they reach the line. Let $p_{n}$ be the probability that the person stops at a point $n$ meters north of $A$. Find the value ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 3,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-e43ed6a12af6370f0f946f36770d94c30c624d4bf46994285366288c1d645593 | A person starts a trip at a speed of 30 km/hr and reaches the midpoint. With what speed should the person traverse the remaining distance to achieve an average speed of 60 km/hr for the entire trip? | [
"\\infty"
] | [
"A person starts a trip at a speed of 30 km/hr and reaches the midpoint. With what speed should the person traverse the remaining distance to achieve an average speed of 60 km/hr for the entire trip?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5bd3128df082f82d1266ad773ff8121269b3d96b0d68563b7b2d8b854fd5be1c | A person starts at the origin and walks along the positive x-axis with speed $\alpha$ at time $t=0$. His dog is initially at $(a,b)$ at time $t=0$ and starts running towards him constantly with speed $\beta$. Find the amount of time needed for the dog to reach the person. | [
"\\(\\frac{\\sqrt{a^2 + b^2}}{\\beta}\\)"
] | [
"A person starts at the origin and walks along the positive x-axis with speed $\\alpha$ at time $t=0$. His dog is initially at $(a,b)$ at time $t=0$ and starts running towards him constantly with speed $\\beta$. Find the amount of time needed for the dog to reach the person."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7d2d80eacfbc7b2832d539250bf8ad5b3a584528f776b04e816f5fb80ce8ba44 | A person starts at the origin and walks along the positive x-axis with speed $\alpha$ at time $t=0$. His dog is initially at $(a,b)$ at time $t=0$ and starts running towards him constantly with speed $\beta$. Find the equation of the path the dog runs before reaching the person. | [
"y = \\frac{b}{2} \\left[ \\left( \\frac{x}{a} \\right)^{1 - \\frac{\\alpha}{\\beta}} - \\left( \\frac{x}{a} \\right)^{1 + \\frac{\\alpha}{\\beta}} \\right]"
] | [
"A person starts at the origin and walks along the positive x-axis with speed $\\alpha$ at time $t=0$. His dog is initially at $(a,b)$ at time $t=0$ and starts running towards him constantly with speed $\\beta$. Find the equation of the path the dog runs before reaching the person."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-145512296035a9f86ee56b3d82e2d84cb1836028bc044465a55e79295e0a4cb1 | A person starts at vertex 1 of a square with vertices labeled as follows: 1 2 4 3 On each turn, the person can move to an adjacent vertex and add the number at that vertex to their total "sum". The initial vertex does not count towards the sum (starting sum is 0). How many unique paths can the person take such that the... | [
"167"
] | [
"A person starts at vertex 1 of a square with vertices labeled as follows:\n\n1 2\n4 3\n\nOn each turn, the person can move to an adjacent vertex and add the number at that vertex to their total \"sum\". The initial vertex does not count towards the sum (starting sum is 0). How many unique paths can the person take... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-1b15e021ccd79727ff418a20c6aad800f1608114ea2a3a4828a90775f2129727 | A person starts from the origin in the $xy$-plane and takes steps of one unit along the positive $x$-axis or $y$-axis. Find the total number of ways the person can reach the point $A(9, 6)$ while avoiding the points $P(3, 3)$ and $Q(6, 4)$. | [
"2025"
] | [
"A person starts from the origin in the $xy$-plane and takes steps of one unit along the positive $x$-axis or $y$-axis. Find the total number of ways the person can reach the point $A(9, 6)$ while avoiding the points $P(3, 3)$ and $Q(6, 4)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-96421cdecc6bce568bb0318162b143c1f359065a9de4df4c077720fe608a1d04 | A person starts walking on a 2-D grid, flipping two coins each time to decide which way to go (left, right, up, or down with equal probability). What is the probability that the person reaches the line x = -3 before reaching the line y = 5? | [
"\\dfrac{5}{8}"
] | [
"A person starts walking on a 2-D grid, flipping two coins each time to decide which way to go (left, right, up, or down with equal probability). What is the probability that the person reaches the line x = -3 before reaching the line y = 5?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-2f7a4ab36bf18a92aa522ea07f0bf4d55607e4c270c088834d5330a5f3dffcca | A person starts with one coin and tosses it repeatedly. If the coin lands right-side up, they get a new coin; if it lands wrong-side up, they lose the coin they tossed. The person continues tossing coins until they have no coins left. What is the expected number of coin tosses until all coins are lost? | [
"\\infty"
] | [
"A person starts with one coin and tosses it repeatedly. If the coin lands right-side up, they get a new coin; if it lands wrong-side up, they lose the coin they tossed. The person continues tossing coins until they have no coins left. What is the expected number of coin tosses until all coins are lost?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-7c850ab33fdf3935c1217a63471a23de9187d5542b89afad3669af2f0c9a34b1 | A person thinks of a specific card and picks a card from a deck of 52 cards, repeating this process 52 times. What is the probability that the card he thought of was chosen at least once? | [
"\\(0.636\\)"
] | [
"A person thinks of a specific card and picks a card from a deck of 52 cards, repeating this process 52 times. What is the probability that the card he thought of was chosen at least once?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-65009bb9168d85331c6029051559393e0d48d47d7bf57c78d0dcb1bb237268a3 | A person travels $2c$ distance north and observes two objects such that the difference in their angles of depression is $\alpha$. The person then travels another $2c$ distance north and observes that the difference in their angles of depression is now $\beta$. If both objects lie along the x-axis (east direction), find... | [
"\\dfrac{6c \\tan \\alpha \\tan \\beta}{2 \\tan \\alpha - \\tan \\beta}"
] | [
