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-c2d71c70eabf39cc3db6043121aa82811a4ad6f36fcc28c5fa5ccc909eb81981 | An organization of 100 people sets up a telephone call system where the initial contact person calls three people, each of whom calls three others, and so on, until all have been contacted. What is the maximum number of people who do not need to make a call? | [
"67"
] | [
"An organization of 100 people sets up a telephone call system where the initial contact person calls three people, each of whom calls three others, and so on, until all have been contacted. What is the maximum number of people who do not need to make a call?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 8
} | 27 | 0 | 0 |
omr-ws-945ca45ac16626d6d4ec3f82c169a9083621216502d43ef4c630f3b062870ff2 | An origin-symmetric parallelogram $P$ (that is, if $(x, y)$ is in $P$, then so is $(-x, -y)$) lies in the coordinate plane. It is given that $P$ has two horizontal sides, with a distance of $2020$ between them, and that there is no point with integer coordinates except the origin inside $P$. Also, $P$ has the maximum p... | [
"\\dfrac{1011}{1010}"
] | [
"An origin-symmetric parallelogram $P$ (that is, if $(x, y)$ is in $P$, then so is $(-x, -y)$) lies in the coordinate plane. It is given that $P$ has two horizontal sides, with a distance of $2020$ between them, and that there is no point with integer coordinates except the origin inside $P$. Also, $P$ has the maxi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-73cf8baf4779ceef39c314f42a2f49acaa9f243c7770f1dabbe32290fdc9e87b | An original two-digit number is added to its reversal to produce a second number. The reversal of the second number is added to the second number to produce a third number. This process continues until the fifth number produced is the first palindromic number. Find the sum of all possible distinct values of the origina... | [
"330"
] | [
"An original two-digit number is added to its reversal to produce a second number. The reversal of the second number is added to the second number to produce a third number. This process continues until the fifth number produced is the first palindromic number. Find the sum of all possible distinct values of the or... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-a9dd0295100c4c454cb6b07c1a3931853c6c53259ab87ee9b31a4efbc62db496 | An outfielder throws a ball toward home plate with an initial velocity of 100 ft/sec. An infielder can position himself \( x \) feet from home plate and throw the ball with an initial velocity of 110 ft/sec. The ball's velocity \( v(t) \) satisfies the differential equation \( \frac{dv}{dt} = -\frac{v}{10} \). Let \( R... | [
"\\( x = 350 \\) and \\( R \\) is minimized when \\( x = 350 \\)."
] | [
"An outfielder throws a ball toward home plate with an initial velocity of 100 ft/sec. An infielder can position himself \\( x \\) feet from home plate and throw the ball with an initial velocity of 110 ft/sec. The ball's velocity \\( v(t) \\) satisfies the differential equation \\( \\frac{dv}{dt} = -\\frac{v}{10} ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b23bb8403296d32abbe80d459979a3e8b0b03f555f0d8dbf568ecc959e639f4b | An oval is constructed using four circular arcs. The arcs on the left and right are identical, as are the arcs above and below. The oval has vertical and horizontal lines of symmetry and fits exactly within a rectangle of dimensions 4 x 8. The radius of the smaller arcs is 1. Determine the radius of the larger arcs. | [
"6"
] | [
"An oval is constructed using four circular arcs. The arcs on the left and right are identical, as are the arcs above and below. The oval has vertical and horizontal lines of symmetry and fits exactly within a rectangle of dimensions 4 x 8. The radius of the smaller arcs is 1. Determine the radius of the larger arc... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-40e1198c63c8ecd7fbac8390b7fe081822ac5ed752bef40ba910873cb5013275 | An overgrown grass patch is in the shape of an equilateral triangle. John has a machine with a blade whose length is half the height of the triangle. John can trim a sector of grass with the radius of the sector being half the height of the triangle. Starting from one of the vertices, find the path John should follow t... | [
"h"
] | [
"An overgrown grass patch is in the shape of an equilateral triangle. John has a machine with a blade whose length is half the height of the triangle. John can trim a sector of grass with the radius of the sector being half the height of the triangle. Starting from one of the vertices, find the path John should fol... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-b081ee66e8ab39366107c5195db8a939ca6877abd7396a19806ddebb21b14a18 | An triangle with coordinates \((x_1,y_1)\), \((x_2, y_2)\), \((x_3,y_3)\) has centroid at \((1,1)\). The ratio between the lengths of the sides of the triangle is \(3:4:5\). Given that \[x_1^3 + x_2^3 + x_3^3 = 3x_1x_2x_3 + 20 \quad \text{and} \quad y_1^3 + y_2^3 + y_3^3 = 3y_1y_2y_3 + 21,\] the area of the triangle ca... | [
"107"
] | [
"An triangle with coordinates \\((x_1,y_1)\\), \\((x_2, y_2)\\), \\((x_3,y_3)\\) has centroid at \\((1,1)\\). The ratio between the lengths of the sides of the triangle is \\(3:4:5\\). Given that \n\\[x_1^3 + x_2^3 + x_3^3 = 3x_1x_2x_3 + 20 \\quad \\text{and} \\quad y_1^3 + y_2^3 + y_3^3 = 3y_1y_2y_3 + 21,\\]\nthe ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 1
} | 23 | 0 | 0 |
omr-ws-21da6ea46c25cdf838a1e61dadace410616c6c5c408c205d364864b54e0fd19f | An unbiased coin is tossed $N$ times. What is the probability that no two consecutive heads appear? | [
"\\(\\frac{\\text{(N+2)th Fibonacci number}}{2^N}\\)"
] | [
"An unbiased coin is tossed $N$ times. What is the probability that no two consecutive heads appear?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-06b109599b74ab316c710c975fcd3e4d308fe2ffe83639a723d230a0155b1b9b | An unbiased coin is tossed $n$ times. What is the expected value of $|H-T|$, where $H$ is the number of heads and $T$ is the number of tails? | [
"\\(\\frac{n\\binom{n-1}{\\lfloor\\frac{n-1}{2}\\rfloor}}{2^{n-1}}\\)"
] | [
"An unbiased coin is tossed $n$ times. What is the expected value of $|H-T|$, where $H$ is the number of heads and $T$ is the number of tails?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-c2369ee22c3de9bc834f1adc829eaffd3ff1f738e90feca14bc7b65d53703909 | An unbiased dice is tossed until a $"6"$ appears. Find the probability (in terms of $n$) that it takes at least $n$ throws for a $"6"$ to appear. | [
"\\((5/6)^{n-1}\\)"
] | [
"An unbiased dice is tossed until a $\"6\"$ appears. Find the probability (in terms of $n$) that it takes at least $n$ throws for a $\"6\"$ to appear."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-613dea588ec7f9e2e6e5271639ca0499b9959a6a77018e1984d9b5b4d03691e3 | An unbiased die is thrown until a number greater than 4 appears. Find the probability that an even number of throws is needed. | [
"\\(\\frac{2}{5}\\)"
] | [
