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-f083bdcd4bee4ce814433b6c8b78dd3dac8f68ca8b11c17c93efb212facafb4c | A box contains 3 black balls, 4 red balls, and 7 green balls. Two balls are drawn at random. What is the probability of drawing two green balls? | [
"\\(\\frac{3}{13}\\)"
] | [
"A box contains 3 black balls, 4 red balls, and 7 green balls. Two balls are drawn at random. What is the probability of drawing two green balls?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-09f5037e378a539bbf8b7aec5f9fc4df6bb006f2abe5c951d2a41ae52fa3c348 | A box contains 3 red balls, 4 white balls, and 3 blue balls. Balls are drawn from the box one at a time at random without replacement. The probability that all three red balls will be drawn before any white ball is obtained is \(\frac{m}{n}\), where \(m\) and \(n\) are coprime natural numbers. Find \(n - m\). | [
"34"
] | [
"A box contains 3 red balls, 4 white balls, and 3 blue balls. Balls are drawn from the box one at a time at random without replacement. The probability that all three red balls will be drawn before any white ball is obtained is \\(\\frac{m}{n}\\), where \\(m\\) and \\(n\\) are coprime natural numbers. Find \\(n - m... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-d9b7f5afd1322adbd663a9cd2ecd533e55bed929bfc5cc6db3ff882ca5efb4a4 | A box contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn? | [
"\\(\\frac{2}{5}\\)",
"\\dfrac{2}{5}"
] | [
"A box contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 23
} | 55 | 0 | 0 |
omr-ws-3a4ea9f47e9dabab5a38d1fdbaf8c23a54fb85f3517aff114e7f77bc68497c56 | A box contains 3 shiny pennies and 4 dull pennies. One by one, pennies are drawn from the box and not replaced. What is the probability that it will take more than four draws until the third shiny penny appears? If this probability is expressed as $\frac{a}{b}$ in lowest terms, find $a + b$. | [
"66"
] | [
"A box contains 3 shiny pennies and 4 dull pennies. One by one, pennies are drawn from the box and not replaced. What is the probability that it will take more than four draws until the third shiny penny appears? If this probability is expressed as $\\frac{a}{b}$ in lowest terms, find $a + b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-221bb173dbbe63e38e4b09832d0ab8b40ef6c2c0f49aa520f4797454e647f7b7 | A box contains 3 shiny pennies and 4 dull pennies. Pennies are drawn one by one without replacement. What is the probability that it will take more than four draws until the third shiny penny appears? Express the probability as a fraction a/b in lowest terms and find a+b. | [
"66"
] | [
"A box contains 3 shiny pennies and 4 dull pennies. Pennies are drawn one by one without replacement. What is the probability that it will take more than four draws until the third shiny penny appears? Express the probability as a fraction a/b in lowest terms and find a+b."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-4f910b3a75210d26dcc0f0bdaf68bb07a8393b4fcd070517ada7e641e725f1e7 | A box contains 3 shiny pennies and 4 dull pennies. Pennies are drawn one by one without replacement. What is the probability that it will take more than four draws until the third shiny penny appears? If this probability is expressed as $\frac{a}{b}$ in lowest terms, find $a+b$. | [
"66"
] | [
"A box contains 3 shiny pennies and 4 dull pennies. Pennies are drawn one by one without replacement. What is the probability that it will take more than four draws until the third shiny penny appears? If this probability is expressed as $\\frac{a}{b}$ in lowest terms, find $a+b$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-153bbb98bf68dfda59510cbb651553233fa4e9411f0ce1095b0e5628ceec7085 | A box contains 30 candies colored red, blue, or green. If we randomly take 25 candies out of the box, we will always have at least 3 red, at least 5 blue, and at least 7 green candies. How many green candies were in the original box? | [
"12"
] | [
"A box contains 30 candies colored red, blue, or green. If we randomly take 25 candies out of the box, we will always have at least 3 red, at least 5 blue, and at least 7 green candies. How many green candies were in the original box?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-371594a2b444b9b2a309baa85e2f9ec8cfa3137d38b2da4d3a8185b58540169f | A box contains 32 balls. Three balls each have $2^1, 2^2, \ldots, 2^{10}$ written on them, and two distinct balls have $2^0$ written on them. In how many ways can a person draw some balls such that the sum of the numbers written on them is 2012? | [
"384"
] | [
"A box contains 32 balls. Three balls each have $2^1, 2^2, \\ldots, 2^{10}$ written on them, and two distinct balls have $2^0$ written on them. In how many ways can a person draw some balls such that the sum of the numbers written on them is 2012?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6a00e7147bb27c409b4460a3df2a3eb943d6343571a67aa721e5e2b0224f9418 | A box contains 4 parts, two of which are defective. Successive extractions of a part are carried out without repositioning. Let \(X\) and \(Y\) be the random variables representing the number of extractions performed to detect the first defective part and the number of additional extractions required to detect the seco... | [
"\\dfrac{3}{2}"
] | [
"A box contains 4 parts, two of which are defective. Successive extractions of a part are carried out without repositioning. Let \\(X\\) and \\(Y\\) be the random variables representing the number of extractions performed to detect the first defective part and the number of additional extractions required to detect... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f1337f77949a7e9d8e4314910aabf6bbdf305303636d5a613e5afa28d6400d93 | A box contains 4 red marbles, 5 white marbles, and 6 yellow marbles. How many ways are there to select four marbles such that not all three colors are represented? | [
"645"
] | [
"A box contains 4 red marbles, 5 white marbles, and 6 yellow marbles. How many ways are there to select four marbles such that not all three colors are represented?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-1b0c2b6ee1e3e857ec2ee7be1a5ba434ed1eb33e38ca15186e09d520d7085971 | A box contains 4 white balls and 4 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors? | [
"\\dfrac{1}{35}"
] | [
"A box contains 4 white balls and 4 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f2af63c5df6757fafc2212d66fa8055eb7ba39e54a13c6dcaace6c2da394ca85 | A box contains 4 white balls and 4 red balls, which are drawn from the box in some order without replacement. Before each draw, we guess the color of the ball being drawn, always guessing the color more likely to occur (if one is more likely than the other). What is the expected number of correct guesses? | [
"\\( \\frac{373}{70} \\)"
] | [
"A box contains 4 white balls and 4 red balls, which are drawn from the box in some order without replacement. Before each draw, we guess the color of the ball being drawn, always guessing the color more likely to occur (if one is more likely than the other). What is the expected number of correct guesses?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 22
} | 43 | 0 | 0 |
omr-ws-b74c05e135c9f74bbb34e423e9fce53bee2c04e0ddbc2efa76e5bbfe8858266b | A box contains 4032 answer scripts, exactly half of which have an odd number of marks. We randomly choose 2 scripts and follow these rules: - If both scripts have odd scores, we add one mark to one of them, put it back in the box, and keep the other script outside. - If both scripts have even scores, we put back one of... | [
"2"
] | [