"A person travels $2c$ distance north and observes two objects such that the difference in their angles of depression is $\\alpha$. The person then travels another $2c$ distance north and observes that the difference in their angles of depression is now $\\beta$. If both objects lie along the x-axis (east direction... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-3a6448f9425254335912050bf2a6b42827c106d4e205d8a4b5ffa354b7444438 | A person travels from hill top A to hill top B, going downhill at 72 kmph, on plains at 63 kmph, and uphill at 56 kmph, taking a total of 4 hours. On the return journey from B to A, with the same speeds, the journey takes 4 hours and 40 minutes. Find the distance between A and B. | [
"273",
"273 km"
] | [
"A person travels from hill top A to hill top B, going downhill at 72 kmph, on plains at 63 kmph, and uphill at 56 kmph, taking a total of 4 hours. On the return journey from B to A, with the same speeds, the journey takes 4 hours and 40 minutes. Find the distance between A and B."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 16
} | 48 | 0 | 0 |
omr-ws-b7f7cf38f9a9aaaa6a4f2aef162412bf8d2cedb75f25664557417e1ef36ddc2a | A person walking along a straight road observes that at two consecutive milestones (1 mile apart), the angle of elevation of a hill in front of him are \(30^\circ\) and \(75^\circ\). Find the height of the hill. | [
"\\(\\frac{\\sqrt{3}+1}{4}\\)"
] | [
"A person walking along a straight road observes that at two consecutive milestones (1 mile apart), the angle of elevation of a hill in front of him are \\(30^\\circ\\) and \\(75^\\circ\\). Find the height of the hill."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-83c52eee9e8bd7342379f6663ea803d615a2cbf6c7671c67ae45b5171f30bceb | A person walks along a straight sidewalk at a constant speed of 6 feet/second. A spotlight is trained on the person from across the street, which is 20 feet wide. How fast is the spotlight turning (in radians/second) when the distance between the person and the spotlight is 31 feet? | [
"\\(\\frac{120}{961}\\)"
] | [
"A person walks along a straight sidewalk at a constant speed of 6 feet/second. A spotlight is trained on the person from across the street, which is 20 feet wide. How fast is the spotlight turning (in radians/second) when the distance between the person and the spotlight is 31 feet?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-4039ea25ab38a023ecf81210078e3540649bbf3761160b612a96320357841e33 | A person walks from point A to point B, rotating their map by an angle $\alpha_1$ after each segment. The total rotation angle $\alpha$ is less than $180^\circ$. If the total distance traveled is denoted by $D$, find the maximum possible value of $D \cdot \cos\frac{\alpha}{2}$ given that $AB = 10$. | [
"10"
] | [
"A person walks from point A to point B, rotating their map by an angle $\\alpha_1$ after each segment. The total rotation angle $\\alpha$ is less than $180^\\circ$. If the total distance traveled is denoted by $D$, find the maximum possible value of $D \\cdot \\cos\\frac{\\alpha}{2}$ given that $AB = 10$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18c637d87e1cd1b531be783b12e4d68182df48310648876e8916475893249030 | A person walks one mile, veers 24 degrees to the right, and walks another mile, repeating this process. How far will the person walk before returning to the original position? | [
"15 miles"
] | [
"A person walks one mile, veers 24 degrees to the right, and walks another mile, repeating this process. How far will the person walk before returning to the original position?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-7b4f6f4b9acb594ff524374d7e615953503faa0804a098484f47e4af4f69230a | A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged places. When did he go out? | [
"4:\\!26\\!\\frac{122}{143}"
] | [
"A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged places. When did he go out?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94dbaa0d0f432a7e61dd99c3cd8b2b5caa4841263f7c041e5e9d36b1f5e46e91 | A person wins $10 each time he rolls a 5 or a 6, and loses $4 each time he rolls a 1, 2, 3, or 4. If he plays the die roll game 30 times, what is the probability that he wins $30 or more? | [
"0.423"
] | [
"A person wins $10 each time he rolls a 5 or a 6, and loses $4 each time he rolls a 1, 2, 3, or 4. If he plays the die roll game 30 times, what is the probability that he wins $30 or more?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-502022ae9bbe8fd049b54e701729bee93c647c0a5f085c43ba6628f2ba8ee4c9 | A person with $ \$256 $ makes $ 8 $ bets and wins exactly four times. The wins and losses occur in random order. If each wager is for half of the money she has at the time of the bet, how much will she win or lose? | [
"81"
] | [
"A person with $ \\$256 $ makes $ 8 $ bets and wins exactly four times. The wins and losses occur in random order. If each wager is for half of the money she has at the time of the bet, how much will she win or lose?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b3549550dd6a86735aa03b2a1447f30a35de8c62b25e897479cc8fd7a0df83cf | A person writes a non-zero digit (1 through 9) on a blackboard, then continues to write another non-zero digit at the right until there are $10^6$ non-zero digits on the blackboard. Let $N$ be the number of times a perfect square occurs as a substring of the sequence of digits. Find the maximum possible value of $N$. | [
"1500000"
] | [
"A person writes a non-zero digit (1 through 9) on a blackboard, then continues to write another non-zero digit at the right until there are $10^6$ non-zero digits on the blackboard. Let $N$ be the number of times a perfect square occurs as a substring of the sequence of digits. Find the maximum possible value of $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-63d48bcf5482733a3d338d172fba6b52d80c88de47f9b8eee762b5c746cfdd08 | A person's age in 1962 was one more than the sum of the digits of the year of his birth. How old is he? | [
"23"
] | [