"An unbiased die is thrown until a number greater than 4 appears. Find the probability that an even number of throws is needed."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-34ea1797a030000a0708bfc8da16788f38f0de201160b6caa11cb0ee1c1336ec | An unbiased die, with faces numbered $1, 2, 3, 4, 5,$ and $6$, is thrown $n$ times. What is the probability that only three of the numbers $1, 2, 3, 4, 5,$ and $6$ appear in the list of $n$ numbers? | [
"\\(\\frac{20\\cdot(3^n+3-3\\cdot2^n)}{6^n}\\)"
] | [
"An unbiased die, with faces numbered $1, 2, 3, 4, 5,$ and $6$, is thrown $n$ times. What is the probability that only three of the numbers $1, 2, 3, 4, 5,$ and $6$ appear in the list of $n$ numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-86214e9af382f91303649de52aeaebc510800978f41cb7dafea5f7535abc0f58 | An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches. If the distance between the points of contact of the belt with the pulleys is $24$ inches, then what is the distance between the centers of the pulleys in inches? | [
"26"
] | [
"An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches. If the distance between the points of contact of the belt with the pulleys is $24$ inches, then what is the distance between the centers of the pulleys in inches?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-dadf175ab1a9fe10c58069282f58c60bba41ae574028209cd4917a4ec681d33e | An underground burrow consists of an infinite sequence of rooms labeled by the integers $(\dots, -3, -2, -1, 0, 1, 2, 3,\dots)$. Initially, there are $k+1$ rabbits in room $k$ for each $k=0, 1, 2, \ldots, 9, 10$, and all other rooms are empty. Each minute, if a room has two or more rabbits, half of the rabbits (roundin... | [
"21"
] | [
"An underground burrow consists of an infinite sequence of rooms labeled by the integers $(\\dots, -3, -2, -1, 0, 1, 2, 3,\\dots)$. Initially, there are $k+1$ rabbits in room $k$ for each $k=0, 1, 2, \\ldots, 9, 10$, and all other rooms are empty. Each minute, if a room has two or more rabbits, half of the rabbits ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b8e40acac3e3a5be391c0d89831989e414714ce342851b348ee56b4293c84085 | An underground burrow consists of an infinite sequence of rooms labeled by the integers $(\dots, -3, -2, -1, 0, 1, 2, 3,\dots)$. Initially, there are $k+1$ rabbits in room $k$ for each $k=0, 1, 2, \ldots, 9, 10$, and all other rooms are empty. Each minute, if a room has two or more rabbits, half of the rabbits (roundin... | [
"\\(-26, -25, \\dots, 28, 30, 31, 32, 33, \\dots, 40\\)"
] | [
"An underground burrow consists of an infinite sequence of rooms labeled by the integers $(\\dots, -3, -2, -1, 0, 1, 2, 3,\\dots)$. Initially, there are $k+1$ rabbits in room $k$ for each $k=0, 1, 2, \\ldots, 9, 10$, and all other rooms are empty. Each minute, if a room has two or more rabbits, half of the rabbits ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 6,
"tir": 5
} | 12 | 0 | 0 |
omr-ws-fda52f2aafc24068e31fbec5210bd5c1446bee565cf8ac300479b7fe643a7f94 | An unfair 6-sided die with faces numbered 1-6 is rolled 3 times, and the numbers showing on the top face are added together. The sums of 3, 4, 5, 16, 17, and 18 all have the same non-zero probability of occurring. When rolling a single die, the probability that a 1 is rolled can be written as \(\frac{a}{b}\), for relat... | [
"37"
] | [
"An unfair 6-sided die with faces numbered 1-6 is rolled 3 times, and the numbers showing on the top face are added together. The sums of 3, 4, 5, 16, 17, and 18 all have the same non-zero probability of occurring. When rolling a single die, the probability that a 1 is rolled can be written as \\(\\frac{a}{b}\\), f... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 4
} | 25 | 0 | 0 |
omr-ws-7b85b576e3c8e7f82813c160b65864ab90af55a48e149aaf8fa65957c86a398a | An unfair coin has a $\frac{1}{3}$ chance of coming up heads and a $\frac{2}{3}$ chance of coming up tails. The coin is flipped until a head interrupts a string of tails or vice versa. What is the expected value of the number of flips? | [
"\\(\\frac{7}{2}\\)"
] | [
"An unfair coin has a $\\frac{1}{3}$ chance of coming up heads and a $\\frac{2}{3}$ chance of coming up tails. The coin is flipped until a head interrupts a string of tails or vice versa. What is the expected value of the number of flips?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-468b44dea5bc2abe4d3d484404379824f70407580927c331a8976224631a0ecd | An unfair coin has a $\frac{2}{3}$ probability of landing on heads. If the coin is flipped 50 times, what is the probability that the total number of heads is even? | [
"\\dfrac{1 + (1/3)^{50}}{2}"
] | [
"An unfair coin has a $\\frac{2}{3}$ probability of landing on heads. If the coin is flipped 50 times, what is the probability that the total number of heads is even?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-572ff2601dacfea59cb986108a764d8e9dcce7b7be77d6d44b46f77237522eb8 | An unfair coin has a \( \frac{2}{3} \) probability of turning up heads. If this coin is tossed 50 times, what is the probability that the total number of heads is even? | [
"\\(\\frac{1 - \\frac{1}{3^{50}}}{2}\\)",
"\\(\\frac{3^{50} + 1}{2 \\cdot 3^{50}}\\)"
] | [
"An unfair coin has a \\( \\frac{2}{3} \\) probability of turning up heads. If this coin is tossed 50 times, what is the probability that the total number of heads is even?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 6,
"tir": 30
} | 52 | 0 | 0 |
omr-ws-5f6bbffb269f3b942c1e09082b34c582f945475ecb98a6bd2b968e64c8c6764a | An unfair coin has a unique property such that the probability of obtaining exactly one head when flipped four times is $\frac{8}{21}$ the probability of obtaining exactly one head when flipped three times. The coin must either land heads or tails every flip, and the sum of the possible probabilities of obtaining heads... | [
"57"
] | [
"An unfair coin has a unique property such that the probability of obtaining exactly one head when flipped four times is $\\frac{8}{21}$ the probability of obtaining exactly one head when flipped three times. The coin must either land heads or tails every flip, and the sum of the possible probabilities of obtaining... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-caaa6c461a217fd32a21f043c6a3b9359475a1e4cede22ffe6858e2d2dfa1be4 | An unfair coin has probability \( p \) of coming up heads on a single toss. Let \( w \) be the probability that, in 5 independent tosses of this coin, heads come up exactly 3 times. If \( w = \frac{144}{625} \), then determine the possible values of \( p \). | [
"\\( p = \\frac{2}{5} \\) or \\( p \\) is a value between \\( \\frac{3}{5} \\) and \\( \\frac{4}{5} \\)"
] | [
"An unfair coin has probability \\( p \\) of coming up heads on a single toss. Let \\( w \\) be the probability that, in 5 independent tosses of this coin, heads come up exactly 3 times. If \\( w = \\frac{144}{625} \\), then determine the possible values of \\( p \\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-1fa87e32f956d73cf128665686b33ae0390d3031c5a3da517b4249845c2ff85c | An unfair coin has probability \( p \) of coming up heads on a single toss. Let \( w \) be the probability that, in 5 independent tosses of this coin, heads come up exactly 3 times. If \( w = \frac{144}{625} \), then determine the value of \( p \). | [
"p is not uniquely determined."
