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-4d7a982cdf2eb77fe51cd77ae09cb1e508bd9ddcfb6aaf82655bb77f5e7ecd2e | A deck of 2007 cards numbered from 1 to 2007 is stacked in random order. Cards are removed from the top and placed on a desk, with the next card moved to the bottom of the deck. This process is repeated until all cards are on the desk. The final order of the cards on the desk is 1, 2, ..., 2006, 2007. How many cards we... | [
"1003"
] | [
"A deck of 2007 cards numbered from 1 to 2007 is stacked in random order. Cards are removed from the top and placed on a desk, with the next card moved to the bottom of the deck. This process is repeated until all cards are on the desk. The final order of the cards on the desk is 1, 2, ..., 2006, 2007. How many car... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
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"cot": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-40ab975caf70e37c9c6d3edb150059b4983d55eb9ee8e087ebf4ff474f2700f0 | A deck of 30 cards consists of five cards numbered each of 1, 2, 3, 4, 5, and 6. If a "hand" of four cards is dealt, what is the probability that it will contain "two pair" (for example, two 3’s and two 6’s)? | [
"\\(\\frac{2}{7}\\)"
] | [
"A deck of 30 cards consists of five cards numbered each of 1, 2, 3, 4, 5, and 6. If a \"hand\" of four cards is dealt, what is the probability that it will contain \"two pair\" (for example, two 3’s and two 6’s)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8494419dfdab36b34705dc9938e373fe13fb831b59239e278b2047b45683b684 | A deck of 30 cards consists of five cards numbered each of 1, 2, 3, 4, 5, and 6. If a "hand" of three cards is dealt, what is the probability that the "hand" contains "one pair" (for example, two 5's plus another card that is not a 5)? | [
"\\dfrac{75}{203}"
] | [
"A deck of 30 cards consists of five cards numbered each of 1, 2, 3, 4, 5, and 6. If a \"hand\" of three cards is dealt, what is the probability that the \"hand\" contains \"one pair\" (for example, two 5's plus another card that is not a 5)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
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omr-ws-4cd3b20e9f9fdd97a1c16be8b7973ee665a314aa60f410fe4b8df2e64b518e6f | A deck of 32 cards has 3 suits and 2 jokers. Each suit has cards numbered from 1 to 10, and each joker is numbered 0. A card numbered \( k \) has a value of \( 2^k \), and the value of a hand is the sum of the values of the cards in the hand. Determine the number of hands of value 2004. | [
"4374"
] | [
"A deck of 32 cards has 3 suits and 2 jokers. Each suit has cards numbered from 1 to 10, and each joker is numbered 0. A card numbered \\( k \\) has a value of \\( 2^k \\), and the value of a hand is the sum of the values of the cards in the hand. Determine the number of hands of value 2004."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-8466ab7649e2548f55479b9c5fbc169a0195c3584627828c6d65f3ab93651625 | A deck of 36 cards contains 9 cards in each of the suits spades, hearts, diamonds, and clubs. An assistant examines the deck without changing the order of the cards and points the arrow on the back of each card either towards or away from the magician, according to some system agreed upon in advance. If the magician ca... | [
"18"
] | [
"A deck of 36 cards contains 9 cards in each of the suits spades, hearts, diamonds, and clubs. An assistant examines the deck without changing the order of the cards and points the arrow on the back of each card either towards or away from the magician, according to some system agreed upon in advance. If the magici... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af6269d8b4b26bb279136f5e66fb81ab386549ab3be6bf5b36d40367e1797e83 | A deck of 52 cards contains four suits, each with cards numbered 1 through 13. The audience selects five cards with distinct numbers. The assistant looks at the five cards, turns exactly one card face down, and arranges the remaining four cards in some order. The magician then enters and, using a prearranged agreement ... | [
"5"
] | [
"A deck of 52 cards contains four suits, each with cards numbered 1 through 13. The audience selects five cards with distinct numbers. The assistant looks at the five cards, turns exactly one card face down, and arranges the remaining four cards in some order. The magician then enters and, using a prearranged agree... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ddc4156e20eccedccb1051f29a0cafaa0dbd630c0b0a84ec6647b78568be6f56 | A deck of \( 16 \) cards contains the four aces, four kings, four queens, and four jacks. The \( 16 \) cards are thoroughly shuffled and my opponent (who always tells the truth) draws two cards simultaneously and at random from the deck. He says "I hold at least one ace". What is the chance that he holds two aces in hi... | [