"A person's age in 1962 was one more than the sum of the digits of the year of his birth. How old is he?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-687c650e8bc3b6af19ec1785dc48d2cb4ef12c7730fb6180357ee92359ba2131 | A person's base area is approximately \(25^2\) square centimeters. In an area of \(30^2\) square centimeters, 30 raindrops will be dropped in one second. What is the probability that the person will dodge all the raindrops in one second? | [
"\\left(\\dfrac{11}{36}\\right)^{30}"
] | [
"A person's base area is approximately \\(25^2\\) square centimeters. In an area of \\(30^2\\) square centimeters, 30 raindrops will be dropped in one second. What is the probability that the person will dodge all the raindrops in one second?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1266626621ccc0e36c5d0c912aeccc33e6005fa9a863c283ef813648e722241 | A person, John Doe, gets infected with a flu virus and becomes contagious after one week. During this week, John Doe meets 10 people, and each person has a 10% chance of getting infected by John Doe. Let \( t = 0 \) be the time when the first person gets infected by John Doe. Define the following: - \( E(t) \): the exp... | [
"\\( E(t) = 1 \\) for \\( t = 1, 2, 3, \\ldots, 12 \\)"
] | [
"A person, John Doe, gets infected with a flu virus and becomes contagious after one week. During this week, John Doe meets 10 people, and each person has a 10% chance of getting infected by John Doe. Let \\( t = 0 \\) be the time when the first person gets infected by John Doe. Define the following:\n- \\( E(t) \\... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-e7867198934b0994c75edf1e2ddd8f6507acc712fd2b4d5d300dd395c4002618 | A person’s age when he died is \(\frac{1}{31}\) of the year he was born in. Given that he was alive in 1890, what birthday would this person have celebrated in 1890? | [
"30"
] | [
"A person’s age when he died is \\(\\frac{1}{31}\\) of the year he was born in. Given that he was alive in 1890, what birthday would this person have celebrated in 1890?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-2768a703dad21f3c2c2ed86f37b45c0976ba1eeab2b1b6c3352b22856c683008 | A pet store has 15 puppies, 6 kittens, and 8 hamsters. Alice, Bob, and Charlie each want to buy a pet. For variety, they each want a different kind of pet. How many ways can Alice, Bob, and Charlie buy pets and leave the store satisfied? | [
"4320"
] | [
"A pet store has 15 puppies, 6 kittens, and 8 hamsters. Alice, Bob, and Charlie each want to buy a pet. For variety, they each want a different kind of pet. How many ways can Alice, Bob, and Charlie buy pets and leave the store satisfied?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e2223c71a397a883effb2e2d7e8b74351acea7964809469c0ce79a5f49b015dc | A petition requires 120,577 valid signatures to qualify for a ballot. Approximately 137,689 signatures were submitted. Of the 88,147 signatures checked so far, 77,637 have been accepted as valid, 867 have been rejected as duplicates of valid signatures, and 9,643 have been rejected for not matching a registered voter's... | [
"The petition is likely to succeed."
] | [
"A petition requires 120,577 valid signatures to qualify for a ballot. Approximately 137,689 signatures were submitted. Of the 88,147 signatures checked so far, 77,637 have been accepted as valid, 867 have been rejected as duplicates of valid signatures, and 9,643 have been rejected for not matching a registered vo... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-4acbb0ed14a5613301c81e98ec7584cd56d04f30c108e72585e19c0a66ab1b89 | A pharmaceutical company claims that no more than 8% of patients who use their new medication will suffer a relapse. A government researcher conducts a hypothesis test with 2300 patients using the new medication and finds that 198 patients have suffered a relapse. What is the p-value for this test? | [
"0.141"
] | [
"A pharmaceutical company claims that no more than 8% of patients who use their new medication will suffer a relapse. A government researcher conducts a hypothesis test with 2300 patients using the new medication and finds that 198 patients have suffered a relapse. What is the p-value for this test?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0781b6aaa5fc8d8e2b1b3eda9298a1bc297c0c4125a1f62000bc0a6b02a34e42 | A pharmacist needs to prepare 15 ml of a special eye drop solution with a 2% active ingredient. The pharmacist has only 10% solution and 1% solution in stock. How much of each type of solution should be used to fill the prescription? | [
"\\( V_1 = \\frac{5}{3} \\text{ ml}, V_2 = \\frac{40}{3} \\text{ ml} \\)"
] | [
"A pharmacist needs to prepare 15 ml of a special eye drop solution with a 2% active ingredient. The pharmacist has only 10% solution and 1% solution in stock. How much of each type of solution should be used to fill the prescription?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 29
} | 41 | 0 | 0 |
omr-ws-e86021dab9e486fa3b0029055e863e4cefee2d841ad48aa26d5d358fa9b9667d | A pharmacist wants to put $155$ ml of liquid into $3$ bottles. There are three bottle choices: - Bottle A: Capacity $5$ ml, price $10,000$ Rp, $20\%$ discount on the next bottle up to the $4$th purchase, or $4$ bottles for the price of $3$. - Bottle B: Capacity $8$ ml, price $15,000$ Rp, $20\%$ discount on $2$ bottles,... | [
"30000"
] | [
"A pharmacist wants to put $155$ ml of liquid into $3$ bottles. There are three bottle choices:\n- Bottle A: Capacity $5$ ml, price $10,000$ Rp, $20\\%$ discount on the next bottle up to the $4$th purchase, or $4$ bottles for the price of $3$.\n- Bottle B: Capacity $8$ ml, price $15,000$ Rp, $20\\%$ discount on $2$... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-e01f1b9157388011a7cb7db71a16bd735c416dc25f24fe9da05c5f450c0e3594 | A phet snorlax is standing on the point (13,37) in the coordinate plane. He makes 5 moves, each time randomly moving 2 units up or right. Then he makes 5 more moves, each time randomly moving 2 units up, right, down, or left. What is the expected value of the sum of the coordinates of the point he ends up at? | [
"60"
] | [