] | [
"An unfair coin has probability \\( p \\) of coming up heads on a single toss. Let \\( w \\) be the probability that, in 5 independent tosses of this coin, heads come up exactly 3 times. If \\( w = \\frac{144}{625} \\), then determine the value of \\( p \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a8db8042fc7f0192993c8ac883426223b65a791495064b1bf74c444e045e562d | An unfair coin has the property that when flipped 5 times, it has the same probability of turning 3 heads and 2 tails in any order as the probability of turning 4 heads and 1 tail in any order. What is the probability of getting a head in any one flip? | [
"0.5"
] | [
"An unfair coin has the property that when flipped 5 times, it has the same probability of turning 3 heads and 2 tails in any order as the probability of turning 4 heads and 1 tail in any order. What is the probability of getting a head in any one flip?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e41f71dd544a2f01f6af8b592424f0cb12080bc974a4f514463b7eb74932918 | An unfair coin is tossed 10 times, with the probability of heads on each toss being $0.4$. Let $a_n$ be the number of heads in the first $n$ tosses. Find the probability that $a_n/n \le 0.4$ for $n = 1, 2, \ldots, 9$ and $a_{10}/10 = 0.4$. | [
"\\(\\frac{23 \\cdot 2^4 \\cdot 3^6}{5^{10}}\\)"
] | [
"An unfair coin is tossed 10 times, with the probability of heads on each toss being $0.4$. Let $a_n$ be the number of heads in the first $n$ tosses. Find the probability that $a_n/n \\le 0.4$ for $n = 1, 2, \\ldots, 9$ and $a_{10}/10 = 0.4$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 11
} | 34 | 0 | 0 |
omr-ws-85d4e5bcedb3d3772f3f435738c3c4b0d8a78825c18453f8fc7c30f0464d2b3b | An unfair coin is tossed 10 times, with the probability of heads on each toss being 0.4. Let \( a_n \) be the number of heads in the first \( n \) tosses. Find the probability that \( \frac{a_n}{n} \leq 0.4 \) for \( n = 1, 2, \ldots, 9 \) and \( \frac{a_{10}}{10} = 0.4 \). | [
"\\(\\frac{16767}{9765625}\\)"
] | [
"An unfair coin is tossed 10 times, with the probability of heads on each toss being 0.4. Let \\( a_n \\) be the number of heads in the first \\( n \\) tosses. Find the probability that \\( \\frac{a_n}{n} \\leq 0.4 \\) for \\( n = 1, 2, \\ldots, 9 \\) and \\( \\frac{a_{10}}{10} = 0.4 \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-073ea145d33e21a8a5c52feb68c172585732959e84733c5f752de91a820809e4 | An unfair coin lands heads with probability $\frac{13-3\sqrt{13}}{26}$. Let $m$ be the expected value of the number of flips it takes to roll "HTHTHT...HT" (2020 HT's). Find the remainder when $m$ is divided by 1000. | [
"260"
] | [
"An unfair coin lands heads with probability $\\frac{13-3\\sqrt{13}}{26}$. Let $m$ be the expected value of the number of flips it takes to roll \"HTHTHT...HT\" (2020 HT's). Find the remainder when $m$ is divided by 1000."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ecfaa18b95f81758bd750be65f9e14ceb3c9d0c7bc4c9ba30a53a2dc87801c0d | An unfair coin lands heads with probability $\frac{1}{17}$ and tails with probability $\frac{16}{17}$. Matt flips the coin repeatedly until he flips at least one head and at least one tail. What is the expected number of times that Matt flips the coin? | [
"\\(\\frac{273}{16}\\)"
] | [
"An unfair coin lands heads with probability $\\frac{1}{17}$ and tails with probability $\\frac{16}{17}$. Matt flips the coin repeatedly until he flips at least one head and at least one tail. What is the expected number of times that Matt flips the coin?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-4fb96f71e12c7222bdaedcf4dc39f7295c71620683d0165bdf1b2c6f2cd25f76 | An unfair coin lands on heads with probability $\frac34$ and tails with probability $\frac14$. A heads flip gains $\$3$, but a tails flip loses $\$8$. What is the expected worth of a coin flip? Express your answer as a decimal rounded to the nearest hundredth. | [
"\\$0.25"
] | [
"An unfair coin lands on heads with probability $\\frac34$ and tails with probability $\\frac14$. A heads flip gains $\\$3$, but a tails flip loses $\\$8$. What is the expected worth of a coin flip? Express your answer as a decimal rounded to the nearest hundredth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8eb0135b74375b87c935d175e54895a80ec49311c1e44f2701a4d62fcf3dc9a0 | An unfair coin lands on tails with probability $\frac{15}{28}$. It is flipped until the number of tails exceeds the number of heads. What is the expected number of flips? | [
"\\(\\frac{43}{15}\\)"
] | [
"An unfair coin lands on tails with probability $\\frac{15}{28}$. It is flipped until the number of tails exceeds the number of heads. What is the expected number of flips?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-119031f868085aa71a3b877b99286f2f96164cdccbe396793669ffeca556fba5 | An unfair die is weighted so that the probability of getting any number $6, 5, 4, 3, 2, 1$ is in the ratio $1:3:6:10:15:21$, respectively. If you have two of these dice, what is the probability that the sum of the top-facing numbers is $8$? | [
"\\(\\frac{81}{3136}\\)"
] | [
"An unfair die is weighted so that the probability of getting any number $6, 5, 4, 3, 2, 1$ is in the ratio $1:3:6:10:15:21$, respectively. If you have two of these dice, what is the probability that the sum of the top-facing numbers is $8$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 28
} | 41 | 0 | 0 |
omr-ws-9f09f9d78388b45c94c49bc40e3f65108c43d5989085943a698f7ae8b1bce15b | An unfair football game is set up as follows: - 22 distinguishable players are split into two teams, "MOPpers" and "Cleaners," with each team having a positive number of players. - A team captain is chosen from each team. Let \( N \) be the number of ways this can be done. Find the remainder when \( N \) is divided by ... | [
"112"
] | [
"An unfair football game is set up as follows:\n- 22 distinguishable players are split into two teams, \"MOPpers\" and \"Cleaners,\" with each team having a positive number of players.\n- A team captain is chosen from each team.\nLet \\( N \\) be the number of ways this can be done. Find the remainder when \\( N \\... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-85ad9798713100b4af8569a24d98dce57152ed19a40e94e7a33579210102025f | An unit cube is dissected into \( n > 1 \) cuboids such that each cuboid's sides are parallel to that of the cube, and each cuboid touches at least one face of the cube. Calculate the minimum number of pins, each parallel to one of the edges of the cube, that must be inserted through the cube to ensure that at least on... | [
"4"
] | [
"An unit cube is dissected into \\( n > 1 \\) cuboids such that each cuboid's sides are parallel to that of the cube, and each cuboid touches at least one face of the cube. Calculate the minimum number of pins, each parallel to one of the edges of the cube, that must be inserted through the cube to ensure that at l... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-d83dcb6504f286aa8279096ee8ac231672255aee5f04cd312b9c7e5029b4f0b3 | An urn $U_1$ contains 2 white balls and 3 black balls. A second urn $U_2$ contains 2 white balls and 3 red balls. A ball is chosen at random from $U_1$ and placed into $U_2$. Then a ball is chosen at random from $U_2$ and placed back into $U_1$. Finally, two balls are drawn from $U_1$. Find the probability that $U_1$ h... | [
"$\\frac{5}{16}$"
] | [
"An urn $U_1$ contains 2 white balls and 3 black balls. A second urn $U_2$ contains 2 white balls and 3 red balls. A ball is chosen at random from $U_1$ and placed into $U_2$. Then a ball is chosen at random from $U_2$ and placed back into $U_1$. Finally, two balls are drawn from $U_1$. Find the probability that $U... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c518b59d7ed33f7081de4e9c9f4c898ae51b21b5d8fe8ae0a592363cb961f4f4 | An urn contains $2$ red and $4$ black balls. (a) A ball is drawn from the urn. If it is black then it is returned to the urn, but if it is red then it is not returned to the urn. A second ball is then drawn from the urn. What is the probability that one of the balls drawn was red and the other black? (b) A ball is draw... | [
"\\text{No}"
] | [
"An urn contains $2$ red and $4$ black balls.\n(a) A ball is drawn from the urn. If it is black then it is returned to the urn, but if it is red then it is not returned to the urn. A second ball is then drawn from the urn. What is the probability that one of the balls drawn was red and the other black?\n(b) A ball ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-deea69cb0df4c33f5fa5c8313e7ab1c1e94029fa9919dba7760497f84e01ec60 | An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$. | [
"144"
] | [
"An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$.\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fb7ed1a42255c4c1ee5e9090d9015dd8d532123ed7a1ccadebc1c104af7f7416 | An urn contains $B$ blue balls and $R$ red balls. Suppose that one extracts successively $n$ balls at random such that when a ball is chosen, it is returned to the urn along with $C$ extra balls of the same color. If $B=5$, $R=7$, and $C=3$, calculate the probability that the $10$-th ball extracted is red. Express your... | [
"\\dfrac{7}{12}"
] | [
"An urn contains $B$ blue balls and $R$ red balls. Suppose that one extracts successively $n$ balls at random such that when a ball is chosen, it is returned to the urn along with $C$ extra balls of the same color. If $B=5$, $R=7$, and $C=3$, calculate the probability that the $10$-th ball extracted is red. Express... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-08fbb529d1828145419ff99864761db3280239c5d0eb4e11ec330476751da900 | An urn contains $\theta$ black balls and $N-\theta$ white balls. A sample of $n$ balls is selected without replacement. Let $Y$ denote the number of black balls in the sample. If $N = 100$, $n = 20$, and the observed value of $Y$ is $8$, calculate the method-of-moments (MOM) estimate of $\theta$. | [
"40"
] | [
"An urn contains $\\theta$ black balls and $N-\\theta$ white balls. A sample of $n$ balls is selected without replacement. Let $Y$ denote the number of black balls in the sample. If $N = 100$, $n = 20$, and the observed value of $Y$ is $8$, calculate the method-of-moments (MOM) estimate of $\\theta$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4feb6b5313105e65e0d49dbb1fe03e718d28daf643522312e95c73904da993ca | An urn contains $a$ white balls and $b$ black balls. After a ball is drawn, it is returned to the urn if it is white, but if it is black, it is replaced by a white ball from another urn. Let $M_n$ denote the expected number of white balls in the urn after the operation has been repeated $n$ times. Calculate the value o... | [
"\\dfrac{586657}{100000}"
] | [
"An urn contains $a$ white balls and $b$ black balls. After a ball is drawn, it is returned to the urn if it is white, but if it is black, it is replaced by a white ball from another urn. Let $M_n$ denote the expected number of white balls in the urn after the operation has been repeated $n$ times. Calculate the va... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf68df4c643ca91211374c91cdda8651c0a2db5e4860c27727a083803760b8a2 | An urn contains $n$ balls labeled $1, 2, \ldots, n$. We draw the balls out one by one (without replacement) until we obtain a ball whose number is divisible by $k$. Find all $k$ such that the expected number of balls removed is $k$. | [
"k \\mid (n + 1)"
] | [
"An urn contains $n$ balls labeled $1, 2, \\ldots, n$. We draw the balls out one by one (without replacement) until we obtain a ball whose number is divisible by $k$. Find all $k$ such that the expected number of balls removed is $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-c4f0cffea79f111b33ed920c818e029d39dce1cf9dee5770dd1d2faea260ce39 | An urn contains $n$ balls labeled $1, 2, \ldots, n$. We draw the balls out one by one (without replacement) until we obtain a ball whose number is divisible by $k$. Find all values of $k$ such that the expected number of balls removed is $k$. | [
"\\( k \\) must be a divisor of \\( n \\)."