"\\dfrac{1}{9}"
] | [
"A deck of \\( 16 \\) cards contains the four aces, four kings, four queens, and four jacks. The \\( 16 \\) cards are thoroughly shuffled and my opponent (who always tells the truth) draws two cards simultaneously and at random from the deck. He says \"I hold at least one ace\". What is the chance that he holds two... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7c1a9d3780dbbcbf8048b4ffd865216685c2905452143247ee15aa7d52389276 | A deck of cards consists of $30$ cards labeled with the integers $1$ to $30$, inclusive. The cards numbered $1$ through $15$ are purple, and the cards numbered $16$ through $30$ are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number $32$. L... | [
"121"
] | [
"A deck of cards consists of $30$ cards labeled with the integers $1$ to $30$, inclusive. The cards numbered $1$ through $15$ are purple, and the cards numbered $16$ through $30$ are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number $3... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37689ed6f974c37f3972e05b529b6600535c9c3a5d32cd46f8666df86cd1d2b9 | A deck of cards consists of 2 purple cards and 6 orange cards. Zak draws a hand of three cards at random from the deck. If all three cards are orange, he returns them to the deck, shuffles the deck, and repeats the process until he has a hand containing at least one purple card. Then he draws one additional card. Compu... | [
"\\(\\frac{1}{6}\\)"
] | [
"A deck of cards consists of 2 purple cards and 6 orange cards. Zak draws a hand of three cards at random from the deck. If all three cards are orange, he returns them to the deck, shuffles the deck, and repeats the process until he has a hand containing at least one purple card. Then he draws one additional card. ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 58
} | 90 | 0 | 0 |
omr-ws-eda16be4382eb8bc4afa5fe0aa483926c72d4d3063356293fef13b0344cb92af | A deck of cards contains 4 suites with 13 numbers in each suit. Albert draws a card first. Evan wins if he draws a card with the same number as Albert’s card or the same suit as Albert’s card. What is the probability that Evan wins? | [
"\\(\\frac{14}{51}\\)"
] | [
"A deck of cards contains 4 suites with 13 numbers in each suit. Albert draws a card first. Evan wins if he draws a card with the same number as Albert’s card or the same suit as Albert’s card. What is the probability that Evan wins?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a4613090c2586a30e829264c027541f24f8825076bc7c19d5dd3fbdd4fe2e10b | A deck of cards has 5 black cards and 5 red cards. Every second, Arthur Wang draws a card at random from the deck. If the card is black, he throws it away. If it is red, he puts it back in the deck. What is the expected number of seconds before the deck has no more black cards? | [
"\\(\\frac{197}{12}\\)"
] | [
"A deck of cards has 5 black cards and 5 red cards. Every second, Arthur Wang draws a card at random from the deck. If the card is black, he throws it away. If it is red, he puts it back in the deck. What is the expected number of seconds before the deck has no more black cards?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-25cde61533232784a2139a894bf3a7038772d6fe610e7ab9f7e74fca79e02a3a | A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is \(\frac{1}{3}\). When 4 black cards are added to the deck, the probability of choosing red becomes \(\frac{1}{4}\). How many cards were in the deck originally? | [
"12"
] | [
"A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is \\(\\frac{1}{3}\\). When 4 black cards are added to the deck, the probability of choosing red becomes \\(\\frac{1}{4}\\). How many cards were in the deck originally?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e9e9c86907030441cfcd46319dc5376a59504d58ecb6dde9793d0cb8277a167c | A deck of forty cards consists of four $1$ 's, four $2$ 's,..., and four $10$ 's. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relative... | [
"758"
] | [
"A deck of forty cards consists of four $1$ 's, four $2$ 's,..., and four $10$ 's. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are ... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-d4323ca03e0ad0c973f8121b88aa3ac4d4284bf74b4db7cbfa952cc0ab3d57aa | A deck of forty cards consists of four $1$'s, four $2$'s,..., and four $10$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relatively ... | [
"758"
] | [
"A deck of forty cards consists of four $1$'s, four $2$'s,..., and four $10$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relati... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed2af8374c5ee150b697eeaea3a5a38395ba45704bfca434482c32900a2b9e34 | A deck of forty cards consists of four 1's, four 2's, ..., and four 10's. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let \( \frac{m}{n} \) be the probability that two randomly selected cards also form a pair, where \( m \) and \( n \) ... | [