"A phet snorlax is standing on the point (13,37) in the coordinate plane. He makes 5 moves, each time randomly moving 2 units up or right. Then he makes 5 more moves, each time randomly moving 2 units up, right, down, or left. What is the expected value of the sum of the coordinates of the point he ends up at?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90c558db787553d06bd7d03b274716e2613febe24af5556f3b9fe766694f0873 | A photo is printed on an 11-inch by 13-inch piece of paper. The photo covers 80 square inches and has a uniform border. What is the width of the border? | [
"1.5"
] | [
"A photo is printed on an 11-inch by 13-inch piece of paper. The photo covers 80 square inches and has a uniform border. What is the width of the border?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-78a5021c995aa876bed3c9e1fffeb85eb4fb54617a0e67a60ac72521cf4274f8 | A photographer took some pictures at a party with $10$ people. Each of the $45$ possible pairs of people appears together on exactly one photo, and each photo depicts two or three people. What is the smallest possible number of photos taken? | [
"19"
] | [
"A photographer took some pictures at a party with $10$ people. Each of the $45$ possible pairs of people appears together on exactly one photo, and each photo depicts two or three people. What is the smallest possible number of photos taken?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 11,
"tir": 2
} | 59 | 0 | 0 |
omr-ws-366657e8c5148847218705b2ce171e83dfe8929ea7bb026fa3addce8f517f8be | A physicist has discovered particles called imons that can be entangled with each other. The physicist can perform two operations: (i) Destroy an imon if it is entangled with an odd number of other imons. (ii) Double the number of imons in the lab by creating a copy of each imon, where each copy $I'$ becomes entangled ... | [
"4"
] | [
"A physicist has discovered particles called imons that can be entangled with each other. The physicist can perform two operations:\n(i) Destroy an imon if it is entangled with an odd number of other imons.\n(ii) Double the number of imons in the lab by creating a copy of each imon, where each copy $I'$ becomes ent... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-f9501eba189465289774ba269d44022fcb2a5e468d912d1c4fe846b95cf6a059 | A physicist has discovered particles called imons that can be entangled with each other. The physicist can perform two operations: (i) Destroy an imon if it is entangled with an odd number of other imons. (ii) Double the number of imons in the lab, creating a copy $I'$ of each imon $I$, such that $I'$ and $J'$ are enta... | [
"6"
] | [
"A physicist has discovered particles called imons that can be entangled with each other. The physicist can perform two operations:\n(i) Destroy an imon if it is entangled with an odd number of other imons.\n(ii) Double the number of imons in the lab, creating a copy $I'$ of each imon $I$, such that $I'$ and $J'$ a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-f74f6f4346524d8ed56fb8250dc1b506da924e2d11e72e10297a04e621e0b068 | A physics club with 22 members needs to select 3 officers: chairman, vice chairman, and sergeant-at-arms. Each person can hold at most one office. Two members, Penelope and Quentin, will only be officers if the other one is also an officer. In how many ways can the club choose its officers? | [
"6960"
] | [
"A physics club with 22 members needs to select 3 officers: chairman, vice chairman, and sergeant-at-arms. Each person can hold at most one office. Two members, Penelope and Quentin, will only be officers if the other one is also an officer. In how many ways can the club choose its officers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-9231e52435b7c71f6e75d253e0d008758bdc9918ef053174912504ec6c817304 | A piece is placed in each unit square of an $ n \times n$ square on an infinite board of unit squares. A move consists of finding two adjacent pieces (in squares which have a common side) so that one of the pieces can jump over the other onto an empty square. The piece jumped over is removed. Moves are made until no fu... | [
"0"
] | [
"A piece is placed in each unit square of an $ n \\times n$ square on an infinite board of unit squares. A move consists of finding two adjacent pieces (in squares which have a common side) so that one of the pieces can jump over the other onto an empty square. The piece jumped over is removed. Moves are made until... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-86c0bd79623aa397a24b08ef0751fa8f8516bdc74706c578dc27c17c49b64aeb | A piece is placed in the lower left-corner cell of a $15 \times 15$ board. It can move to adjacent cells by sides or corners, alternating between horizontal and diagonal moves (the first move must be diagonal). What is the maximum number of moves it can make without stepping on the same cell twice? | [
"196"
] | [
"A piece is placed in the lower left-corner cell of a $15 \\times 15$ board. It can move to adjacent cells by sides or corners, alternating between horizontal and diagonal moves (the first move must be diagonal). What is the maximum number of moves it can make without stepping on the same cell twice?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-99e544b5fba72d0da531d14d5c6628fdb21ed0a6a1dd753cc7c2b191a303a69e | A piece is separated at random from each of three identical rods. Find the probability that an acute triangle can be formed from the three pieces. | [
"\\( 1 - \\frac{\\pi}{4} \\)"
] | [
"A piece is separated at random from each of three identical rods. Find the probability that an acute triangle can be formed from the three pieces."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 3,
"tir": 19
} | 30 | 0 | 0 |
omr-ws-55e279d1720ba2ea46597fe855e9034b87c0b46de31bf7dc333b779ced7260b5 | A piece of a copper-zinc alloy weighing 24 kg was immersed in water and lost $2\frac{8}{9}$ kg in weight. Determine the amount of copper and zinc in the alloy, if it is known that in water, copper loses $11\frac{1}{9}\%$ and zinc, $14\frac{2}{7}\%$ of its weight. | [
"17 kg of copper, 7 kg of zinc"
] | [