"A box contains 4032 answer scripts, exactly half of which have an odd number of marks. We randomly choose 2 scripts and follow these rules:\n- If both scripts have odd scores, we add one mark to one of them, put it back in the box, and keep the other script outside.\n- If both scripts have even scores, we put back... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f82bcfe7303ab0d0152aceda1b6f17a5f873772a594e919e72ac30f973e5c36 | A box contains 4032 scripts, exactly half of which have an odd number of marks. We randomly choose 2 scripts and follow these rules: - If both scripts have odd scores, we add one mark to one of them, put it back in the box, and keep the other script outside. - If both scripts have even scores, we put back one of the sc... | [
"2"
] | [
"A box contains 4032 scripts, exactly half of which have an odd number of marks. We randomly choose 2 scripts and follow these rules:\n- If both scripts have odd scores, we add one mark to one of them, put it back in the box, and keep the other script outside.\n- If both scripts have even scores, we put back one of... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8bc3b3f07745ee5ee216ce5af34df5e92ed197bcc2e5159b7a4714edd3443908 | A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4. What is the probability that 3 draws are required? | [
"\\(\\frac{1}{5}\\)"
] | [
"A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4. What is the probability that 3 draws are required?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 58
} | 90 | 0 | 0 |
omr-ws-02423fc6071dd075f64c91be7e79265a3f5e27e7ef11db8d24a3c877e0712019 | A box contains 5 fair coins and 5 biased coins, where each biased coin has a probability of heads of $\frac{4}{5}$. A coin is drawn at random from the box and tossed. Given that the first coin shows heads, what is the conditional probability that the second coin drawn (without replacement) is a fair coin? | [
"\\(\\frac{20}{39}\\)"
] | [
"A box contains 5 fair coins and 5 biased coins, where each biased coin has a probability of heads of $\\frac{4}{5}$. A coin is drawn at random from the box and tossed. Given that the first coin shows heads, what is the conditional probability that the second coin drawn (without replacement) is a fair coin?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-c158811051e091b3ca9e82d0a08ac7549bf4463991532742fa6cf78f9de8acae | A box contains 5 fair coins and 5 biased coins, where each biased coin has a probability of heads of $\frac{4}{5}$. A coin is drawn at random from the box and tossed. Then a second coin is drawn at random from the box (without replacing the first one). Given that the first coin has shown heads, what is the conditional ... | [
"\\(\\frac{20}{39}\\)"
] | [
"A box contains 5 fair coins and 5 biased coins, where each biased coin has a probability of heads of $\\frac{4}{5}$. A coin is drawn at random from the box and tossed. Then a second coin is drawn at random from the box (without replacing the first one). Given that the first coin has shown heads, what is the condit... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-daa7433371615dd28d6490e4f430fa99de2394c367fa91cde97bfaa8e8cdfcc2 | A box contains 5 radio tubes, of which some are defective. The tubes are tested one after the other until 2 defective tubes are discovered. Find the probability that the process stops on the second test. | [
"\\(\\frac{1}{10}\\)"
] | [
"A box contains 5 radio tubes, of which some are defective. The tubes are tested one after the other until 2 defective tubes are discovered. Find the probability that the process stops on the second test."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-199ab7eb7a62a008fa818c06ec64c7edc3e622b859b0e8b63af17535b4c048b2 | A box contains 5 radio tubes, of which some are defective. The tubes are tested one after the other until 2 defective tubes are discovered. Find the probability that the process stops on the third test. | [
"\\(\\frac{2}{10}\\)"
] | [
"A box contains 5 radio tubes, of which some are defective. The tubes are tested one after the other until 2 defective tubes are discovered. Find the probability that the process stops on the third test."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 6,
"tir": 7
} | 59 | 0 | 0 |
omr-ws-3fcba04bfeea6fb5070203005a7be90e222145b7555abc9921f1f936b70732e0 | A box contains 5 red balls and 4 blue balls. In how many ways can 4 balls be chosen such that there are at most 3 balls of each color? | [
"124"
] | [
"A box contains 5 red balls and 4 blue balls. In how many ways can 4 balls be chosen such that there are at most 3 balls of each color?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6cd5b8f17215a521b6a31344efe3194107fd71cf0fbaebaecd8e1a32fd8ea264 | A box contains 5 red balls, 10 blue balls, and 5 yellow balls. Four balls are selected at random without replacement. If a red or yellow ball is selected, 3 new balls of the same color are added to the box. What is the probability that the first two balls are red, the third ball is blue, and the fourth ball is yellow? | [
"\\(\\frac{175}{24288}\\)"
] | [
"A box contains 5 red balls, 10 blue balls, and 5 yellow balls. Four balls are selected at random without replacement. If a red or yellow ball is selected, 3 new balls of the same color are added to the box. What is the probability that the first two balls are red, the third ball is blue, and the fourth ball is yel... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 32
} | 46 | 0 | 0 |
omr-ws-39b1238cb8bf1bc346d01f3416977ee515b48ee7db1d63621bc8c524a3a4de4c | A box contains 5 red balls, 4 blue balls, and 3 green balls. If three balls are randomly drawn from the box without replacement, what is the probability that all three balls are the same color? | [
"\\(\\frac{3}{44}\\)"
] | [
"A box contains 5 red balls, 4 blue balls, and 3 green balls. If three balls are randomly drawn from the box without replacement, what is the probability that all three balls are the same color?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-6e7072e1b3daea324d89ec9ae1a29c055707556776018fe29a8cac0fcd588076 | A box contains 5 white balls and 6 black balls. A ball is drawn out of the box at random. What is the probability that the ball is white? | [
"\\dfrac{5}{11}"
] | [
"A box contains 5 white balls and 6 black balls. A ball is drawn out of the box at random. What is the probability that the ball is white?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-44feae8f30afd25374a8669bf7585e15ff73ba8a0732c7a914304ea1d51191af | A box contains 5 white balls and 6 black balls. Five balls are drawn out of the box at random. What is the probability that they all are white? | [
"\\dfrac{1}{462}"
] | [
"A box contains 5 white balls and 6 black balls. Five balls are drawn out of the box at random. What is the probability that they all are white?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e9241c7077f9811fdbe496e5e5a2b5d39c411a756da5b41f4c47dcc78e38d4d5 | A box contains 5 white balls and 6 black balls. Two balls are drawn out of the box at random. What is the probability that they both are white? | [
"\\dfrac{2}{11}"
] | [
"A box contains 5 white balls and 6 black balls. Two balls are drawn out of the box at random. What is the probability that they both are white?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cafbcedce79b77bf1a8c872f4114b4d935424a768b6a666c4cd2b5ec4b09784a | A box contains 5 white, 2 blue, and 3 red balls. If 6 balls are drawn with replacement, find the probability that there are 2 white, 3 blue, and 1 red ball. | [
"0"
] | [
"A box contains 5 white, 2 blue, and 3 red balls. If 6 balls are drawn with replacement, find the probability that there are 2 white, 3 blue, and 1 red ball."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eab88b9314c21df6507df260e30e03e106b1e481942016826f64b17bc94c1799 | A box contains 50 red, 30 green, and 20 blue balls. Balls are drawn successively at random with replacement from the box. Let \( N \) denote the minimum number of draws required to obtain balls of all three colors. Compute \( P(N > n) \) for all positive integers \( n \). | [