"758"
] | [
"A deck of forty cards consists of four 1's, four 2's, ..., and four 10's. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let \\( \\frac{m}{n} \\) be the probability that two randomly selected cards also form a pair, where \\( m \\) an... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-becc61555f7f7127105e6ff79498202155b0e05ba3f32baaf6d2a62e146e361c | A deck of seven cards numbered $0$ to $6$ are shuffled randomly. A card is turned face-up. If the card is a zero, the player loses. If the card is $k>0$, then the next $k$ cards in the deck are discarded facedown (or all of the cards, if fewer than $k$ remain). The process is repeated until either a zero is turned face... | [
"\\(\\frac{151}{210}\\)"
] | [
"A deck of seven cards numbered $0$ to $6$ are shuffled randomly. A card is turned face-up. If the card is a zero, the player loses. If the card is $k>0$, then the next $k$ cards in the deck are discarded facedown (or all of the cards, if fewer than $k$ remain). The process is repeated until either a zero is turned... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 16
} | 17 | 0 | 0 |
omr-ws-5a03766cbbe769e091fd2b8ada40b038f9cd662b7849404ac8afc8200f81e3c3 | A decorator plans to place a rug in a room that is 9 meters by 12 meters such that a uniform strip of flooring around the rug will remain uncovered. If the rug is to cover half the floor space, what should the dimensions of the rug be? | [
"6m by 9m"
] | [
"A decorator plans to place a rug in a room that is 9 meters by 12 meters such that a uniform strip of flooring around the rug will remain uncovered. If the rug is to cover half the floor space, what should the dimensions of the rug be?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 32
} | 46 | 0 | 0 |
omr-ws-901dbbfe418ce073b2837bd86fd82efdaad857b679b287d7c0f1cf23b03026e1 | A decreasing positive sequence \( f_n \) satisfies the recurrence relation \( 12(n+1)f_{n+2} = 14(2n+1)f_{n+1} - (16n - \frac{1}{n})f_n \). Find the value of \( \frac{f_2}{f_1} \). | [
"1.5"
] | [
"A decreasing positive sequence \\( f_n \\) satisfies the recurrence relation \\( 12(n+1)f_{n+2} = 14(2n+1)f_{n+1} - (16n - \\frac{1}{n})f_n \\). Find the value of \\( \\frac{f_2}{f_1} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ebf21d8e52a6b2c7680413257d0b7b855dc3b547b73b8e2d01ac1f3b60dc954 | A decreasing, continuous function \( f(x) \) satisfies the following conditions: 1. For all positive reals \( x \), \( f(x) > 0 \). 2. For all positive reals \( t \), the triangle with vertices \( (0,0) \), \( (t, f(t)) \), \( (t+1, f(t+1)) \) has area \( \frac{t+1}{t} \). 3. \( \int_{1}^2 \frac{f(x)}{x} \, dx = 2 \). ... | [
"\\(\\frac{64}{63}\\)."
] | [
"A decreasing, continuous function \\( f(x) \\) satisfies the following conditions:\n1. For all positive reals \\( x \\), \\( f(x) > 0 \\).\n2. For all positive reals \\( t \\), the triangle with vertices \\( (0,0) \\), \\( (t, f(t)) \\), \\( (t+1, f(t+1)) \\) has area \\( \\frac{t+1}{t} \\).\n3. \\( \\int_{1}^2 \\... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 4
} | 17 | 0 | 0 |
omr-ws-d93e5314d55196d3623d0f85743c01098e66ceb21ac5241917433f0d3a49ae41 | A defective digital watch displays only the numbers from $1$ to $12$. After one minute from $n:59$, it displays $2n:00$ (mod $12$) instead of $(n+1):00$. If we set the watch to an arbitrary time, what is the probability that the hour part displays $4$ after exactly one day? Note: One day consists of 24 hours, which is ... | [
"\\(\\frac{1}{3}\\)"
] | [
"A defective digital watch displays only the numbers from $1$ to $12$. After one minute from $n:59$, it displays $2n:00$ (mod $12$) instead of $(n+1):00$. If we set the watch to an arbitrary time, what is the probability that the hour part displays $4$ after exactly one day?\n\nNote: One day consists of 24 hours, w... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-dbb2989f5cbb8ee113eacdf67e47fb7bd223c7182c2971ec1df2120e2572fe08 | A degree 6 polynomial \( f(x) \) satisfies \( f(x) = \frac{1}{x} \) for \( x = 1, 2, 3, 4, 5, 6, 7 \). Find \( f(8) \). Note that \( f(8) \neq \frac{1}{8} \). | [
"\\(\\frac{1}{4}\\)"
] | [
"A degree 6 polynomial \\( f(x) \\) satisfies \\( f(x) = \\frac{1}{x} \\) for \\( x = 1, 2, 3, 4, 5, 6, 7 \\). Find \\( f(8) \\). Note that \\( f(8) \\neq \\frac{1}{8} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-55d72b434f8de2a7fd2fee5d13422232b1c9543269f53a061e42b3dc72709cd1 | A delegation of 4 students is to be chosen from among 12 for a conference. How many ways can this be done if 2 particular students refuse to attend the meeting together? | [
"240"
] | [