"A piece of a copper-zinc alloy weighing 24 kg was immersed in water and lost $2\\frac{8}{9}$ kg in weight. Determine the amount of copper and zinc in the alloy, if it is known that in water, copper loses $11\\frac{1}{9}\\%$ and zinc, $14\\frac{2}{7}\\%$ of its weight."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c8df710c4caa2f57c7fea5e48eccba25919dfc64d559decdb168ae652d46e69d | A piece of a rail making 72\% of the rail length is cut off. The remaining part weighs 45.2 kg. Determine the weight of the cut-off piece. | [
"The weight of the cut-off piece is (approximately) equal to 116.2 kg"
] | [
"A piece of a rail making 72\\% of the rail length is cut off. The remaining part weighs 45.2 kg. Determine the weight of the cut-off piece."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7ca776196bd0bdf7569c93488ebf8572cb8f401bfc2247b7f04bd83d8964da2d | A piece of a silver-copper alloy weighs 2 kg. The weight of silver comes to $14\frac{2}{7}\%$ of that of copper. How much silver is there in this piece? | [
"The weight of the silver is $\\frac{1}{4}$ kg"
] | [
"A piece of a silver-copper alloy weighs 2 kg. The weight of silver comes to $14\\frac{2}{7}\\%$ of that of copper. How much silver is there in this piece?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-efe1fd290622b5afc5661d9b414a6cad2e19158c4c9bf712eda25c293fc93011 | A piece of an alloy of two metals weighs $P$ kg and loses $A$ kg in weight when immersed in water. A portion of $P$ kg of one of the metals loses $B$ kg in water and a portion of the same weight of the other metal loses $C$ kg. Find the weights of the components of the alloy and test the solvability of the problem depe... | [
"x=\\frac{A-C}{B-C}P\n\nAdditionally, the conditions for solvability are:\n1. \\( B \\neq C \\)\n2. \\( 0 < \\frac{A-C}{B-C} < 1 \\)\n3. Either \\( B > A > C \\) or \\( C > A > B \\)"
] | [
"A piece of an alloy of two metals weighs $P$ kg and loses $A$ kg in weight when immersed in water. A portion of $P$ kg of one of the metals loses $B$ kg in water and a portion of the same weight of the other metal loses $C$ kg. Find the weights of the components of the alloy and test the solvability of the problem... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a83586b4e747182e57e5e3435052d0b05a6c4e2e9ff80cc73963f39b562cf303 | A piece of ancient charcoal contains only 15% as much radioactive carbon as a piece of modern charcoal. If the half-life of Carbon-14 (C^14) is 5730 years, how long ago was the tree burned to make the ancient charcoal? | [
"15687 years"
] | [
"A piece of ancient charcoal contains only 15% as much radioactive carbon as a piece of modern charcoal. If the half-life of Carbon-14 (C^14) is 5730 years, how long ago was the tree burned to make the ancient charcoal?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-94ae8b5d5d410386a8378dfc523e37a3f55b72034e56ef374cac04d58a3141ba | A piece of cardboard has the shape of a pentagon $ABCDE$ in which $BCDE$ is a square and $ABE$ is an isosceles triangle with a right angle at $A$. If the side length of the square $BCDE$ is $s$, calculate the area of the largest right isosceles triangle that can be formed by rearranging the pentagon into three parts in... | [
"\\dfrac{5s^2}{4}"
] | [
"A piece of cardboard has the shape of a pentagon $ABCDE$ in which $BCDE$ is a square and $ABE$ is an isosceles triangle with a right angle at $A$. If the side length of the square $BCDE$ is $s$, calculate the area of the largest right isosceles triangle that can be formed by rearranging the pentagon into three par... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31a916ec630e83e1e3fd93e882424af6fc8924a180597186e571084f89b114f8 | A piece of cheese is located at $(12, 10)$ in a coordinate plane. A mouse is at $(4, -2)$ and is running up the line $y = -5x + 18$. At the point $(a, b)$, the mouse starts getting farther from the cheese rather than closer to it. What is $a + b$? | [
"10"
] | [
"A piece of cheese is located at $(12, 10)$ in a coordinate plane. A mouse is at $(4, -2)$ and is running up the line $y = -5x + 18$. At the point $(a, b)$, the mouse starts getting farther from the cheese rather than closer to it. What is $a + b$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9df311703dffcd72699b3a3c92211d244fd25cfe93074969e58e8182d908ce76 | A piece of cheese is located at $(12,10)$ in a coordinate plane. A mouse is at $(4,-2)$ and is running up the line $y=-5x+18$. At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $a + b$? | [
"10"
] | [
"A piece of cheese is located at $(12,10)$ in a coordinate plane. A mouse is at $(4,-2)$ and is running up the line $y=-5x+18$. At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $a + b$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e4bf6eeaa809e714b57a8c44a2376f95f66554cafa0a16cd4a188a4a412a656d | A piece of equipment costs a factory Rs600,000. It depreciates in value by 15% in the first year, 13.5% in the next year, 12% in the third year, and so on. What will be its value at the end of 10 years, with all percentages applying to the original cost? | [
"105,000"
] | [
"A piece of equipment costs a factory Rs600,000. It depreciates in value by 15% in the first year, 13.5% in the next year, 12% in the third year, and so on. What will be its value at the end of 10 years, with all percentages applying to the original cost?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-3bcebda1e670248b2ad6117be1b4ef46c958de653781fc33866031394a4bc373 | A piece of graph paper is folded once so that $(0,2)$ is matched with $(4,0)$ and $(7,3)$ is matched with $(m,n)$. Find $m + n$. | [
"6.8"
] | [
"A piece of graph paper is folded once so that $(0,2)$ is matched with $(4,0)$ and $(7,3)$ is matched with $(m,n)$. Find $m + n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-d6a470be591b53a0d61e6df53ab0c68e173b260eb6d615ddc717fed0e4db2083 | A piece of graph paper is folded once so that $(0,2)$ is matched with $(4,0)$, and $(7,3)$ is matched with $(m,n)$. Find $m+n$. | [
"$\\frac{34}{5}$",
"6.8"
] | [
"A piece of graph paper is folded once so that $(0,2)$ is matched with $(4,0)$, and $(7,3)$ is matched with $(m,n)$. Find $m+n$."