] | [
"An urn contains $n$ balls labeled $1, 2, \\ldots, n$. We draw the balls out one by one (without replacement) until we obtain a ball whose number is divisible by $k$. Find all values of $k$ such that the expected number of balls removed is $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-aa6415f8eced445c7d3e16ebe0dd09f5a67d5d875daba1b59793d8144095e50f | An urn contains $n$ balls of which one is special. If $k$ of these balls are withdrawn one at a time, with each selection being equally likely to be any of the balls that remain at that time, what is the probability that the special ball is chosen? | [
"\\(\\frac{k}{n}\\)"
] | [
"An urn contains $n$ balls of which one is special. If $k$ of these balls are withdrawn one at a time, with each selection being equally likely to be any of the balls that remain at that time, what is the probability that the special ball is chosen?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-6298393dbe4d3ef6f13b5fbd0db888861a2b3fbe13570281b49e39f8fd56a214 | An urn contains $n$ white balls and $n$ black balls. The balls are drawn one by one without replacement until only balls of one color remain. What is the order of magnitude of the number of balls left? | [
"\\( O(1) \\)"
] | [
"An urn contains $n$ white balls and $n$ black balls. The balls are drawn one by one without replacement until only balls of one color remain. What is the order of magnitude of the number of balls left?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-eb2f79797ed74fa5be79241b3241094eb6fc87e1ac3e20233826dc16ed4d43c2 | An urn contains $w$ white balls, $b$ black balls, and $r$ red balls, where $w, b, r > 0$. Balls are drawn one by one without replacement until all balls are exhausted. Find the probability that the white balls are exhausted first among the three colors. | [
"\\(\\frac{br}{(r+w)(b+r+w)} + \\frac{br}{(b+w)(b+r+w)}\\)"
] | [
"An urn contains $w$ white balls, $b$ black balls, and $r$ red balls, where $w, b, r > 0$. Balls are drawn one by one without replacement until all balls are exhausted. Find the probability that the white balls are exhausted first among the three colors."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dcfdc266caf80a567ee18bab59a2b54ddc269c8e884ccc5db26692adf97d936e | An urn contains 10 black, 10 white, and 10 blue balls. Balls are drawn randomly with replacement. Find the number of ways to have at least one ball of each of the three colors in at least \( n \) draws. | [
"30^n - 3 \\cdot 20^n + 3 \\cdot 10^n"
] | [
"An urn contains 10 black, 10 white, and 10 blue balls. Balls are drawn randomly with replacement. Find the number of ways to have at least one ball of each of the three colors in at least \\( n \\) draws."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-2154c0688fd34d4c389239d6e65c141120c197678fa493b7ae9c87933547d5b2 | An urn contains 16 balls of two colors, red and blue. Four balls are drawn from the urn without replacement. The probability of getting exactly 2 red balls and 2 blue balls is 1/20. The urn contains no more red than blue balls. How many blue balls were originally in the urn? | [
"14"
] | [
"An urn contains 16 balls of two colors, red and blue. Four balls are drawn from the urn without replacement. The probability of getting exactly 2 red balls and 2 blue balls is 1/20. The urn contains no more red than blue balls. How many blue balls were originally in the urn?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-74e6c8b8f67a17a65ae94a297f889b7384f74c7571b24b76c410f39770749643 | An urn contains 2 Black and 4 Red balls. A sample of 2 balls is drawn and then without replacement, two more balls are pulled out. a) Check whether the events: P: The balls of the first sample have the same color. Q: The balls of the second draw have the same color. Are independent? b) Find the probabilities for each o... | [
"\\dfrac{1}{2}"
] | [
"An urn contains 2 Black and 4 Red balls. A sample of 2 balls is drawn and then without replacement, two more balls are pulled out.\na) Check whether the events:\nP: The balls of the first sample have the same color.\nQ: The balls of the second draw have the same color.\nAre independent?\nb) Find the probabilities ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-973049a45b558cf9cf0dc0e8b7499bcb6e073bda27ba15d3f65bf76669f7aa62 | An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white, it is not replaced into the urn. Otherwise, it is replaced with another ball of the same color. The process is repeated. Find the probability that the third ball drawn is black. | [
"\\(\\frac{49}{72}\\)"
] | [
"An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white, it is not replaced into the urn. Otherwise, it is replaced with another ball of the same color. The process is repeated. Find the probability that the third ball drawn is black."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-4424985169c74b5bf7da11a184e5f101cc3c0ab8c7abfc68fabafed3c0dee6e8 | An urn contains 25 balls numbered from 1 to 25. A card shows the numbers $\{1,2,3,4,5,7,8,9,11,15,16,19,21,23,24\}$. Fifteen balls are drawn from the urn without replacement. Let $X$ be the number of drawn balls whose numbers appear on the card. Find the distribution of $X$. | [
"\\( P(X = k) = \\frac{\\binom{15}{k} \\binom{10}{15-k}}{\\binom{25}{15}} \\) for \\( k = 0, 1, 2, \\ldots, 15 \\)."