"A delegation of 4 students is to be chosen from among 12 for a conference. How many ways can this be done if 2 particular students refuse to attend the meeting together?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1847472363276f72b4e17001700096694ad6b67545d84f302c14782d9e43040 | A delegation of 4 students is to be chosen from among 12 for a conference. How many ways can this be done if 2 students who are married to each other will attend only if they both can go? | [
"255"
] | [
"A delegation of 4 students is to be chosen from among 12 for a conference. How many ways can this be done if 2 students who are married to each other will attend only if they both can go?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-6662d9c1866528389ef97909ec4002476ddd8eff0e6b552677975a94fdf6da91 | A delicious circular pie with diameter $12\text{ cm}$ is cut into three equal-sized sector-shaped pieces. Let $l$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is $l^2$? | [
"108"
] | [
"A delicious circular pie with diameter $12\\text{ cm}$ is cut into three equal-sized sector-shaped pieces. Let $l$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is $l^2$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-83e400f9eea6e33005a490bb222bb1869bca45dc8caa5116bf1c8db6f271ae73 | A delivery guy delivers 5 different types of meals: a, b, c, d, e. He must deliver food to a party of 8 people but has forgotten what each person ordered. What is the least number of meals he should bring to ensure that everyone from the party gets at least their first or second choice? | [
"32"
] | [
"A delivery guy delivers 5 different types of meals: a, b, c, d, e. He must deliver food to a party of 8 people but has forgotten what each person ordered. What is the least number of meals he should bring to ensure that everyone from the party gets at least their first or second choice?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-35facecd1476fb69e75dae842f60c755d1d263fea1946f5f8372954f490dfda1 | A demon has a magic box that generates a number uniformly between 0 and 1 each time a button is pressed. The demon presses the button 10 times. You need to guess the 7th largest number generated. If your guess is within 0.01 of the actual 7th largest number, you win. What is your strategy for guessing the 7th largest n... | [
"\\(\\frac{1}{3}\\)"
] | [
"A demon has a magic box that generates a number uniformly between 0 and 1 each time a button is pressed. The demon presses the button 10 times. You need to guess the 7th largest number generated. If your guess is within 0.01 of the actual 7th largest number, you win. What is your strategy for guessing the 7th larg... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-d13b340e53cf2c2e17b4540f4f7f581f938c90bbe8ddfd8afabb4e280b9f96dc | A density curve looks like an inverted letter V. The first segment goes from point (0, 0.6) to the point (0.5, 1.4). The second segment goes from (0.5, 1.4) to (1, 0.6). What percent of the observations lie below 0.3? | [
"25.2\\%"
] | [
"A density curve looks like an inverted letter V. The first segment goes from point (0, 0.6) to the point (0.5, 1.4). The second segment goes from (0.5, 1.4) to (1, 0.6). What percent of the observations lie below 0.3?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-558be7e291624ffa466aa92bf834ed5bd7b169f27144034bf5bddb9707f7eb6d | A density curve looks like an inverted letter V. The first segment goes from point (0, 0.6) to the point (0.5, 1.4). The second segment goes from (0.5, 1.4) to (1, 0.6). What percent of the observations lie between 0.3 and 0.7? | [
"49.6\\%"
] | [
"A density curve looks like an inverted letter V. The first segment goes from point (0, 0.6) to the point (0.5, 1.4). The second segment goes from (0.5, 1.4) to (1, 0.6). What percent of the observations lie between 0.3 and 0.7?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8dc504e1eb729965de22de037d283f4263ec2bd5cc0571995030faee3c066ec0 | A department store offers two promotions. Promotion A says, "Buy one pair of shoes, get the second pair for half the price." Promotion B says, "Buy one pair of shoes, get $\$10$ off the second pair." Jane wants to buy two pairs of shoes that cost $\$30$ each. She can only use one of the promotions, A or B. Jane decides... | [
"5"
] | [
"A department store offers two promotions. Promotion A says, \"Buy one pair of shoes, get the second pair for half the price.\" Promotion B says, \"Buy one pair of shoes, get $\\$10$ off the second pair.\" Jane wants to buy two pairs of shoes that cost $\\$30$ each. She can only use one of the promotions, A or B. J... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7d1b75f139d10d4df5287273ad79f16952b7ed7978b8654f40a5302f4905b409 | A department store sells two brands of calculators. The store pays $6 for each brand A calculator and $8 for each brand B calculator. The demand equations are given by: \[ x = 116 - 30p + 20q \] for brand A, \[ y = 144 + 16p - 24q \] for brand B, where \( p \) is the selling price for brand A and \( q \) is the selling... | [