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 58
} | 89 | 0 | 0 |
omr-ws-05d43f919380afc96ab48768fbb1e852faafcc1f7ce9760852c6f6728880be72 | A piece of paper in the shape of a right triangle is folded such that the smaller leg rests on the hypotenuse. If the segment $a = 9$ and segment $b = \frac{24}{5}$, calculate the lengths of the sides of the paper. | [
"\\(9, \\frac{24}{5}, \\frac{51}{5}\\)"
] | [
"A piece of paper in the shape of a right triangle is folded such that the smaller leg rests on the hypotenuse. If the segment $a = 9$ and segment $b = \\frac{24}{5}$, calculate the lengths of the sides of the paper."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1fe70d96b27ebf202e0c9b653d6be4f4cc7baf968da378a8416663bbf81fca50 | A piece of paper in the shape of an equilateral triangle \(ABC\) has \(AB = 8\). When point \(A\) is folded over to any point on \(\overline{BC}\), a crease of length \(X\) is formed which joins some point of \(\overline{AC}\) to some point of \(\overline{AB}\). As \(A\) is moved along \(\overline{BC}\) from \(B\) to \... | [
"(4\\sqrt{3}, 4)"
] | [
"A piece of paper in the shape of an equilateral triangle \\(ABC\\) has \\(AB = 8\\). When point \\(A\\) is folded over to any point on \\(\\overline{BC}\\), a crease of length \\(X\\) is formed which joins some point of \\(\\overline{AC}\\) to some point of \\(\\overline{AB}\\). As \\(A\\) is moved along \\(\\over... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-2d33a2c7c44bd0bf3e9155db7b03601baf2c3e6353d54eec54b47f5286e215b9 | A piece of paper is black on one side and white on the other. A regular dodecagon $ABCDEFGHIJKL$ is cut out with $AG=8$. On the white side, color $AE$ red, $CF$ green, and $DH$ blue. Fold along the red line, then fold along the visible part of the green line, and finally fold along the visible part of the blue line, en... | [
"196"
] | [
"A piece of paper is black on one side and white on the other. A regular dodecagon $ABCDEFGHIJKL$ is cut out with $AG=8$. On the white side, color $AE$ red, $CF$ green, and $DH$ blue. Fold along the red line, then fold along the visible part of the green line, and finally fold along the visible part of the blue lin... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d18c7b73a2107f5045f95dce54759f702453fdad9cd57287ea64f2a51684fc3b | A piece of paper is black on one side and white on the other. Cut out a regular dodecagon \(ABCDEFGHIJKL\) with \(AG = 8\). Color \(AE\) red, \(CF\) green, and \(DH\) blue. Fold along the red line, then the green line, and finally the blue line. If the area of the visible black region is \(a + \sqrt{b}\) where \(a\) an... | [
"56"
] | [
"A piece of paper is black on one side and white on the other. Cut out a regular dodecagon \\(ABCDEFGHIJKL\\) with \\(AG = 8\\). Color \\(AE\\) red, \\(CF\\) green, and \\(DH\\) blue. Fold along the red line, then the green line, and finally the blue line. If the area of the visible black region is \\(a + \\sqrt{b}... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-79d614c60331f9999f2826f2826493deda9f813288a78819675004437349e195 | A piece of paper is folded in half, then folded again at an angle $\phi$ ($0^\circ < \phi < 90^\circ$) to the first fold, and a cut is made. When the paper is unfolded, the resulting hole is a polygon with one vertex at $O$. All other vertices of the hole lie on a circle centered at $O$, and $\angle XOY = 144^\circ$, w... | [
"72"
] | [
"A piece of paper is folded in half, then folded again at an angle $\\phi$ ($0^\\circ < \\phi < 90^\\circ$) to the first fold, and a cut is made. When the paper is unfolded, the resulting hole is a polygon with one vertex at $O$. All other vertices of the hole lie on a circle centered at $O$, and $\\angle XOY = 144... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-f367e830407ff175d447fe6fa1a4ff9e577c91403698ebbbf5bd01530df18b24 | A piece of paper is the square \(ABCD\). We fold it by placing the vertex \(D\) on the point \(D'\) of the side \(BC\). We assume that \(AD\) moves on the segment \(A'D'\) and that \(A'D'\) intersects \(AB\) at \(E\). If the side length of the square is \(s\), find the value of \(x\) such that the perimeter of the tria... | [
"2"
] | [
"A piece of paper is the square \\(ABCD\\). We fold it by placing the vertex \\(D\\) on the point \\(D'\\) of the side \\(BC\\). We assume that \\(AD\\) moves on the segment \\(A'D'\\) and that \\(A'D'\\) intersects \\(AB\\) at \\(E\\). If the side length of the square is \\(s\\), find the value of \\(x\\) such tha... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-74c3ace56d3d9896175e95f4dfa01e0d304e23ece857b653547e766549336659 | A piece of paper was bent and crumpled, and then unfolded, marking the fold lines. Four such lines formed four angles. The two adjacent angles were $57^\circ$ and $83^\circ$. Find the other two angles. | [
"123^\\circ \\text{ and } 97^\\circ"
] | [
"A piece of paper was bent and crumpled, and then unfolded, marking the fold lines. Four such lines formed four angles. The two adjacent angles were $57^\\circ$ and $83^\\circ$. Find the other two angles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-a14114556fd981a0c600232add8515dead037d87d56d607ca6c781040f773aa4 | A piece of rectangular paper $20 \times 19$, divided into unit squares, is cut into several square pieces. An odd square is defined as a square piece with an odd side length. What is the minimum possible number of odd squares? | [
"4"
] | [
"A piece of rectangular paper $20 \\times 19$, divided into unit squares, is cut into several square pieces. An odd square is defined as a square piece with an odd side length. What is the minimum possible number of odd squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-f0366ab99a1881b14b694dd13b20c1c6a7bee9ef85a2214cd2f6bdacc1e2760c | A piece of rectangular paper $20 \times 19$, divided into unit squares, is cut into several square pieces. An odd square is defined as a square piece with an odd side length. What is the smallest value that the sum of the perimeters of the odd squares can take? | [
"80"
] | [