] | [
"An urn contains 25 balls numbered from 1 to 25. A card shows the numbers $\\{1,2,3,4,5,7,8,9,11,15,16,19,21,23,24\\}$. Fifteen balls are drawn from the urn without replacement. Let $X$ be the number of drawn balls whose numbers appear on the card. Find the distribution of $X$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 8
} | 22 | 0 | 0 |
omr-ws-13a7875ec46e880e67119553dfc4dda693aa5b5ebdc8dfe3dfe830d76a25bccb | An urn contains 25 balls numbered from 1 to 25. A card shows the numbers $\{1,2,3,4,5,7,8,9,11,15,16,19,21,23,24\}$. Fifteen balls are drawn from the urn without replacement. Let $X$ be the number of drawn balls whose numbers appear on the card. Find the expected value $\mathbb{E}(X)$ and the variance $\text{Var}(X)$. | [
"\\(\\mathbb{E}(X) = 9, \\text{Var}(X) = 1.5\\)"
] | [
"An urn contains 25 balls numbered from 1 to 25. A card shows the numbers $\\{1,2,3,4,5,7,8,9,11,15,16,19,21,23,24\\}$. Fifteen balls are drawn from the urn without replacement. Let $X$ be the number of drawn balls whose numbers appear on the card. Find the expected value $\\mathbb{E}(X)$ and the variance $\\text{V... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-da316b4cbf8beee924ee5426c81d635dcafecff17d1469fdc8598c9ca29ff638 | An urn contains 3 red and 7 black balls. Players A and B withdraw balls consecutively until a red ball is selected. Find the probability that A selects the red ball. (A draws the first ball, then B, and so on. There is no replacement of the balls drawn.) | [
"\\(\\frac{7}{12}\\)"
] | [
"An urn contains 3 red and 7 black balls. Players A and B withdraw balls consecutively until a red ball is selected. Find the probability that A selects the red ball. (A draws the first ball, then B, and so on. There is no replacement of the balls drawn.)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-ed90b430d3c18c482efffda9158fbcf6a91a92d1262285fd49691f8b57d4f0f6 | An urn contains 4 red and 6 black balls, and another urn contains 5 red and 2 black balls. A ball is taken out from the second urn and put into the first urn. If a ball is now taken out from the first urn, what is the probability that it is black? | [
"\\(\\frac{4}{7}\\)"
] | [
"An urn contains 4 red and 6 black balls, and another urn contains 5 red and 2 black balls. A ball is taken out from the second urn and put into the first urn. If a ball is now taken out from the first urn, what is the probability that it is black?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-27102eebe39ae392f7a68fc5284940cabb5b9e1887c8c05dee4d5c742b38fcdb | An urn contains 4 red, 6 black, and 6 blue balls. If 4 balls are drawn at random without replacement, what is the probability that the 4 balls will contain at least one ball of each color? | [
"\\(\\frac{18}{35}\\)"
] | [
"An urn contains 4 red, 6 black, and 6 blue balls. If 4 balls are drawn at random without replacement, what is the probability that the 4 balls will contain at least one ball of each color?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 34
} | 50 | 0 | 0 |
omr-ws-b07f6d2064d1e5765225a5ea08f9b9fcb4b24e07193d40eb6278050ccfd13ba3 | An urn contains 4 white and 9 red balls. Balls are drawn with replacement. Let \( P(r) \) be the probability that no two white balls appear in succession when \( r \) balls are drawn. Find the recursion relation for \( P(r) \). | [
"\\( P(r) = \\frac{9}{13} P(r-1) + \\frac{36}{169} P(r-2) \\)"
] | [
"An urn contains 4 white and 9 red balls. Balls are drawn with replacement. Let \\( P(r) \\) be the probability that no two white balls appear in succession when \\( r \\) balls are drawn. Find the recursion relation for \\( P(r) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-79d707b1701ef7a87b6369b367522ad979a2e9434867175a8a33d596d14d1bfa | An urn contains 5 balls. Two balls are drawn and found to be white. If the probability that all balls in the urn are white is $P$, then find the value of $6P-1$. | [
"2"
] | [
"An urn contains 5 balls. Two balls are drawn and found to be white. If the probability that all balls in the urn are white is $P$, then find the value of $6P-1$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3c6023d414ffd26f868bb7a94327d0ffcab4c231456c373652931ccbad40c756 | An urn contains 5 black balls, 5 white balls, and 10 red balls, all 20 balls being distinct. Seven balls are chosen from the urn with replacement. Calculate the probability that at least one ball of each color is drawn. | [
"\\dfrac{3003}{4096}"
] | [
"An urn contains 5 black balls, 5 white balls, and 10 red balls, all 20 balls being distinct. Seven balls are chosen from the urn with replacement. Calculate the probability that at least one ball of each color is drawn."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8b5ae0fedc62525eadaab39fa65edd0bd60bd6c758434c555db710e99931761e | An urn contains 5 black balls, 5 white balls, and 10 red balls, all 20 balls being distinct. Seven balls are chosen from the urn without replacement. Calculate the probability that at least one ball of each color is drawn. | [
"\\(\\frac{2155}{2584}\\)"
] | [
"An urn contains 5 black balls, 5 white balls, and 10 red balls, all 20 balls being distinct. Seven balls are chosen from the urn without replacement. Calculate the probability that at least one ball of each color is drawn."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-61b5042752fe9c63f2cd52cad031b025de0d570dc8f60472c9958fd6b1b7a1a7 | An urn contains 5 red balls, 8 green balls, and 7 blue balls. If four balls are drawn simultaneously, what is the probability that balls of all three colors are obtained? | [
"\\(\\frac{2}{7}\\)"
] | [
"An urn contains 5 red balls, 8 green balls, and 7 blue balls. If four balls are drawn simultaneously, what is the probability that balls of all three colors are obtained?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7d143732bc569f730f902c3c66cf51cf94cdd8bb28d8be224eafb0f48e9ae1b0 | An urn contains 5 red, 6 blue, and 8 green balls. If a set of 3 balls is randomly selected with replacement, what is the probability that each of the balls will be of the same color? | [
"\\(\\frac{853}{6859}\\)"
] | [
"An urn contains 5 red, 6 blue, and 8 green balls. If a set of 3 balls is randomly selected with replacement, what is the probability that each of the balls will be of the same color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-823f9031bc332853a00c055563c6f8abb022a08c9b7dbbbf8d919c70ebc52242 | An urn contains 5 red, 6 blue, and 8 green balls. If a set of 3 balls is randomly selected without replacement, what is the probability that each of the balls will be of the same color? | [
"\\(\\frac{86}{969}\\)"
] | [
"An urn contains 5 red, 6 blue, and 8 green balls. If a set of 3 balls is randomly selected without replacement, what is the probability that each of the balls will be of the same color?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-c2a7588ac492a50bd1ecd9d81055e1a58777b46fc092ff4706e3c026962ce327 | An urn contains 5 red, 6 blue, and 8 green balls. If a set of 3 balls is randomly selected, what is the probability that each of the balls will be of the same color? | [
"\\(\\frac{86}{969}\\)"
] | [
"An urn contains 5 red, 6 blue, and 8 green balls. If a set of 3 balls is randomly selected, what is the probability that each of the balls will be of the same color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-84579a264a72602143f8d6ee83981274bbc04e45870f2e8d6bfda4ef3b4a30b9 | An urn contains 6 balls of different colors. The probability of P telling the truth is \( p \) and the probability of Q telling the truth is \( q \). A ball is picked from the urn, and both P and Q assert that it is red. Find the probability that the ball is actually red. | [
"\\(\\frac{pq}{pq + 5(1-p)(1-q)}\\)"
] | [
"An urn contains 6 balls of different colors. The probability of P telling the truth is \\( p \\) and the probability of Q telling the truth is \\( q \\). A ball is picked from the urn, and both P and Q assert that it is red. Find the probability that the ball is actually red."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 4,
"tir": 4
} | 17 | 0 | 0 |
omr-ws-4afd78daeb466d2c7644348de9d3f41c686b81a6267a982c25de1157618ff244 | An urn contains 8 white balls, 7 black balls, and 4 green balls. If 3 balls are removed without replacement, determine the probability that they are of different colors. | [
"\\(\\frac{224}{969}\\)"
] | [
"An urn contains 8 white balls, 7 black balls, and 4 green balls. If 3 balls are removed without replacement, determine the probability that they are of different colors."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-0ccfdf480826ecd5f687534a54d05b97a55af66395e93ec60fb9003264337e40 | An urn contains \( N \) balls, each of which is either black or white. The expected number of black balls is \( M \). If a ball is drawn at random from the urn, calculate the expected value of the number of black balls drawn. | [
"\\dfrac{M}{N}"
] | [
"An urn contains \\( N \\) balls, each of which is either black or white. The expected number of black balls is \\( M \\). If a ball is drawn at random from the urn, calculate the expected value of the number of black balls drawn."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-73d620b0e6221f085937bb393fb61a2145229f2bca8d182a4a10cb058351e364 | An urn contains \( b \) blue and \( r \) red balls. Balls are removed at random until the first blue ball is drawn. Calculate the expected number of balls drawn when \( b = 5 \) and \( r = 10 \). | [
"\\dfrac{8}{3}"
] | [
"An urn contains \\( b \\) blue and \\( r \\) red balls. Balls are removed at random until the first blue ball is drawn. Calculate the expected number of balls drawn when \\( b = 5 \\) and \\( r = 10 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f563b3b81ded2554569dcc38587c1c6b1ef50d2e7ebe066974c4130095a78b5f | An urn contains \( b \) blue and \( r \) red balls. The balls are replaced, and then removed at random until all the remaining balls are of the same color. Find the expected number of balls remaining in the urn. | [