"-1527.272"
] | [
"A department store sells two brands of calculators. The store pays $6 for each brand A calculator and $8 for each brand B calculator. The demand equations are given by:\n\\[ x = 116 - 30p + 20q \\] for brand A,\n\\[ y = 144 + 16p - 24q \\] for brand B,\nwhere \\( p \\) is the selling price for brand A and \\( q \\... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a0f9a65110cd0df3045c2477cf9d2bfee2ef8a946d6dadeff6f90fb9d8de51e0 | A department's employees were divided into two groups (Women and Men) and underwent a training program. The Women's group (A) consists of 10 people with scores: 48, 40, 37, 33, 44, 30, 35, 42, 44, 41. The Men's group (B) consists of 9 people with scores: 44, 38, 30, 32, 39, 43, 40, 45, 31. Assume the scores follow norm... | [
"\\text{Fail to reject } H_0"
] | [
"A department's employees were divided into two groups (Women and Men) and underwent a training program. The Women's group (A) consists of 10 people with scores: 48, 40, 37, 33, 44, 30, 35, 42, 44, 41. The Men's group (B) consists of 9 people with scores: 44, 38, 30, 32, 39, 43, 40, 45, 31. Assume the scores follow... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1a16b92eff30266b746bc5af5114bd4452032872e7b094f21d1a3025837134ef | A deposit of $N$ roubles was made in a savings bank with a $p\%$ annual interest. At the end of each year the customer withdraws $M$ roubles. How many years after a withdrawal of a corresponding sum will the remainder be thrice as large as the initial deposit? | [
"In the number of years which is larger than $\\log \\left(\\frac{3Np-100M}{Np-100M}\\right)/\\log \\left(1+\\frac{p}{100}\\right)$"
] | [
"A deposit of $N$ roubles was made in a savings bank with a $p\\%$ annual interest. At the end of each year the customer withdraws $M$ roubles. How many years after a withdrawal of a corresponding sum will the remainder be thrice as large as the initial deposit?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7cafa398e8f0cdcd4ead9449a3aa4ced879c66f252f69e422913a69b42e39c57 | A derivable function \( f: \mathbb{R^+} \rightarrow \mathbb{R} \) satisfies the condition \( f(x) - f(y) \geq \ln(x/y) + x - y \) for all \( x, y \in \mathbb{R^+} \). Determine the form of \( f(x) \). | [
"\\( f(x) = x + \\ln x + a \\)"
] | [
"A derivable function \\( f: \\mathbb{R^+} \\rightarrow \\mathbb{R} \\) satisfies the condition \\( f(x) - f(y) \\geq \\ln(x/y) + x - y \\) for all \\( x, y \\in \\mathbb{R^+} \\). Determine the form of \\( f(x) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-b0f19b21348b4818213e56897f6b8b97f605c1fa60fdffb46f2cf8cdf3f23b0f | A desert expedition needs to provide one liter of drinking water for another member residing $n$ days of walking from the camp. Each member can carry at most 3 liters of water and must drink one liter of water every day spent in the desert. All members must return to the camp. How much water do they need (at least) to ... | [
"\\(3^n\\)"
] | [
"A desert expedition needs to provide one liter of drinking water for another member residing $n$ days of walking from the camp. Each member can carry at most 3 liters of water and must drink one liter of water every day spent in the desert. All members must return to the camp. How much water do they need (at least... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 31,
"genselect": 7
} | 38 | 0 | 0 |
omr-ws-8de9f77ca11ec27d698fe20fe48a9283d1fd319195c78b8275088575d732e360 | A design $X$ is an array of the digits $1, 2, \ldots, 9$ in the shape of an $X$. We will say that a design $X$ is balanced if the sum of the numbers on each of the diagonals match. Determine the number of balanced designs $X$. | [
"32640"
] | [
"A design $X$ is an array of the digits $1, 2, \\ldots, 9$ in the shape of an $X$. We will say that a design $X$ is balanced if the sum of the numbers on each of the diagonals match. Determine the number of balanced designs $X$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-23f6ec1638adebd5773f28ff4cad449ded09a0d764a1ff7a00df8f56515c2f3b | A designer has 3 fabric colors he may use for a dress: red, green, and blue. Four different patterns are available for the dress. If each dress design requires exactly one color and one pattern, how many different dress designs are possible? | [
"12"
] | [
"A designer has 3 fabric colors he may use for a dress: red, green, and blue. Four different patterns are available for the dress. If each dress design requires exactly one color and one pattern, how many different dress designs are possible?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ad31c5091ae7969c8bc8b291437f00b1cd6e343e18694575a3573cb7b631f902 | A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible? | [
"729",
"\\(3^6\\)"
] | [
"A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?"