"A piece of rectangular paper $20 \\times 19$, divided into unit squares, is cut into several square pieces. An odd square is defined as a square piece with an odd side length. What is the smallest value that the sum of the perimeters of the odd squares can take?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-0bdc52cf8a480b077480c66cee3580720fa25e1536328ccdd7393eb68dff2472 | A piece of rectangular paper 20 × 19, divided into square units, is cut into several pieces of square shape, with cuts made along the sides of the unit squares. An odd square is defined as a square piece with an odd number side length. What is the minimum possible number of odd squares? | [
"20"
] | [
"A piece of rectangular paper 20 × 19, divided into square units, is cut into several pieces of square shape, with cuts made along the sides of the unit squares. An odd square is defined as a square piece with an odd number side length. What is the minimum possible number of odd squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-eba0f7041e7da956c7f6456c0261432750c6edc1baaf2e20487bf3042bcd8478 | A piece of rectangular paper 20 × 19, divided into square units, is cut into several pieces of square shape, with cuts made along the sides of the unit squares. An odd square is defined as a square piece with an odd number side length. What is the smallest value that the sum of the perimeters of the odd squares can tak... | [
"80"
] | [
"A piece of rectangular paper 20 × 19, divided into square units, is cut into several pieces of square shape, with cuts made along the sides of the unit squares. An odd square is defined as a square piece with an odd number side length. What is the smallest value that the sum of the perimeters of the odd squares ca... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-b5ca4fc019b004bbe51fa56e901b043e74ff22fda380f59fe0ce4bb9ff6545d5 | A piece of rock contains $10^{25}$ atoms. Each atom has an exponentially distributed lifetime with a half-life of one century. How many centuries must pass before there is just about a 50% chance that at least one atom still remains? | [
"83.6"
] | [
"A piece of rock contains $10^{25}$ atoms. Each atom has an exponentially distributed lifetime with a half-life of one century. How many centuries must pass before there is just about a 50% chance that at least one atom still remains?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b5810b14805fc80e274b3c89763aae0a9ce336b8fd4639b8963ff76418a20904 | A piece of spaghetti that is $7$ inches long is cracked into $3$ pieces of positive length. What is the probability that no piece is below $1$ inch long? Express the probability as $\frac{m}{n}$ and find $m+n$. | [
"65"
] | [
"A piece of spaghetti that is $7$ inches long is cracked into $3$ pieces of positive length. What is the probability that no piece is below $1$ inch long? Express the probability as $\\frac{m}{n}$ and find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-06a2a57a4809a50356234efb209844d415dc9e1d05ce4eb47c7c111a2efc3902 | A piece of string fits exactly once around the perimeter of a square whose area is 144. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string? | [
"183"
] | [
"A piece of string fits exactly once around the perimeter of a square whose area is 144. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7874fb9dc18e71df54c3856f689ab4c713bdc51023a363d0cb81342a7783a1c2 | A piece of string is cut in two at a point selected at random. What is the probability that the longer piece is at least $x$ times as large as the shorter piece? | [
"\\(\\frac{2}{x+1}\\)"
] | [
"A piece of string is cut in two at a point selected at random. What is the probability that the longer piece is at least $x$ times as large as the shorter piece?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 2,
"tir": 8
} | 20 | 0 | 0 |
omr-ws-fdafc7e6b240d5989c29beec0fc4341419a23d2d33c3d3fee5ea762a92b59e84 | A piece of string is cut in two at a random point. What is the probability that the larger piece is at least $x$ times as large as the shorter piece, where $x \geq 1$? | [
"\\(\\frac{2}{x+1}\\)",
"\\dfrac{2}{x + 1}"
] | [
"A piece of string is cut in two at a random point. What is the probability that the larger piece is at least $x$ times as large as the shorter piece, where $x \\geq 1$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 29
} | 61 | 0 | 0 |
omr-ws-5561224fa74a89f6e45557fdd8509b6e58d3e46693568c831010a66791f2919e | A piece of wire 20 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How much wire should be used for the square in order to maximize the total area? | [
"20"
] | [
"A piece of wire 20 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How much wire should be used for the square in order to maximize the total area?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f6d5ce0bd6b42a9abc43bceab7d2f0c2cf69e6834ca18f98583214e22b7b0978 | A piece of wire 20 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How much wire should be used for the square in order to minimize the total area? | [
"\\(\\frac{240\\sqrt{3} - 320}{11}\\)"
] | [
"A piece of wire 20 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How much wire should be used for the square in order to minimize the total area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-f9332b02b2a6ef53e7e8047b5c83f26d182975277583f8b4634e20091d3bc601 | A piece of wire 30 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. How much wire should be used for the circle in order to maximize the total area? | [
"30.00"
] | [
"A piece of wire 30 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. How much wire should be used for the circle in order to maximize the total area?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 6
} | 26 | 0 | 0 |
omr-ws-86990f19c37ce95e41cb8c243b4655f3889e2ebc292230f670ebbe5c67363171 | A piece of wire 30 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. How much wire should be used for the circle in order to minimize the total area? | [
"13.20"
] | [
"A piece of wire 30 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. How much wire should be used for the circle in order to minimize the total area?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-0a6fb14a067a6cddf067ec7331caec6d56a7a89a96b9bb4022648946a7fb5bc9 | A piece of wire 6 m long is cut into two pieces. One piece is bent into a square and the other into an equilateral triangle. How much wire (in meters) should be used for the square to minimize the total area? | [