"\\dfrac{b}{r + 1} + \\dfrac{r}{b + 1}"
] | [
"An urn contains \\( b \\) blue and \\( r \\) red balls. The balls are replaced, and then removed at random until all the remaining balls are of the same color. Find the expected number of balls remaining in the urn."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-fe0f3bed0fece5c42e20ba533d7e98be90e2dd4cd13c11cb64941375a7ec729a | An urn contains \( n + m \) balls, of which \( n \) are red and \( m \) are black. Balls are withdrawn one at a time without replacement. Let \( X \) be the number of red balls removed before the first black ball is chosen. Express \( X \) in terms of the random variables \( X_i \), where \( X_i = 1 \) if red ball \( i... | [
"\\( \\frac{n}{m+1} \\)"
] | [
"An urn contains \\( n + m \\) balls, of which \\( n \\) are red and \\( m \\) are black. Balls are withdrawn one at a time without replacement. Let \\( X \\) be the number of red balls removed before the first black ball is chosen. Express \\( X \\) in terms of the random variables \\( X_i \\), where \\( X_i = 1 \... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-661d844845250510da4b9c0069e38924d80b785c8e215ed52f254df01dc9d8ca | An urn contains \( n + m \) balls, of which \( n \) are red and \( m \) are black. Balls are withdrawn one at a time without replacement. Let \( X \) be the number of red balls removed before the first black ball is chosen. Find the expected value \( E(X) \). | [
"\\( \\frac{n}{m+1} \\)"
] | [
"An urn contains \\( n + m \\) balls, of which \\( n \\) are red and \\( m \\) are black. Balls are withdrawn one at a time without replacement. Let \\( X \\) be the number of red balls removed before the first black ball is chosen. Find the expected value \\( E(X) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-aa9c2605e714860d9ec63776fdfff55d6638f7fa14393b0634281dd913596f23 | An urn contains \( n \) balls, with exactly one ball being white. Two persons, A and B, have probabilities \( P_1 \) and \( P_2 \) of telling the truth, respectively. A ball is taken out of the urn and both A and B report it as white. What is the probability that the ball is actually white? | [
"\\(\\frac {P_{1}P_{2}}{P_{1}P_{2} + (n - 1)(1 - P_{1})(1 - P_{2})}\\)"
] | [
"An urn contains \\( n \\) balls, with exactly one ball being white. Two persons, A and B, have probabilities \\( P_1 \\) and \\( P_2 \\) of telling the truth, respectively. A ball is taken out of the urn and both A and B report it as white. What is the probability that the ball is actually white?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 10
} | 24 | 0 | 0 |
omr-ws-e7b1a7a7b18150450a4de98957f4e2634ffdfdde46f6632f96c54f7ee0868608 | An urn contains \( x \) white and \( b \) black balls. After a ball is drawn, it is returned to the urn if it is white; if it is black, it is replaced by a white ball from another urn. Let \( M_{n} \) denote the expected number of white balls in the urn after the operation has been repeated \( n \) times. a) Derive the... | [
"1 - \\frac{b}{a + b} \\left(1 - \\frac{1}{a + b}\\right)^n"
] | [
"An urn contains \\( x \\) white and \\( b \\) black balls. After a ball is drawn, it is returned to the urn if it is white; if it is black, it is replaced by a white ball from another urn. Let \\( M_{n} \\) denote the expected number of white balls in the urn after the operation has been repeated \\( n \\) times.\... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-8257fbd278a8ec8caee5bb0028bea7a5f085099264510ed5738987543247ba1e | An urn contains \(2009! + 1\) black balls and \(2009! + 1\) red balls. In every step, we take out 2009 balls and discard all of them if they are all of the same color, or keep the red ones and discard the black ones if they are of different colors. Given that 1 ball remains at the end after a finite number of steps, fi... | [
"1"
] | [
"An urn contains \\(2009! + 1\\) black balls and \\(2009! + 1\\) red balls. In every step, we take out 2009 balls and discard all of them if they are all of the same color, or keep the red ones and discard the black ones if they are of different colors. Given that 1 ball remains at the end after a finite number of ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 29,
"tir": 1
} | 30 | 0 | 0 |
omr-ws-3ff6f9ee58cc1b69abf99ea5b57689501f287b353b8b7d8d2c2d739e1679cf28 | An urn contains \(n\) balls, each randomly coloured with one of \(k\) colours. \(d\) balls are drawn randomly from the urn and their colours observed; then they are placed back in the urn. What is the probability \( p(a_i,x) \) that the urn contains exactly \(x\) balls of colour \(a_i\)? | [
"\\binom{n}{x} \\left( \\frac{1}{k} \\right)^x \\left( 1 - \\frac{1}{k} \\right)^{n - x}"
] | [
"An urn contains \\(n\\) balls, each randomly coloured with one of \\(k\\) colours. \\(d\\) balls are drawn randomly from the urn and their colours observed; then they are placed back in the urn. What is the probability \\( p(a_i,x) \\) that the urn contains exactly \\(x\\) balls of colour \\(a_i\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-72414c1c221d06e7547907b038f31d9007e0095ea384a95040580e253c496274 | An urn contains a red ball, a green ball, and a blue ball. A ball is drawn at random and returned to the urn. This process is repeated until every ball has been drawn at least once. Let \(N\) be the number of draws required, and let \(P(N)\) be the probability that \(N\) draws were required to draw each ball at least o... | [
"\\dfrac{2^{N-1} - 1}{3^{N-2}}"
] | [
"An urn contains a red ball, a green ball, and a blue ball. A ball is drawn at random and returned to the urn. This process is repeated until every ball has been drawn at least once. Let \\(N\\) be the number of draws required, and let \\(P(N)\\) be the probability that \\(N\\) draws were required to draw each ball... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8687b4a86c4df0afe3dbfc10d6f909cf2be562d66ae0a67bd1ffd018c4daac37 | An urn contains an unknown number of red and blue balls. The probability of selecting 4 balls from the urn without replacement and having all of them be blue is 1/2. What are the possible values for the total number of balls in the urn? | [
"8"
] | [
"An urn contains an unknown number of red and blue balls. The probability of selecting 4 balls from the urn without replacement and having all of them be blue is 1/2. What are the possible values for the total number of balls in the urn?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dde30b1db5357c15823385629d3bcd6e18082e02ad65f29725a4005946fb6abf | An urn contains balls of $k$ different colors, with $n_i$ balls of the $i$-th color. Balls are selected at random from the urn, one by one, without replacement, until $m$ balls of the same color appear. Find the greatest number of selections needed to ensure that $m$ balls of the same color are selected. | [
"\\( 1 + \\sum_{i=1}^k \\min\\left\\{(m-1), n_i\\right\\} \\)"
] | [
"An urn contains balls of $k$ different colors, with $n_i$ balls of the $i$-th color. Balls are selected at random from the urn, one by one, without replacement, until $m$ balls of the same color appear. Find the greatest number of selections needed to ensure that $m$ balls of the same color are selected."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-7861ae25395658018fc4278298de1f39fd9bba6e132b0a3ffb8e17a73b92c25c | An urn contains five white and four black balls. Four balls are transferred to a second urn. A ball is then drawn from this second urn, and it happens to be black. Find the probability of drawing a white ball from among the remaining three balls in the second urn. | [
"\\dfrac{5}{8}"
] | [
"An urn contains five white and four black balls. Four balls are transferred to a second urn. A ball is then drawn from this second urn, and it happens to be black. Find the probability of drawing a white ball from among the remaining three balls in the second urn."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3112c4cfc2447cfd6c1576c2035f8e5e72321ab06e915fe5a830e65b78f52b05 | An urn contains four white and six black balls. A random sample of size 4 is chosen, and let \( X \) denote the number of white balls in the sample. An additional ball is now selected from the remaining six balls in the urn, and let \( Y \) equal 1 if this ball is white and 0 if it is black. Find \( E[X|Y = 1] \). | [
"\\(\\frac{4}{3}\\)"
] | [
"An urn contains four white and six black balls. A random sample of size 4 is chosen, and let \\( X \\) denote the number of white balls in the sample. An additional ball is now selected from the remaining six balls in the urn, and let \\( Y \\) equal 1 if this ball is white and 0 if it is black. Find \\( E[X|Y = 1... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-7111febe65b949f69b1384367cca0320f4f18e19ff2deb98c5360744dafeac43 | An urn contains four white and six black balls. A random sample of size 4 is chosen, and let \( X \) denote the number of white balls in the sample. An additional ball is now selected from the remaining six balls in the urn, and let \( Y \) equal 1 if this ball is white and 0 if it is black. Find \( E[Y|X=2] \). | [
"\\(\\frac{1}{3}\\)"
] | [
"An urn contains four white and six black balls. A random sample of size 4 is chosen, and let \\( X \\) denote the number of white balls in the sample. An additional ball is now selected from the remaining six balls in the urn, and let \\( Y \\) equal 1 if this ball is white and 0 if it is black. Find \\( E[Y|X=2] ... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-0e2bbeb5482230dc8011bee13bbcc2181f654598feaf4e2b5567db611fc3909b | An urn contains four white and six black balls. A random sample of size 4 is chosen, and let \( X \) denote the number of white balls in the sample. An additional ball is now selected from the remaining six balls in the urn, and let \( Y \) equal 1 if this ball is white and 0 if it is black. Find \( \text{Var}(X|Y=1) \... | [