] | [
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 20
} | 51 | 0 | 0 |
omr-ws-5f8ef76af6db2d453e9fed1a2e5bbc9b64a760b5526bb66cadabf907831c647d | A detective arrives at a crime scene at 2:00 pm and checks the body temperature of a murder victim and finds it to be 35°C. Two hours later, the temperature is now 30°C. If the room temperature is 20°C, and we assume the normal body temperature of a living person to be 37°C, at what time did the murder occur? Given the... | [
"1:23 pm"
] | [
"A detective arrives at a crime scene at 2:00 pm and checks the body temperature of a murder victim and finds it to be 35°C. Two hours later, the temperature is now 30°C. If the room temperature is 20°C, and we assume the normal body temperature of a living person to be 37°C, at what time did the murder occur?\n\nG... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-58158021d1d2de1ec2f14843ab01186fc907a94c049a01b54c58aef912cd459b | A diagonal of a 100-gon is called good if it divides the 100-gon into two polygons each with an odd number of sides. A 100-gon was split into triangles with non-intersecting diagonals, exactly 49 of which are good. The triangles are colored into two colors such that no two triangles that border each other are colored w... | [
"0"
] | [
"A diagonal of a 100-gon is called good if it divides the 100-gon into two polygons each with an odd number of sides. A 100-gon was split into triangles with non-intersecting diagonals, exactly 49 of which are good. The triangles are colored into two colors such that no two triangles that border each other are colo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-42e65b199e17d13ffc32dbadbf33510108e809cebc7e8871a5a2a7e75d01c88a | A diagonal of a polygon is a segment joining two nonconsecutive vertices of the polygon. How many diagonals does a regular octagon have? | [
"20"
] | [
"A diagonal of a polygon is a segment joining two nonconsecutive vertices of the polygon. How many diagonals does a regular octagon have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f7c2054c7b3c209572b7f37319c72ff600faa59ae59093e646d1a3c966119b89 | A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals. Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the maximum possible number of isosceles t... | [
"1003"
] | [
"A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals. Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the maximum possible number of isosce... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8989d0ad97c2166d441d8ff8e5d6070450b35ae42eee643295063de755e62a3a | A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides of the 2006-gon are also regarded as odd diagonals. Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Define a good triangle as an is... | [
"1003"
] | [
"A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides of the 2006-gon are also regarded as odd diagonals. Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Define a good triangle as ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-968229bca4fd3f2191a004ba0bc53196b4becf3e6d6961b4d0d820b2a297b7bd | A diameter $AB$ of a circle intersects a chord $CD$ at the point $E$. If $CE = 7$, $DE = 1$, and $\angle{BED} = 45^{\circ}$, determine the radius of the circle. | [
"5"
] | [
"A diameter $AB$ of a circle intersects a chord $CD$ at the point $E$. If $CE = 7$, $DE = 1$, and $\\angle{BED} = 45^{\\circ}$, determine the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fdf4e7a088c7f31528599b33aff82ebde7716c1c8d8f0b2a1660422529f2b5a7 | A diameter $CD$ of a circle is extended through $D$ to an external point $P$. The measure of secant $CP$ is $77$. From $P$, another secant is drawn intersecting the circle first at $A$, then at $B$. The measure of secant $PB$ is $33$. The diameter of the circle measures $74$. Find the measure of the angle formed by the... | [
"30 degrees"
] | [
"A diameter $CD$ of a circle is extended through $D$ to an external point $P$. The measure of secant $CP$ is $77$. From $P$, another secant is drawn intersecting the circle first at $A$, then at $B$. The measure of secant $PB$ is $33$. The diameter of the circle measures $74$. Find the measure of the angle formed b... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43fbf2c41be7fb56b4f1f939985d743c21ba31bc990e0520c9f53e6593f0dc39 | A dice is thrown $n$ times. Let $A_i \in \{\{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{6\}\}$ denote the value obtained on the $i$th throw. What is the probability that the union of the values obtained in the first $n-1$ throws contains exactly 5 distinct values, and the union of the values obtained in all $n$ throws contains... | [
"\\dfrac{5^{n-1} - 5 \\cdot 4^{n-1} + 10 \\cdot 3^{n-1} - 10 \\cdot 2^{n-1} + 5}{6^{n-1}}"
] | [