"2.6098 meters"
] | [
"A piece of wire 6 m long is cut into two pieces. One piece is bent into a square and the other into an equilateral triangle. How much wire (in meters) should be used for the square to minimize the total area?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-5c0ca40d342dba36bf36be4c12b2edc3521de15419c203045ffcc95ad5816641 | A piece of wire of 1500 cm long is cut into \( n \) parts and arranged in ascending order. The length of each part is 5 cm longer than the preceding part, and the sum of the lengths of the first \( \frac{n}{2} \) parts is 390 cm. Find the length of the first part of the wire. | [
"5"
] | [
"A piece of wire of 1500 cm long is cut into \\( n \\) parts and arranged in ascending order. The length of each part is 5 cm longer than the preceding part, and the sum of the lengths of the first \\( \\frac{n}{2} \\) parts is 390 cm. Find the length of the first part of the wire."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a369cd848ba50e62f00938f17f235aa7ac73cdcc95d0312e2f909d8a912242b | A piece of wire of 1500 cm long is cut into \( n \) parts and arranged in ascending order. The length of each part is 5 cm longer than the preceding part, and the sum of the lengths of the first \( \frac{n}{2} \) parts is 390 cm. Find the value of \( n \). | [
"24"
] | [
"A piece of wire of 1500 cm long is cut into \\( n \\) parts and arranged in ascending order. The length of each part is 5 cm longer than the preceding part, and the sum of the lengths of the first \\( \\frac{n}{2} \\) parts is 390 cm. Find the value of \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5d98db1e50345616ba6b27abb14cd205d5711ddbd866c780e22ac2c4d2fa3f47 | A piece of wood has a lower edge that resembles the curve \( y = x^2 \). A 90-degree angle with infinite legs is placed such that the legs touch the edges of the wood. As the angle is moved from left to right while keeping the legs in contact with the wood, a cut is made in the paper underneath. Determine the function ... | [
"\\( y = -\\frac{1}{4} \\)"
] | [
"A piece of wood has a lower edge that resembles the curve \\( y = x^2 \\). A 90-degree angle with infinite legs is placed such that the legs touch the edges of the wood. As the angle is moved from left to right while keeping the legs in contact with the wood, a cut is made in the paper underneath. Determine the fu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 14
} | 60 | 0 | 0 |
omr-ws-fea40bca9686c389cb50ad63c64c855281780e52848a73a4fe81fa06c96ce13e | A piece stands on the leftmost cell of a $1 \times 100$ board. Two players alternately move the piece by $1$, $10$, or $11$ cells to the right. The player who cannot move loses. Who can win? | [
"The first player can win."
] | [
"A piece stands on the leftmost cell of a $1 \\times 100$ board. Two players alternately move the piece by $1$, $10$, or $11$ cells to the right. The player who cannot move loses. Who can win?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 26
} | 40 | 0 | 0 |
omr-ws-32c27c294570feca4eba522c24cc152bc8f39bcf77ec1aac255f8336dcc305d3 | A piggy bank contains 24 coins, all of which are nickels, dimes, or quarters. If the total value of the coins is two dollars, what combinations of coins are possible? | [
"(8, 16, 0), (11, 12, 1), (14, 8, 2), (17, 4, 3), (20, 0, 4)"
] | [
"A piggy bank contains 24 coins, all of which are nickels, dimes, or quarters. If the total value of the coins is two dollars, what combinations of coins are possible?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 26
} | 31 | 0 | 0 |
omr-ws-facfc523fd8ab54c7e5452660648a8b2b2f73eaf0256ee0024edf8d0adadf8f3 | A piggy bank contains quarters, nickels, and dimes. If there are 23 coins worth a total of 154 cents, how many dimes are there? | [
"8"
] | [
"A piggy bank contains quarters, nickels, and dimes. If there are 23 coins worth a total of 154 cents, how many dimes are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-867dd51faacff29d6d1c2c30c72f327050be9b08572ec2e1502d1e578bb1e2b7 | A pile contains $2021^{2021}$ stones. In a move, any pile can be divided into two piles such that the numbers of stones in them differ by a power of $2$ with a non-negative integer exponent. After some moves, it turned out that the number of stones in each pile is a power of $2$ with a non-negative integer exponent. Ca... | [
"no_valid_answer_found"
] | [
"A pile contains $2021^{2021}$ stones. In a move, any pile can be divided into two piles such that the numbers of stones in them differ by a power of $2$ with a non-negative integer exponent. After some moves, it turned out that the number of stones in each pile is a power of $2$ with a non-negative integer exponen... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c81bfaec4af269399cf08a83481dda11aa466512181a99e72c55978012557546 | A pile of 2001 pebbles is arranged in a vertical column. A pebble is movable if it is at the top of a column that has at least two more pebbles than the column immediately to its right. At each step, a movable pebble is selected and moved to the top of the column to its right. The process stops when no pebbles can be m... | [
"64593"
] | [
"A pile of 2001 pebbles is arranged in a vertical column. A pebble is movable if it is at the top of a column that has at least two more pebbles than the column immediately to its right. At each step, a movable pebble is selected and moved to the top of the column to its right. The process stops when no pebbles can... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3205f57ed0df39d83583c02b719f2270473a83d0589660253448b1e6ab556515 | A pile of \( n \) pebbles is placed in a vertical column. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right (consider a column with 0 pebbles if there are no pebbles to the right). At each stage, choose a pebble from among those that ... | [
"63"
] | [
"A pile of \\( n \\) pebbles is placed in a vertical column. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right (consider a column with 0 pebbles if there are no pebbles to the right). At each stage, choose a pebble from among thos... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
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