"0"
] | [
"An urn contains four white and six black balls. A random sample of size 4 is chosen, and let \\( X \\) denote the number of white balls in the sample. An additional ball is now selected from the remaining six balls in the urn, and let \\( Y \\) equal 1 if this ball is white and 0 if it is black. Find \\( \\text{Va... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e1d9435afd18395ae4d67e4f695991ba2c47f2f6483bcf23bae458852e80bece | An urn contains four white and six black balls. A random sample of size 4 is chosen, and let \( X \) denote the number of white balls in the sample. An additional ball is now selected from the remaining six balls in the urn, and let \( Y \) equal 1 if this ball is white and 0 if it is black. Find \( \text{Var}(Y|X = 0)... | [
"\\(\\frac{2}{9}\\)"
] | [
"An urn contains four white and six black balls. A random sample of size 4 is chosen, and let \\( X \\) denote the number of white balls in the sample. An additional ball is now selected from the remaining six balls in the urn, and let \\( Y \\) equal 1 if this ball is white and 0 if it is black. Find \\( \\text{Va... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-413557482a9b210582e8c5f483ba9ff42ca347114a0ea4b4615554aa3165bd00 | An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely: (a) the selection of four red marbles, (b) the selection of one white and three red marbles, (c) the selection of one white, one blue, and two red marbles, and (... | [
"46"
] | [
"An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely: (a) the selection of four red marbles, (b) the selection of one white and three red marbles, (c) the selection of one white, one blue, and two red marbles, ... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8356bb29633aa86e7204f2987227f4242efbc9309c35e65d8b1990fa6862d17f | An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely: (a) the selection of four red marbles; (b) the selection of one white and three red marbles; (c) the selection of one white, one blue, and two red marbles; and (... | [
"21"
] | [
"An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely:\n\n(a) the selection of four red marbles;\n(b) the selection of one white and three red marbles;\n(c) the selection of one white, one blue, and two red marb... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-8e15309f599b678384b09c47b9af79a8a6c2efb41419669f9f8e00dc05f2e093 | An urn contains one hundred balls numbered from 1 to 100. Four balls are removed at random (one at a time) without replacement. Find the probability that the number on the last ball is smaller than the number on the first ball. | [
"\\(\\frac{1}{2}\\)"
] | [
"An urn contains one hundred balls numbered from 1 to 100. Four balls are removed at random (one at a time) without replacement. Find the probability that the number on the last ball is smaller than the number on the first ball."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-539888f6f89136bca110a99d576ece6446874b88fde88ece4611d542d2b72817 | An urn contains one red ball and one blue ball. A box of extra red and blue balls is nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations, the urn... | [
"\\(\\frac{1}{5}\\)"
] | [
"An urn contains one red ball and one blue ball. A box of extra red and blue balls is nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations, th... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-9e1db9a0525f5397d46b3a2787ae115767a5c14721b8d66a19731f07167c60a5 | An urn contains three white, six red, and five black balls. Six balls are randomly selected without replacement. Let \(X\) and \(Y\) denote the number of white and black balls selected, respectively. Compute \(E[X \mid Y = 1]\). | [
"\\(\\frac{5}{4}\\)"
] | [
"An urn contains three white, six red, and five black balls. Six balls are randomly selected without replacement. Let \\(X\\) and \\(Y\\) denote the number of white and black balls selected, respectively. Compute \\(E[X \\mid Y = 1]\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3651e076202ee1a66423a7bdec6abbcf32488c79a87726952a7b55177370b308 | An urn contains three white, six red, and five black balls. Six balls are randomly selected without replacement. Let \(X\) and \(Y\) denote the number of white and black balls selected, respectively. Compute the conditional probability function of \(X\) given that \(Y = 3\). | [
"\\[\nP(X = k \\mid Y = 3) = \\begin{cases}\n\\frac{5}{21} & \\text{if } k = 0 \\\\\n\\frac{15}{28} & \\text{if } k = 1 \\\\\n\\frac{3}{14} & \\text{if } k = 2 \\\\\n\\frac{1}{84} & \\text{if } k = 3 \\\\\n0 & \\text{otherwise}\n\\end{cases}\n\\]"
] | [
"An urn contains three white, six red, and five black balls. Six balls are randomly selected without replacement. Let \\(X\\) and \\(Y\\) denote the number of white and black balls selected, respectively. Compute the conditional probability function of \\(X\\) given that \\(Y = 3\\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 2,
"tir": 30
} | 42 | 0 | 0 |
omr-ws-e6063834269694a42429916e9386bcb699647d6f51cac7de2070b70b0f7d7f04 | An urn contains two balls. It was filled by tossing a fair coin twice, putting a white ball for each head and a black ball for each tail. A ball drawn from the urn is found to be white. What is the probability that the other ball in the urn is also white? | [
"\\(\\frac{1}{2}\\)"
] | [
"An urn contains two balls. It was filled by tossing a fair coin twice, putting a white ball for each head and a black ball for each tail. A ball drawn from the urn is found to be white. What is the probability that the other ball in the urn is also white?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-9ee7700c9081cdb2f6d487c7df85c7de7c736124db19e74b8336cc58d65bfc29 | An urn has $m$ black balls. At each stage, a black ball is removed and a new ball, which is black with probability $p$ and white with probability $1-p$, is put in its place. Find the expected number of stages needed until there are no black balls in the urn. | [
"\\dfrac{m}{1 - p}"
] | [
"An urn has $m$ black balls. At each stage, a black ball is removed and a new ball, which is black with probability $p$ and white with probability $1-p$, is put in its place. Find the expected number of stages needed until there are no black balls in the urn."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-add8dde613b2b68231dc0a75b2049b8fb9806f02fd841eed17045929ae368036 | An urn has 2 red balls and 1 black ball. A ball is drawn from the urn, and then returned to the urn if and only if it is black; this is then repeated for an indefinite number of times. Let \( P(N) \) be the probability the Nth ball drawn is red. (a) Find \( P(2) \), \( P(3) \), \( P(4) \). (b) Find a formula for \( P(N... | [
"12"
] | [
"An urn has 2 red balls and 1 black ball. A ball is drawn from the urn, and then returned to the urn if and only if it is black; this is then repeated for an indefinite number of times. Let \\( P(N) \\) be the probability the Nth ball drawn is red.\n(a) Find \\( P(2) \\), \\( P(3) \\), \\( P(4) \\).\n(b) Find a for... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-216a5cfc9d20dba63c9579265df9961865e910894a0f41d14626ce176038e810 | An urn has 5 balls numbered 1, 2, 3, 4, 5. Two balls are drawn (without replacement) from the urn and \( S \) is the sum of the numbers on the two balls. Find the probability \( S \) is an even number. | [
"\\(\\frac{2}{5}\\)"
] | [
"An urn has 5 balls numbered 1, 2, 3, 4, 5. Two balls are drawn (without replacement) from the urn and \\( S \\) is the sum of the numbers on the two balls. Find the probability \\( S \\) is an even number."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-f38f1cb6ed11138c3c0fb15ff25d072d65cf724b79800a361141d0c37a8e99b4 | An urn has \( N \) (\( N > 2 \)) balls numbered \( 1, 2, 3, \ldots, N \). Three balls are drawn from the urn (no balls are put back in the urn). Let \( P(N) \) be the probability that one of the three numbers drawn is greater than or equal to the sum of the other two. Find \( P(3) \), \( P(4) \), \( P(5) \), \( P(6) \)... | [
"\\dfrac{13}{20}"
] | [
"An urn has \\( N \\) (\\( N > 2 \\)) balls numbered \\( 1, 2, 3, \\ldots, N \\). Three balls are drawn from the urn (no balls are put back in the urn). Let \\( P(N) \\) be the probability that one of the three numbers drawn is greater than or equal to the sum of the other two. Find \\( P(3) \\), \\( P(4) \\), \\( ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e3f97c382d4ddf9034edbcca9f0e480d4a571b8b2677a6621f574fe5f83e20d | An urn has \( N \) balls numbered 1, 2, ..., \( N \). Three balls are drawn (without replacement) from the urn and \( S \) is the sum of the numbers on the three balls. Find the probability \( S \) is an even number. | [
"\\(\\frac{\\binom{N-k}{3} + \\binom{N-k}{1} \\cdot \\binom{k}{2}}{\\binom{N}{3}}\\)"
] | [
"An urn has \\( N \\) balls numbered 1, 2, ..., \\( N \\). Three balls are drawn (without replacement) from the urn and \\( S \\) is the sum of the numbers on the three balls. Find the probability \\( S \\) is an even number."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b95be66d48b971feaaa1dc0b81193b2e479d313305b9ca330409a8dc02aea977 | An urn has \( N \) balls numbered 1, 2, ..., \( N \). Two balls are drawn (without replacement) from the urn and \( S \) is the sum of the numbers on the two balls. Find the probability \( S \) is an even number. | [
"\\( \\frac{k-1}{2k-1} \\) if \\( N = 2k \\) (even), \\( \\frac{k}{2k+1} \\) if \\( N = 2k+1 \\) (odd)."