"A dice is thrown $n$ times. Let $A_i \\in \\{\\{1\\}, \\{2\\}, \\{3\\}, \\{4\\}, \\{5\\}, \\{6\\}\\}$ denote the value obtained on the $i$th throw. What is the probability that the union of the values obtained in the first $n-1$ throws contains exactly 5 distinct values, and the union of the values obtained in all... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-435077397817cb5e5904528f6cab4254b1f9a1d4201d2008b234464285a0d472 | A dice is thrown 3 times and the sum of the 3 numbers thrown is 15. Find the probability that the first throw was a four. | [
"\\(\\frac{1}{5}\\)"
] | [
"A dice is thrown 3 times and the sum of the 3 numbers thrown is 15. Find the probability that the first throw was a four."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-ce8201fbfab2d36e8e3ffc26784c13f544844078e898d865191101618a123aa0 | A dice is tossed twice. Let \( a \) and \( b \) be the numbers that appear on the first and second tosses, respectively. The probability that there are real numbers \( x \), \( y \), and \( z \) that satisfy the equations \( x + y + z = a \) and \( x^2 + y^2 + z^2 = b \) is... | [
"\\dfrac{4}{9}"
] | [
"A dice is tossed twice. Let \\( a \\) and \\( b \\) be the numbers that appear on the first and second tosses, respectively. The probability that there are real numbers \\( x \\), \\( y \\), and \\( z \\) that satisfy the equations \\( x + y + z = a \\) and \\( x^2 + y^2 + z^2 = b \\) is..."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f23fae0a85f3cc211a8692fa1b40f9694da562de6bfe354a16d3894f662cf6f2 | A dice maker has an infinite number of labels for the numbers $1$ through $6$, inclusive. A set contains two six-sided dice for each possible configuration using the labels on the sides such that rotations are indistinguishable (there can be multiple identical labels on the dice). What is the remainder when the number ... | [
"452"
] | [
"A dice maker has an infinite number of labels for the numbers $1$ through $6$, inclusive. A set contains two six-sided dice for each possible configuration using the labels on the sides such that rotations are indistinguishable (there can be multiple identical labels on the dice). What is the remainder when the nu... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-9e317562273f5c02d7489cdf5eb9f2ae97bbd03ed7f890a0f2defd591aeb2b41 | A dice with its center at the origin of a coordinate system and faces perpendicular to the axes can be rotated 90 degrees about any of the axes in either direction. Determine all possible numbers of steps required for the dice to return to its initial position with each number in its original place. | [
"\\(4k\\) where \\(k\\) is a positive integer."
] | [
"A dice with its center at the origin of a coordinate system and faces perpendicular to the axes can be rotated 90 degrees about any of the axes in either direction. Determine all possible numbers of steps required for the dice to return to its initial position with each number in its original place."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-71a24d89e1772bbb3304140a551930dd8cbbf21a0d70c85aef360a6f635e6034 | A die has $\mu$ faces marked $1$, $\nu$ faces marked $-1$, and the rest marked $0$. The die is repeatedly thrown until the sum of the numbers turned up no longer lies between the limits $m$ and $-n$. If the probability of reaching either extreme of these limits is the same, find the value of $\frac{\mu^{m+n} + \nu^{m+n... | [
"2"
] | [
"A die has $\\mu$ faces marked $1$, $\\nu$ faces marked $-1$, and the rest marked $0$. The die is repeatedly thrown until the sum of the numbers turned up no longer lies between the limits $m$ and $-n$. If the probability of reaching either extreme of these limits is the same, find the value of $\\frac{\\mu^{m+n} +... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23396305b4ad577ea2432119827746da72a7bcacec8fbec782285a947bf2ca89 | A die has 2 faces with the number 1 and 4 faces with the number 2. The die is tossed 6 consecutive times. What is the probability that at least two consecutive throws result in the same number? | [
"\\(\\frac{713}{729}\\)"
] | [
"A die has 2 faces with the number 1 and 4 faces with the number 2. The die is tossed 6 consecutive times. What is the probability that at least two consecutive throws result in the same number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-0bdc8f137ebc8bc93b3ccc76c78862b6fb81b0a33f7d80e1546d38094387b975 | A die is a cube with side length 1 with the numbers from 1 to 6 on each of its faces, so that each number appears only once, and the sum of the numbers on opposite faces is 7. A cube of $3 \times 3$ using 27 dice, and then the sum of all the numbers that appear above its faces is calculated. Determine the possible valu... | [
"\\( 27 \\text{ to } 162 \\)"
] | [