"\\( P(N > n) = \\left( \\frac{7}{10} \\right)^n + \\left( \\frac{4}{5} \\right)^n - \\left( \\frac{1}{5} \\right)^n - \\left( \\frac{3}{10} \\right)^n \\)"
] | [
"A box contains 50 red, 30 green, and 20 blue balls. Balls are drawn successively at random with replacement from the box. Let \\( N \\) denote the minimum number of draws required to obtain balls of all three colors. Compute \\( P(N > n) \\) for all positive integers \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-d8aa9e8cfe21bc7ed82593a41b2d56f78c447aa51f3037349bd24ca6b255c97e | A box contains 6 balls: 2 green, 2 red, and 2 gold. Balls are removed from the box two at a time until all balls are removed. What is the probability of drawing two balls of the same color in a single draw? | [
"\\(\\frac{1}{5}\\)"
] | [
"A box contains 6 balls: 2 green, 2 red, and 2 gold. Balls are removed from the box two at a time until all balls are removed. What is the probability of drawing two balls of the same color in a single draw?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-6e86865ce2e6ca09fa538ff438cf550f550029de3fe87dbc5496777ff2823507 | A box contains 6 red balls and 3 blue balls. Bill picks one ball at a time without replacement until all the red balls or all the blue balls are out. Find the probability that the last ball Bill picks is blue. | [
"\\(\\frac{2}{3}\\)"
] | [
"A box contains 6 red balls and 3 blue balls. Bill picks one ball at a time without replacement until all the red balls or all the blue balls are out. Find the probability that the last ball Bill picks is blue."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-6af46f039a30017e84abc945547182529bfe789128e694d47bce2cfd8e1cbeca | A box contains 6 red balls and 4 yellow balls. Two balls are picked randomly and returned to the box. This process is repeated 8 times. Determine the probability of picking 1 red ball exactly 5 times. | [
"\\( 56 \\left( \\frac{8}{15} \\right)^5 \\left( \\frac{7}{15} \\right)^3 \\)"
] | [
"A box contains 6 red balls and 4 yellow balls. Two balls are picked randomly and returned to the box. This process is repeated 8 times. Determine the probability of picking 1 red ball exactly 5 times."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-71a97ae50bf5fddbd0d044f5affdc5b630b8eb644a5cb04acbf08901c42e062f | A box contains 6 red balls and 4 yellow balls. Two balls are picked randomly and returned to the box. This process is repeated 8 times. Determine the probability of picking 2 yellow balls exactly 4 times. | [
"\\( 70 \\left( \\frac{2}{15} \\right)^4 \\left( \\frac{13}{15} \\right)^4 \\)"
] | [
"A box contains 6 red balls and 4 yellow balls. Two balls are picked randomly and returned to the box. This process is repeated 8 times. Determine the probability of picking 2 yellow balls exactly 4 times."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-357dc53b5e30a4d45f45367a65a600e270ef1dfc6b189cb7986ba549d85d2563 | A box contains 60 black and 40 white balls. Balls are randomly picked and removed from the box without replacement until the box is empty. A color change occurs when a ball is immediately followed by a ball of the other color. What is the expected number of color changes? | [
"48"
] | [
"A box contains 60 black and 40 white balls. Balls are randomly picked and removed from the box without replacement until the box is empty. A color change occurs when a ball is immediately followed by a ball of the other color. What is the expected number of color changes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-5ed3583dfa9d56467d961d6af1350ca844c91f00097272f560df6450ef941da3 | A box contains 7 identical white balls and 5 identical black balls. They are drawn at random, one at a time without replacement, until the box is empty. Find the probability that the 6th ball drawn is white, while exactly 3 black balls are drawn before that. | [
"\\(\\frac{25}{132}\\)"
] | [
"A box contains 7 identical white balls and 5 identical black balls. They are drawn at random, one at a time without replacement, until the box is empty. Find the probability that the 6th ball drawn is white, while exactly 3 black balls are drawn before that."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-890aab5261c5c2103d9d1ba5c902fe034bb59f5c8884addbb7120e17bbdd9ded | A box contains 7 identical white balls and 5 identical black balls. They are drawn randomly, one at a time without replacement, until the box is empty. Find the probability that the 6th ball drawn is white, while before that exactly 3 black balls are drawn. | [
"\\(\\frac{25}{132}\\)"
] | [
"A box contains 7 identical white balls and 5 identical black balls. They are drawn randomly, one at a time without replacement, until the box is empty. Find the probability that the 6th ball drawn is white, while before that exactly 3 black balls are drawn."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-d89ff8c5989d7ddcdbf0d5c02af2d83122538ed5179909770328c24c323f4d69 | A box contains 7 white balls and 5 black balls. Balls are drawn randomly one at a time until the box is empty. Find the probability that the 6th ball drawn is white, given that exactly 3 black balls are drawn before the 6th ball. | [
"\\(\\frac{25}{132}\\)"
] | [
"A box contains 7 white balls and 5 black balls. Balls are drawn randomly one at a time until the box is empty. Find the probability that the 6th ball drawn is white, given that exactly 3 black balls are drawn before the 6th ball."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-70ef65b80b44e7031dc357e3ac3c71be09c50a65ac82968db9c133d1f88c965e | A box contains 75 balls numbered from 1 to 75. If 10 balls are drawn with replacement, what is the probability that at least two of them have the same number? | [
"\\( 1 - \\frac{74!}{65! \\times 75^9} \\)"
] | [
"A box contains 75 balls numbered from 1 to 75. If 10 balls are drawn with replacement, what is the probability that at least two of them have the same number?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 19
} | 30 | 0 | 0 |
omr-ws-41eb7910dcf38a4ead1d537e7e0b55e571a344ec95ad243e49e301e45f68089b | A box contains 9 cards labeled from 1 to 9. Two cards are chosen at a time, and the smaller number among them is denoted as $X$. The cards are then returned to the box, and two cards are chosen again, with the smaller number among them denoted as $Y$. Find the probability that $X = Y$. | [
"\\(\\frac{17}{108}\\)"
] | [
"A box contains 9 cards labeled from 1 to 9. Two cards are chosen at a time, and the smaller number among them is denoted as $X$. The cards are then returned to the box, and two cards are chosen again, with the smaller number among them denoted as $Y$. Find the probability that $X = Y$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-95e377baaa7bb62aadd72584c214af8b1bd92db1c083ed18abe4c73999a73e28 | A box contains 9 yellow balls and \( n \) pink balls. Each ball is marked with a distinct integer from 1 to 100 inclusive. The sum of the numbers on two balls is the value of that pair of balls. What is the least number of pink balls in the box such that at least 2 pairs, consisting of one yellow and one pink ball, hav... | [
"22"
] | [
"A box contains 9 yellow balls and \\( n \\) pink balls. Each ball is marked with a distinct integer from 1 to 100 inclusive. The sum of the numbers on two balls is the value of that pair of balls. What is the least number of pink balls in the box such that at least 2 pairs, consisting of one yellow and one pink ba... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-e02882ce8dc24873ce6f082a35b45430c365524fab95c0823cb9ca2e6198a3a5 | A box contains 900 cards numbered from 100 to 999. How many cards must Paulo take out of the box to be sure of finding at least three cards whose digit sums are the same? | [