] | [
"An urn has \\( N \\) balls numbered 1, 2, ..., \\( N \\). Two balls are drawn (without replacement) from the urn and \\( S \\) is the sum of the numbers on the two balls. Find the probability \\( S \\) is an even number."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 14
} | 23 | 0 | 0 |
omr-ws-0c2d0f01a37e7fde6c1a88b5fdb56b7a4f6b322c9d171e45f18de45cab9cf75d | An urn initially contains 1 white ball and 1 black ball. Every minute, a ball is randomly drawn and replaced along with another ball of some color. Determine the probability that when there are $n$ balls of the same color, exactly $i$ of them are white, where $1 \le i \le n-1$. | [
"\\dfrac{1}{n-1}"
] | [
"An urn initially contains 1 white ball and 1 black ball. Every minute, a ball is randomly drawn and replaced along with another ball of some color. Determine the probability that when there are $n$ balls of the same color, exactly $i$ of them are white, where $1 \\le i \\le n-1$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-20026ff51ca77bb6060bc18a00fc9a105cac9edebd00efb89b6ac506cf1989df | An urn initially contains one red ball and one blue ball. In each step, a ball is chosen uniformly at random from the urn. If the chosen ball is red, another red ball and another blue ball are added to the urn. If the chosen ball is blue for the $k$-th time, a blue ball and $2k + 1$ red balls are added to the urn. Let ... | [
"\\dfrac{4}{3}"
] | [
"An urn initially contains one red ball and one blue ball. In each step, a ball is chosen uniformly at random from the urn. If the chosen ball is red, another red ball and another blue ball are added to the urn. If the chosen ball is blue for the $k$-th time, a blue ball and $2k + 1$ red balls are added to the urn.... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9577fa18b5e9814fea038fa5b0b2de42c93ed5edf511b132e749e5e8dc6b9f62 | Ana and Banana are playing a game. Ana secretly picks a number $1 \leq A \leq 10^6$. In each turn, Banana picks a positive integer $B$, and Ana reveals the most common digit(s) in the product $A \cdot B$. Banana's goal is to determine the number $A$ with certainty after some number of turns. What is the minimum number ... | [
"2"
] | [
"Ana and Banana are playing a game. Ana secretly picks a number $1 \\leq A \\leq 10^6$. In each turn, Banana picks a positive integer $B$, and Ana reveals the most common digit(s) in the product $A \\cdot B$. Banana's goal is to determine the number $A$ with certainty after some number of turns. What is the minimum... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9772dd3b441aabef720b905f3e42dc689611ce5e3b60d427e638edde673e3202 | Ana and Banana are playing tennis in the deuce phase (a player needs two consecutive points to win the game). The probability of Ana winning any point is \( P \). Find the probability of Ana winning the game. | [
"\\(\\frac{p^2}{1 - 2p + 2p^2}\\)"
] | [
"Ana and Banana are playing tennis in the deuce phase (a player needs two consecutive points to win the game). The probability of Ana winning any point is \\( P \\). Find the probability of Ana winning the game."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-a70f82ec618f078ba057a60e41671c53b7ba69a3e3c28c103989eb1582e0242a | Ana and Banana play a game on a 100 by 100 grid of unit squares, each labelled $(R,C)$. Ana picks a set of at least 400 unit squares which forms a rectangle and doesn't tell Banana which squares she picks. Banana can ask $K$ questions of the form "is the square $(R,C)$ in the rectangle picked?", to which Ana must answe... | [
"199"
] | [
"Ana and Banana play a game on a 100 by 100 grid of unit squares, each labelled $(R,C)$. Ana picks a set of at least 400 unit squares which forms a rectangle and doesn't tell Banana which squares she picks. Banana can ask $K$ questions of the form \"is the square $(R,C)$ in the rectangle picked?\", to which Ana mus... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-24d1e419aedebc5056d4ce791ff9ecaafd2d957305a9c1be8e0de356aa628ad1 | Ana and Banana play a game on a plane. In each turn, Ana chooses a point on the plane and Banana decides whether it is blue or red. Ana wins when there exist three points of the same color forming an equilateral triangle. What is the maximum number of turns Banana can prevent Ana from winning, assuming Ana tries her be... | [
"6"
] | [
"Ana and Banana play a game on a plane. In each turn, Ana chooses a point on the plane and Banana decides whether it is blue or red. Ana wins when there exist three points of the same color forming an equilateral triangle. What is the maximum number of turns Banana can prevent Ana from winning, assuming Ana tries h... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-5ce904858cc903ad0d3f4fd4d2b1f19983d6fe13752942ec4fe9e5e31e1cc676 | Ana and Banana take turns filling in three squares with integers from the set \(\{1, 2, 3, 4, 5\}\) such that no integer is used more than once. Ana's goal is to minimize the minimum value of the polynomial \( a_1x^2 + a_2x + a_3 \) over all real \( x \), where \( a_1, a_2, a_3 \) are the integers written in the square... | [
"312"
] | [
"Ana and Banana take turns filling in three squares with integers from the set \\(\\{1, 2, 3, 4, 5\\}\\) such that no integer is used more than once. Ana's goal is to minimize the minimum value of the polynomial \\( a_1x^2 + a_2x + a_3 \\) over all real \\( x \\), where \\( a_1, a_2, a_3 \\) are the integers writte... | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-ed35f01bc93d099243f1204918357d49b9515dfa8e4508206215addfd6ecfba6 | Ana and Beatriz take turns drawing squares on an infinite board, starting with a square of side 1. Each move consists of drawing a new square such that it does not overlap with the existing figure and one of its sides coincides with one of the sides of the existing figure. The winner is the player who completes a figur... | [
"Beatriz"
] | [
"Ana and Beatriz take turns drawing squares on an infinite board, starting with a square of side 1. Each move consists of drawing a new square such that it does not overlap with the existing figure and one of its sides coincides with one of the sides of the existing figure. The winner is the player who completes a ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
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