"53"
] | [
"A box contains 900 cards numbered from 100 to 999. How many cards must Paulo take out of the box to be sure of finding at least three cards whose digit sums are the same?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 23
} | 71 | 0 | 0 |
omr-ws-e88e79ee99d7be97fa1bd0c778f66e19fea4db3ae3f3d21cf0e9d256d2ff4729 | A box contains 900 cards, labeled from 100 to 999. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits? | [
"53"
] | [
"A box contains 900 cards, labeled from 100 to 999. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 22
} | 70 | 0 | 0 |
omr-ws-155bdc8a3e738b1e0fb3b356f946cd8b3a44573b3b3741582e1d0623b1f12fa9 | A box contains \( a \) white balls and \( b \) black balls, and a large pile of black balls lies beside the box. Two balls are chosen at random and taken out of the box. If they are of the same color, a black ball from the pile is put into the box; otherwise, the white ball is put back into the box. The procedure is re... | [
"\\dfrac{1 - (-1)^a}{2}"
] | [
"A box contains \\( a \\) white balls and \\( b \\) black balls, and a large pile of black balls lies beside the box. Two balls are chosen at random and taken out of the box. If they are of the same color, a black ball from the pile is put into the box; otherwise, the white ball is put back into the box. The proced... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ab7765dd3dd298ce30493e2f33e9bfa5b9b1ba030a8ad7f99b288e678af4b228 | A box contains \( m \) tickets numbered 1, 2, ..., \( m \). The tickets are drawn one at a time at random without replacement. Let \( A_i \) be the event "the number on the \( i \)-th ticket drawn is \( i \)". Find the limiting probability \( \lim_{m \to \infty} P(\cup_{i=1}^{m} A_i) \). | [
"\\(\\frac{e-1}{e}\\)"
] | [
"A box contains \\( m \\) tickets numbered 1, 2, ..., \\( m \\). The tickets are drawn one at a time at random without replacement. Let \\( A_i \\) be the event \"the number on the \\( i \\)-th ticket drawn is \\( i \\)\". Find the limiting probability \\( \\lim_{m \\to \\infty} P(\\cup_{i=1}^{m} A_i) \\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-b776311703b34ab8c1742bf89459dc78f7db31a6da3906468d7487d25379f13c | A box contains \( m \) tickets numbered 1, 2, ..., \( m \). The tickets are drawn one at a time at random without replacement. Let \( A_i \) be the event "the number on the \( i \)-th ticket drawn is \( i \)". Find the probability \( P(\cup_{i=1}^{m} A_i) \) when \( m = 3 \). | [
"\\(\\frac{2}{3}\\)"
] | [
"A box contains \\( m \\) tickets numbered 1, 2, ..., \\( m \\). The tickets are drawn one at a time at random without replacement. Let \\( A_i \\) be the event \"the number on the \\( i \\)-th ticket drawn is \\( i \\)\". Find the probability \\( P(\\cup_{i=1}^{m} A_i) \\) when \\( m = 3 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-cd3698a20b6612dbe058ab43c36f11e510c990d67527f45e181265e372b84a87 | A box contains \( n \) identical balls numbered 1 through \( n \). Suppose \( k \) balls are drawn in succession. What is the probability that \( m \) is the largest number drawn? | [
"\\dfrac{\\dbinom{m-1}{k-1}}{\\dbinom{n}{k}}"
] | [
"A box contains \\( n \\) identical balls numbered 1 through \\( n \\). Suppose \\( k \\) balls are drawn in succession. What is the probability that \\( m \\) is the largest number drawn?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-75b03e5408e408bba288873a28db798dfcc1a6f4ca4c88ab882bc18e2940c355 | A box contains \( n \) identical balls numbered 1 through \( n \). Suppose \( k \) balls are drawn in succession. What is the probability that the largest number drawn is less than or equal to \( m \)? | [
"\\(\\frac{\\binom{m}{k}}{\\binom{n}{k}}\\)"
] | [
"A box contains \\( n \\) identical balls numbered 1 through \\( n \\). Suppose \\( k \\) balls are drawn in succession. What is the probability that the largest number drawn is less than or equal to \\( m \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-397c7751743f03a2d395fb668fa78a006de001431183860eb9613b91e1ba8265 | A box contains \( n \) red balls and \( n \) black balls. We repeatedly remove two balls at a time and either discard them if they are of the same color or discard the black one and return the red one if they are of different colors. What is the probability that this process will terminate with one red ball in the box? | [
"\\(\\frac{1}{2} - \\frac{1}{2}(-1)^n\\)"
] | [
"A box contains \\( n \\) red balls and \\( n \\) black balls. We repeatedly remove two balls at a time and either discard them if they are of the same color or discard the black one and return the red one if they are of different colors. What is the probability that this process will terminate with one red ball in... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 14
} | 26 | 0 | 0 |
omr-ws-93a2f4e465c1b92dda3a494cea9629d15bf3936dadb792b5f1ccae6e5f0881c1 | A box contains a collection of $9$ cent stamps and $14$ cent stamps. The total value of the $9$ cent stamps is twice the total value of the $14$ cent stamps and the total value of all the stamps is less than $\$38.00$. What is the maximum number of stamps in the box? | [
"370"
] | [
"A box contains a collection of $9$ cent stamps and $14$ cent stamps. The total value of the $9$ cent stamps is twice the total value of the $14$ cent stamps and the total value of all the stamps is less than $\\$38.00$. What is the maximum number of stamps in the box?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-461e86568dc9538d74d59df5c366ea9da41061bf25c3b44956e9e4dd03808f3c | A box contains chips, each of which is red, white, or blue. The number of blue chips is at least half the number of white chips, and at most one third the number of red chips. The number of white or blue chips is at least 55. What is the minimum number of red chips? | [
"57"
] | [
"A box contains chips, each of which is red, white, or blue. The number of blue chips is at least half the number of white chips, and at most one third the number of red chips. The number of white or blue chips is at least 55. What is the minimum number of red chips?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-16f583305013e0b8a85f6861df94446b9f9b844dc0a74fcf6baad79c21ec960f | A box contains coupons labeled $1, 2, 3, \ldots, 99, 100$. Five coupons are picked at random one after another without replacement. Let the coupons be $x_1, x_2, x_3, x_4, x_5$ in order. The probability that $x_1 > x_2 > x_3 < x_4 < x_5$ is $\frac{a}{b}$. Find $\boxed{\frac{a+b}{7}}$. | [
"3"
] | [
"A box contains coupons labeled $1, 2, 3, \\ldots, 99, 100$. Five coupons are picked at random one after another without replacement. Let the coupons be $x_1, x_2, x_3, x_4, x_5$ in order. The probability that $x_1 > x_2 > x_3 < x_4 < x_5$ is $\\frac{a}{b}$. Find $\\boxed{\\frac{a+b}{7}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-c8d3af6c1a20c6fe5d246b33fcc968ed400c39ecf7467644d0591cc8250b29c4 | A box contains coupons labeled 1, 2, 3, ..., 99, 100. Five coupons are picked at random one after another without replacement. Let the coupons be \( x_1, x_2, x_3, x_4, x_5 \) in order. The probability that \( x_1 > x_2 > x_3 < x_4 < x_5 \) is \( \frac{a}{b} \). Find \( \frac{a+b}{7} \). | [
"3"
] | [
"A box contains coupons labeled 1, 2, 3, ..., 99, 100. Five coupons are picked at random one after another without replacement. Let the coupons be \\( x_1, x_2, x_3, x_4, x_5 \\) in order. The probability that \\( x_1 > x_2 > x_3 < x_4 < x_5 \\) is \\( \\frac{a}{b} \\). Find \\( \\frac{a+b}{7} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e9bfe2f22ef1e8057bf829599660f76d73ad3810c906d97b14bc73d0db011360 | A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white? | [
"\\(\\frac{3}{5}\\)"
] | [
"A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?"
] | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 9
} | 9 | 0 | 0 |
omr-ws-6457bc5528a468e2953b879d4ce4c7bfb4297985a0b62bf4ec6f83ae689edf0f | A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected? | [
"\\(\\frac{3}{10}\\)"
] | [
"A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-87174d765b117d04ccfdca4af6f8eb600ee9e8f21d25e9fc87ca91061195c52c | A box contains five marbles, which are colored red, green, blue, black, and white. Hue removes marbles from the box, one marble at a time, until he removes both the black and white marbles. Compute the number of ways for Hue to complete this task. | [
"98"
] | [
"A box contains five marbles, which are colored red, green, blue, black, and white. Hue removes marbles from the box, one marble at a time, until he removes both the black and white marbles. Compute the number of ways for Hue to complete this task."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fa10905df6d1579e40eba19dd6cf3138a3526a67ec250e0575f2ce87d76e0311 | A box contains five red and five blue balls. A ball is picked from the box and is replaced by a ball of the other colour. This process is repeated ten times. What is the probability that a ball picked from the box after these ten replacements is red? | [
"\\(\\frac{1}{2}\\)"
] | [
"A box contains five red and five blue balls. A ball is picked from the box and is replaced by a ball of the other colour. This process is repeated ten times. What is the probability that a ball picked from the box after these ten replacements is red?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-77e0b143f06c55e1eb1bc2c88d879056f2d508b38cfadce6dda9081be128ec0c | A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven p... | [
"0"
] | [
"A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among se... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fe5442c2f2e2143fbef229ece586da528ba78d28846ca2559d1638aa756c12fe | A box contains marbles, each of which is red, white, or blue. The number of blue marbles is at least half the number of white marbles and at most one third the number of red marbles. The number of white or blue marbles is at least 55. Find the minimum possible number of red marbles. | [
"57"
] | [
"A box contains marbles, each of which is red, white, or blue. The number of blue marbles is at least half the number of white marbles and at most one third the number of red marbles. The number of white or blue marbles is at least 55. Find the minimum possible number of red marbles."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-bf70550cbc00c32436cd30f2e6ceb33d8fb016285a8b50264e287b6e9b60c5f4 | A box contains nine reams of paper, with each ream having 2000 sheets. How many 200-page books can be printed with one box? | [
"90"
] | [
"A box contains nine reams of paper, with each ream having 2000 sheets. How many 200-page books can be printed with one box?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 2
} | 15 | 0 | 0 |
omr-ws-a0daab9eec1b73d318f732fd476ad41fb8a25ef5fe31af0fcf407f3812a59723 | A box contains red, green, blue, and white balls, totaling 111 balls. What is the smallest number of balls you must take out without looking to guarantee that among them there will always be balls of at least 3 different colors? | [
"88"
] | [
"A box contains red, green, blue, and white balls, totaling 111 balls. What is the smallest number of balls you must take out without looking to guarantee that among them there will always be balls of at least 3 different colors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-71f2a08ba920c80223c9730cee32b40fdab32141e718950ddd406cec9bd8b90e | A box contains seven cards numbered from $301$ to $307$. Graham picks three cards from the box and then Zoe picks two cards from the remainder. Graham looks at his cards and then says "I know that the sum of the numbers on your cards is even". What is the sum of the numbers on Graham's cards? | [
"912"
] | [
"A box contains seven cards numbered from $301$ to $307$. Graham picks three cards from the box and then Zoe picks two cards from the remainder. Graham looks at his cards and then says \"I know that the sum of the numbers on your cards is even\". What is the sum of the numbers on Graham's cards?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-fc02dfecf0f1f84b9f6f9139c8d5e66ca037b42248d1acfd376e53f7692ad81d | A box contains six cards. Three of the cards are black on both sides, one card is black on one side and red on the other, and two of the cards are red on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red, what is the probability that the other sid... | [
"\\frac{4}{5}"
] | [
"A box contains six cards. Three of the cards are black on both sides, one card is black on one side and red on the other, and two of the cards are red on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red, what is the probability that the ot... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-35c069c3226ffe653904dbcc000c1ca5b7e863199885e8430eab6fc85a4cd1c0 | A box contains some identical tennis balls. The ratio of the total volume of the tennis balls to the volume of empty space surrounding them in the box is 1 : k, where k is an integer greater than 1. A prime number of tennis balls is removed from the box, and the ratio of the total volume of the remaining tennis balls t... | [
"5"
] | [
"A box contains some identical tennis balls. The ratio of the total volume of the tennis balls to the volume of empty space surrounding them in the box is 1 : k, where k is an integer greater than 1. A prime number of tennis balls is removed from the box, and the ratio of the total volume of the remaining tennis ba... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-d9641f142d0b7a9fa0fd9adbef9fce7188f52a61b98c52d772687765779b1c1d | A box contains some identical tennis balls. The ratio of the total volume of the tennis balls to the volume of empty space surrounding them in the box is 1 : k, where k is an integer greater than 1. A prime number of tennis balls is removed from the box, and the ratio of the total volume of the remaining tennis balls t... | [
"k + 1"
] | [
"A box contains some identical tennis balls. The ratio of the total volume of the tennis balls to the volume of empty space surrounding them in the box is 1 : k, where k is an integer greater than 1. A prime number of tennis balls is removed from the box, and the ratio of the total volume of the remaining tennis ba... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-a2017588744bf8dd2ea2ecc16edfae887fcbe3984e8ab21c7509d0bc75c691e6 | A box contains ten balls labeled 0 through 9. Balls are picked randomly with replacement, and the number on each ball is written down. This process is repeated five times, resulting in five numbers. Let \( x \) be the random variable representing the maximum number written down. Find the distribution probability of the... | [
"\\( P(X = k) = \\left(\\frac{k+1}{10}\\right)^5 - \\left(\\frac{k}{10}\\right)^5 \\) for \\( k = 0, 1, 2, \\ldots, 9 \\)."
] | [
"A box contains ten balls labeled 0 through 9. Balls are picked randomly with replacement, and the number on each ball is written down. This process is repeated five times, resulting in five numbers. Let \\( x \\) be the random variable representing the maximum number written down. Find the distribution probability... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-b953cd77b2b4c41eefad4f8c6a92b461002d711e4767ee4cb1a770920731e2b1 | A box contains three red balls and \( N \) white balls. One ball is randomly drawn and then a white ball is added to the box. This process is repeated \( k \) times. Find the expected value of the number of red balls remaining in the box after \( k \) draws. | [
"\\( 3 \\left( \\frac{n+2}{n+3} \\right)^k \\)"
] | [
"A box contains three red balls and \\( N \\) white balls. One ball is randomly drawn and then a white ball is added to the box. This process is repeated \\( k \\) times. Find the expected value of the number of red balls remaining in the box after \\( k \\) draws."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-fbdd4ebb3d67cbb9d7a3ce0fe305dc11bf62e2ff5fa0b75dafea6a722deb8830 | A box contains tickets serially numbered from 1 to \( N \). Two tickets are drawn using simple random sampling without replacement (SRSWOR). If \( X \) and \( Y \) are the numbers on the tickets, find \( E(X \mid X > Y) \). | [
"\\(\\frac{2}{3}(N+1)\\)"
] | [
"A box contains tickets serially numbered from 1 to \\( N \\). Two tickets are drawn using simple random sampling without replacement (SRSWOR). If \\( X \\) and \\( Y \\) are the numbers on the tickets, find \\( E(X \\mid X > Y) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-909eef3105f70f0e07927f6cdd995a72812b8cc9a023f67c1e5a8b2b810873aa | A box contains tiles, numbered 1, 2, 3,..., 49, 50. Only tiles which are marked with a number congruent to $2 \pmod{5}$ are blue. One tile is chosen randomly from the box. What is the probability that the tile is blue? | [
" \\frac{1}{5} "
] | [
"A box contains tiles, numbered 1, 2, 3,..., 49, 50. Only tiles which are marked with a number congruent to $2 \\pmod{5}$ are blue. One tile is chosen randomly from the box. What is the probability that the tile is blue?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-092b464d33d431ee23ddfc9a5c1a546a53270531cd1aba0d1254c63cf0167138 | A box contains two orange balls and 5 amethyst balls. When a ball is drawn from the box, it is replaced by two balls of the opposite color. What is the probability that the 2nd ball drawn is amethyst? | [
"\\(\\frac{17}{28}\\)"
] | [
"A box contains two orange balls and 5 amethyst balls. When a ball is drawn from the box, it is replaced by two balls of the opposite color. What is the probability that the 2nd ball drawn is amethyst?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-978eb9da27ae92bf20d5eb0d5d369c098d9d0ce5783a99c99e31994c24cff3fb | A box contains two white socks, three blue socks, and four grey socks. Three of these socks have holes, but the colors of these socks are unknown. How many socks must Rachel take from the box without looking to be certain she has a pair of the same color without holes? | [
"7"
] | [
"A box contains two white socks, three blue socks, and four grey socks. Three of these socks have holes, but the colors of these socks are unknown. How many socks must Rachel take from the box without looking to be certain she has a pair of the same color without holes?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-15885dc2d0b2049fc3c50da79d077a56a1837d759bd3f7f218481a099ddd4964 | A box has 10 balls numbered 1 through 10. A ball is picked at random, and then a second ball is picked at random from the remaining nine balls. Find the probability that the numbers on the two selected balls differ by two or more. | [
"\\(\\frac{4}{5}\\)"
] | [
"A box has 10 balls numbered 1 through 10. A ball is picked at random, and then a second ball is picked at random from the remaining nine balls. Find the probability that the numbers on the two selected balls differ by two or more."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-00d40a74f983e4a98c59bcc8848ac9cd9205a90c1e8767a2af34ea20992c92dc | A box has 6 slips of paper numbered 1 through 6. 6 people, also conveniently numbered 1 through 6, each draw a slip of paper from the box, but out of sheer luck, none of them choose their own number. Each person has an object, and each second, they give their current object to the person whose number they chose. After ... | [
"299"
] | [
"A box has 6 slips of paper numbered 1 through 6. 6 people, also conveniently numbered 1 through 6, each draw a slip of paper from the box, but out of sheer luck, none of them choose their own number. Each person has an object, and each second, they give their current object to the person whose number they chose. A... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-51ca9c602e093a8f9f6582d8a282290a136df58b0c9b3d82e4024a5b50c2fda5 | A box has face diagonals of lengths 4, 6, and 7 decimeters. If one ball can fit into the box, will all 13 identical balls fit into the box? | [
"No"
] | [
"A box has face diagonals of lengths 4, 6, and 7 decimeters. If one ball can fit into the box, will all 13 identical balls fit into the box?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee2ccc21b237365189d7cb4c97eac713170524b46dcccc951ca5a09f709d3351 | A box is called $prime$ if the areas of its faces are all prime numbers. If the total surface area of a $prime$ box is $2020$, what is the maximum possible volume of the box? | [
"\\(\\sqrt{507982}\\)"
] | [
"A box is called $prime$ if the areas of its faces are all prime numbers. If the total surface area of a $prime$ box is $2020$, what is the maximum possible volume of the box?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 22,
"tir": 23
} | 45 | 0 | 0 |
omr-ws-e39a6d64d696856f799dbd396d0eb3886f4fcbabc543526002f00e5f1b04bfd3 | A box is to be built in the form of a rectangular parallelepiped with base area 1 cm$^2$. The sum of the lengths of all edges of the parallelepiped must equal 20 cm. For which box dimensions is the total surface area a maximum? | [
"2 cm, 1/2 cm and 5/2 cm"
] | [
"A box is to be built in the form of a rectangular parallelepiped with base area 1 cm$^2$. The sum of the lengths of all edges of the parallelepiped must equal 20 cm. For which box dimensions is the total surface area a maximum?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b7407636c4d8f3d2d77365d1e1010b438c116ede210135ad895b130abf16ee9e | A box of 100 personalized pencils costs $\$30$. How many dollars does it cost to buy 2500 pencils? | [
"\\$750"
] | [
"A box of 100 personalized pencils costs $\\$30$. How many dollars does it cost to buy 2500 pencils?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d148fc49f872c2c7c8b8c213b51a0a2c0f5cfa587242fdfd05e8d9310bea6d7f | A box of 25 chocolate candies costs $\$6$. How many dollars does it cost to buy 600 chocolate candies? | [
"144"
] | [
"A box of 25 chocolate candies costs $\\$6$. How many dollars does it cost to buy 600 chocolate candies?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90c9b122bbab711300b6c503f9f123dad41a396d9eef8d9269583ac6bb74b0ec | A box of chocolates has 36 spots for chocolates. There are 10 different chocolates to choose from, and each type must be used at least once. How many different ways can the box be filled if the order of the chocolates does not matter, only the quantity of each type? | [
"\\dbinom{35}{9}"
] | [
"A box of chocolates has 36 spots for chocolates. There are 10 different chocolates to choose from, and each type must be used at least once. How many different ways can the box be filled if the order of the chocolates does not matter, only the quantity of each type?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-b34d1452dd9773ec5cef3f2982ec7c209cbcd42c4402514de64ce2e558b19550 | A box of dominoes contains all possible dominoes with numbers from $0$ to $24$. A chain of dominoes with length $n$ is a series of dominoes where adjacent numbers on different dominoes are the same. Given that one of the dominoes has a $6$ and the other one has a $9$, the probability that a chain involving all dominoes... | [
"1"
] | [
"A box of dominoes contains all possible dominoes with numbers from $0$ to $24$. A chain of dominoes with length $n$ is a series of dominoes where adjacent numbers on different dominoes are the same. Given that one of the dominoes has a $6$ and the other one has a $9$, the probability that a chain involving all dom... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-419d92b56ad812493b7a3f57846d13477acf7630656a2815a49d953949d3e7f6 | A box of pencils costs the same as 11 erasers and 7 pencils. A box of erasers costs the same as 6 erasers and a pencil. A box of empty boxes and an eraser costs the same as a pencil. Given that boxes cost a penny and each of the boxes contain an equal number of objects, how much does it cost to buy a box of pencils and... | [
"\\dfrac{113}{11}"
] | [
"A box of pencils costs the same as 11 erasers and 7 pencils. A box of erasers costs the same as 6 erasers and a pencil. A box of empty boxes and an eraser costs the same as a pencil. Given that boxes cost a penny and each of the boxes contain an equal number of objects, how much does it cost to buy a box of pencil... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-bef40bfbb2d601cfcc5e3b834dd6b8ab739af754027d03d0d84ed59ad41ba560 | A box whose shape is a parallelepiped can be completely filled with cubes of side $1$. If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40\%$ of the volume of the box is occupied. Determine the possible dimensions of the box. | [
"\\(\\{2, 5, 3\\}\\) or \\(\\{2, 5, 6\\}\\)"
] | [
"A box whose shape is a parallelepiped can be completely filled with cubes of side $1$. If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40\\%$ of the volume of the box is occupied. Determine the possible dimensions of the box."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 15
} | 24 | 0 | 0 |
omr-ws-12f4e7f92cd417d6def3504c653c81a6b6a365a73f153f721a8eff5aac254761 | A box with a square base and an open top is constructed by cutting out four congruent squares from the corners of a 10x10 square of cardboard and folding up the flaps. Find the maximum volume of such a box and give its dimensions. | [
"\\(\\frac{2000}{27}\\) for the volume and \\(\\frac{20}{3} \\times \\frac{20}{3} \\times \\frac{5}{3}\\) for the dimensions."
] | [
"A box with a square base and an open top is constructed by cutting out four congruent squares from the corners of a 10x10 square of cardboard and folding up the flaps. Find the maximum volume of such a box and give its dimensions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 20
} | 23 | 0 | 0 |
omr-ws-2259e97302e02ffc6c3134a4c823cf5355cc2cec75da82a869a31f9829eaba01 | A box with a square base and no lid is to be made from a square piece of metal by cutting squares from the corners and folding up the sides. The cut-off squares are 5 cm on a side. If the volume of the box is $100c^3$, find the dimensions of the original piece of metal. | [
"10 + 2\\sqrt{5}"
] | [
"A box with a square base and no lid is to be made from a square piece of metal by cutting squares from the corners and folding up the sides. The cut-off squares are 5 cm on a side. If the volume of the box is $100c^3$, find the dimensions of the original piece of metal."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-996c3e89f0c88f00696bd28bcd84d240337e3d88352ef8261a983a17663c1106 | A box with a square base and no lid is to be made from a square piece of metal by cutting squares from the corners and folding up the sides. The cut-off squares are 5 cm on a side. If the volume of the box is 100 cm³, find the dimensions of the original piece of metal. | [
"\\( 10 + 2\\sqrt{5} \\)"
] | [
"A box with a square base and no lid is to be made from a square piece of metal by cutting squares from the corners and folding up the sides. The cut-off squares are 5 cm on a side. If the volume of the box is 100 cm³, find the dimensions of the original piece of metal."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-861643676809eb4b0bbcbb9972f425d8c3923050844111525acd38e4c1064296 | A box with a square base and no lid is to be made from a square piece of metal by cutting squares from the corners and folding up the sides. The cut-off squares are 5 cm on a side. If the volume of the box is \(100 \text{ cm}^3\), find the dimensions of the original piece of metal. | [
"10 + 2\\sqrt{5} \\text{ cm}"
] | [
"A box with a square base and no lid is to be made from a square piece of metal by cutting squares from the corners and folding up the sides. The cut-off squares are 5 cm on a side. If the volume of the box is \\(100 \\text{ cm}^3\\), find the dimensions of the original piece of metal."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c695031882b069a655e0b9f4957bc2f6abc1e62a3722c5973247461b603b3ec5 | A box with a volume of 16 $\text{cm}^3$ can hold 50 paperclips. How many paperclips could a box with a volume of 48 $\text{cm}^3$ hold? | [
"150"
] | [
"A box with a volume of 16 $\\text{cm}^3$ can hold 50 paperclips. How many paperclips could a box with a volume of 48 $\\text{cm}^3$ hold?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dc3229f07f185dee35dc0c42488d964745446304a8f34e06c25dbc4abb38cea9 | A box with dimensions $5 \text{ cm} \times 6 \text{ cm} \times 12 \text{ cm}$ is to have an equal increase to each edge to increase its volume by $1080 \text{ cm}^3$. Find the increase in each edge. | [
"4"
] | [
"A box with dimensions $5 \\text{ cm} \\times 6 \\text{ cm} \\times 12 \\text{ cm}$ is to have an equal increase to each edge to increase its volume by $1080 \\text{ cm}^3$. Find the increase in each edge."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-7f0f9950f7821be7c97598acdf5ce3dd9c908e4680fa6fd12c9d66c407758556 | A box with dimensions \( a \times b \times c \) is paved by 3D domino blocks of size \( 1 \times 1 \times 2 \) such that each side of the box has an equal number of blocks paralleling it. Determine the possible dimensions of the box. | [
"4k \\times 4k \\times 4k"
] | [
"A box with dimensions \\( a \\times b \\times c \\) is paved by 3D domino blocks of size \\( 1 \\times 1 \\times 2 \\) such that each side of the box has an equal number of blocks paralleling it. Determine the possible dimensions of the box."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-deb0ba08ae7d1444d82d721c78d0199eaa84c1d813abf6b129c7a84736a5bf72 | A box with edge lengths $l, m, n \in \mathbb{N}$ is divided into $lmn$ unit cubes. Determine how many of these unit cubes a main diagonal of the box intersects. | [
"\\( l + m + n - \\gcd(l, m) - \\gcd(m, n) - \\gcd(n, l) + \\gcd(l, m, n) \\)"
] | [
"A box with edge lengths $l, m, n \\in \\mathbb{N}$ is divided into $lmn$ unit cubes. Determine how many of these unit cubes a main diagonal of the box intersects."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-1777fbcfc8d8483329c7d4e50c34d846bb8f14d0da1d10b892c082ab027c9de4 | A box with height \((x+5)\) cm has a square base with side \(x\) cm. A second box with height \((x+2)\) cm has a square base with side \((x+1)\) cm. If the two boxes have the same volume, find \(x\). | [
"\\( x = 2.5 + \\sqrt{8.25} \\)"
] | [
"A box with height \\((x+5)\\) cm has a square base with side \\(x\\) cm. A second box with height \\((x+2)\\) cm has a square base with side \\((x+1)\\) cm. If the two boxes have the same volume, find \\(x\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-d567065eb1b2db6bba887b1f4390cec6b566e9788e9462baf4be2440ea1b2da1 | A boy and his teacher are at opposite ends of a 100-meter circular pool. The teacher runs around the circumference of the pool at a speed six times that of the boy's swimming speed. If the boy starts swimming directly towards the center of the pool and then swims towards the edge at an angle $\theta$ to escape, find th... | [
"9.59"
] | [
"A boy and his teacher are at opposite ends of a 100-meter circular pool. The teacher runs around the circumference of the pool at a speed six times that of the boy's swimming speed. If the boy starts swimming directly towards the center of the pool and then swims towards the edge at an angle $\\theta$ to escape, f... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-a506f3ed3b44dfe247bb856115f5c2393300970c996d99b28e8b1fdaf37fd4bd | A boy at point $A$ wants to get water at a circular lake and carry it to point $B$. Find the point $C$ on the lake such that the distance walked by the boy is the shortest possible, given that the line $AB$ and the lake are exterior to each other. | [
"C"
] | [
"A boy at point $A$ wants to get water at a circular lake and carry it to point $B$. Find the point $C$ on the lake such that the distance walked by the boy is the shortest possible, given that the line $AB$ and the lake are exterior to each other."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-b5ffb17b7f051cfac0d01e9c8565428f78b1443fbe03c204a803c78f7c5a6d13 | A boy bought six pairs of jeans, each with a leg length of \( x \) cm, and four pairs of shorts, each with a leg length of \( (x-2) \) cm. His father shortened all the trousers by 4 cm. Find the variance of the leg lengths after the alteration. | [
"\\dfrac{24}{25}"
] | [
"A boy bought six pairs of jeans, each with a leg length of \\( x \\) cm, and four pairs of shorts, each with a leg length of \\( (x-2) \\) cm. His father shortened all the trousers by 4 cm. Find the variance of the leg lengths after the alteration."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-354db43f523c427f5c042338aff825e655df61ac14e150e02956502530d2391e | A boy flies a kite at a constant height of 300 ft. The wind carries the kite horizontally at 25 ft/sec. How quickly is the string lengthening when the kite is 500 feet from the boy? | [
"20 ft/s"
] | [
"A boy flies a kite at a constant height of 300 ft. The wind carries the kite horizontally at 25 ft/sec. How quickly is the string lengthening when the kite is 500 feet from the boy?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-8452105f44948b51f52a6f62de0e1f1cf1105f3e2b29889d4e90d654445a4c3e | A boy has a $5 \times 5$ square grid. Initially, every cell of the grid is uncolored, except for the corner cells, which have four different colors: red, yellow, blue, and green. He wants to color some (possibly none) of the uncolored cells with red, yellow, blue, and green such that, for each color, the cells of that ... | [
"550"
] | [
"A boy has a $5 \\times 5$ square grid. Initially, every cell of the grid is uncolored, except for the corner cells, which have four different colors: red, yellow, blue, and green. He wants to color some (possibly none) of the uncolored cells with red, yellow, blue, and green such that, for each color, the cells of... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-5e0154a37729d809f0a17489fbe41862ff88555044dacb22b9e842d615ceea4e | A boy has a box of cookies which he doesn't know the number of. He decides that, in order to find out how many cookies are in the box, he will empty the box and arrange the cookies into a rectangle such that he can multiply the number of columns by the number of rows to find the number of cookies that were in the box w... | [
"29"
] | [
"A boy has a box of cookies which he doesn't know the number of. He decides that, in order to find out how many cookies are in the box, he will empty the box and arrange the cookies into a rectangle such that he can multiply the number of columns by the number of rows to find the number of cookies that were in the ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4cf261c860fec9ea5164b2bc4be6f3b99d90e43ea7ac9c7514c7c892613dd9a9 | A boy has a certain number of sweets. If he eats 16 sweets a day, they will last him 2 days longer than if he eats 18 sweets a day. How many sweets does he have? | [
"288"
] | [
"A boy has a certain number of sweets. If he eats 16 sweets a day, they will last him 2 days longer than if he eats 18 sweets a day. How many sweets does he have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-0c93da15f5b7992bc55bdf766a47128d0d5f2edf94926672c066180d7aa9fde1 | A boy has a collection of blue and green marbles. The number of blue marbles belongs to the set $\{1, 2, 3, \ldots, 13\}$. If two marbles are chosen simultaneously at random from his collection, the probability that they have different colors is $\frac{1}{2}$. Determine the possible number of blue marbles. | [
"\\(\\{1, 3, 6, 10\\}\\)"
] | [
"A boy has a collection of blue and green marbles. The number of blue marbles belongs to the set $\\{1, 2, 3, \\ldots, 13\\}$. If two marbles are chosen simultaneously at random from his collection, the probability that they have different colors is $\\frac{1}{2}$. Determine the possible number of blue marbles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 17
} | 18 | 0 | 0 |
omr-ws-06f3f239efe66f17639847034bd85353d64bf30d21c5ee971d3fa157a8f9b7f8 | A boy has four red marbles and eight blue marbles. He arranges his twelve marbles randomly in a ring. What is the probability that no two red marbles are adjacent? | [
"\\(\\frac{7}{33}\\)"
] | [
"A boy has four red marbles and eight blue marbles. He arranges his twelve marbles randomly in a ring. What is the probability that no two red marbles are adjacent?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 10
} | 23 | 0 | 0 |
omr-ws-0d4a81055a317e89c5f97f8fcba1849163e2e5b2457ea14edc1d3f4a75846be1 | A boy has the following seven coins in his pocket: $2$ pennies, $2$ nickels, $2$ dimes, and $1$ quarter. He takes out two coins, records the sum of their values, and then puts them back with the other coins. He continues to take out two coins, record the sum of their values, and put them back. How many different sums c... | [
"9"
] | [
"A boy has the following seven coins in his pocket: $2$ pennies, $2$ nickels, $2$ dimes, and $1$ quarter. He takes out two coins, records the sum of their values, and then puts them back with the other coins. He continues to take out two coins, record the sum of their values, and put them back. How many different s... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8cacef72cb7bbef3b1bd773135ea521e8212d0393e50055b542dfecc49d96e2a | A boy is allowed to play with 32 balls, where there are 8 balls of each color: Red, Blue, Yellow, and Green. Find the number of ways the boy can select 6 balls such that there is at least one ball of each color. | [
"1560"
] | [
"A boy is allowed to play with 32 balls, where there are 8 balls of each color: Red, Blue, Yellow, and Green. Find the number of ways the boy can select 6 balls such that there is at least one ball of each color."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cc3ccabb6da2550033297e27b760e8e69ca3815575c8a577aad522f9374b4695 | A boy is selected at random from among the children belonging to families with $n$ children. What is the probability that the boy has $k-1$ brothers? | [
"\\(\\frac{\\binom{n-1}{k-1}}{2^{n-1}}\\)"
] | [
"A boy is selected at random from among the children belonging to families with $n$ children. What is the probability that the boy has $k-1$ brothers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
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