"A die is a cube with side length 1 with the numbers from 1 to 6 on each of its faces, so that each number appears only once, and the sum of the numbers on opposite faces is 7. A cube of $3 \\times 3$ using 27 dice, and then the sum of all the numbers that appear above its faces is calculated. Determine the possibl... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-5d124386a45865b203897fe1bcdc6167ff16ce2ed5eb464d36c7b0ed34b4c0cb | A die is placed on a table with the 6 on the lower face. In each step, the die is rolled so that one of the four side faces becomes the upper face with equal probability. How many steps, on average, are necessary for the 6 to reach the upper face? | [
"6"
] | [
"A die is placed on a table with the 6 on the lower face. In each step, the die is rolled so that one of the four side faces becomes the upper face with equal probability. How many steps, on average, are necessary for the 6 to reach the upper face?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9ca15cc02f5232524afc995841d44c01d24780366c9ba18a1632c6085e5e41e | A die is thrown \(n\) times. Let \(x\) be the number of times the throw results in 1, and \(y\) be the number of times the throw results in 6. Find the expected value of their product, \(M(xy)\). | [
"\\(\\frac{n^2}{36}\\)"
] | [
"A die is thrown \\(n\\) times. Let \\(x\\) be the number of times the throw results in 1, and \\(y\\) be the number of times the throw results in 6. Find the expected value of their product, \\(M(xy)\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c06dc458bbd3ebbb441cf01880d9444d93903c65efa91e09ee141f637ab7687d | A die with 6 sides is repeatedly rolled, and the numbers that come up on each roll are added. Once the sum exceeds 12, the rolling is stopped. What sum is expected to occur most frequently? | [
"13"
] | [
"A die with 6 sides is repeatedly rolled, and the numbers that come up on each roll are added. Once the sum exceeds 12, the rolling is stopped. What sum is expected to occur most frequently?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a6c1a9a1e0e9d20f9884e5067eec74dacf260c699e3e030f3db36f71f027a1f0 | A different set of natural numbers has a median of 20 and a mean of 17. What will be the largest number in this series? Given: - The median is 20. - The mean is 17. - The number of terms is odd, so the number of terms is 39. - The sum of the terms is 663. Determine the largest number in the series. | [
"69"
] | [
"A different set of natural numbers has a median of 20 and a mean of 17. What will be the largest number in this series?\n\nGiven:\n- The median is 20.\n- The mean is 17.\n- The number of terms is odd, so the number of terms is 39.\n- The sum of the terms is 663.\n\nDetermine the largest number in the series."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a0d8c068e62101ccccd1a8c4c6e1560725b4c3b1e68aae7c18b68360984bd5fa | A differentiable function $g$ satisfies $\int_{0}^{x} g(x-t+1) g(t) \, dt = x^4 + x^2$ for all $x \geq 0$. Find $g(t)$. | [
"\\sqrt{4t^3 - 12t^2 + 14t - 6}"
] | [
"A differentiable function $g$ satisfies $\\int_{0}^{x} g(x-t+1) g(t) \\, dt = x^4 + x^2$ for all $x \\geq 0$. Find $g(t)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eb2aab95fe455a9e2c592ef31e74f9a60db8df25d83cb63eaef58a1345d4cfcb | A differentiable scalar field \( f \) has, at the point \((1,2)\), a directional derivative of \(+2\) in the direction toward \((2,2)\) and \(-2\) in the direction toward \((1,1)\). Determine the gradient vector at \((1,2)\) and compute the directional derivative in the direction towards \((4,6)\). | [
"\\(\\frac{14}{5}\\)"
] | [
"A differentiable scalar field \\( f \\) has, at the point \\((1,2)\\), a directional derivative of \\(+2\\) in the direction toward \\((2,2)\\) and \\(-2\\) in the direction toward \\((1,1)\\). Determine the gradient vector at \\((1,2)\\) and compute the directional derivative in the direction towards \\((4,6)\\).... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-d049756530ea53ff641af5c61e62ee580ef089dff9c72bdb8cf9c0d8dad27df5 | A differentiable single variable function \(f\) satisfies \(f'(\sin^2 x) = \cos 2x + \tan^2 x\) and \(f(0) = 2\). Find \(f\left(\frac{1}{2}\right)\). | [
"\\(\\ln 2 - \\frac{1}{4}\\)"
] | [
"A differentiable single variable function \\(f\\) satisfies \\(f'(\\sin^2 x) = \\cos 2x + \\tan^2 x\\) and \\(f(0) = 2\\). Find \\(f\\left(\\frac{1}{2}\\right)\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-29c0ee40c72d843cbaa9c44acd3322554ea5cfd1600876c7b9b6fe58a7026ddb | A digit is written to the right of the units digit of $757$. If the resulting four-digit number is divisible by $3$, how many possibilities are there for the digit that was written? | [
"3"
] | [
"A digit is written to the right of the units digit of $757$. If the resulting four-digit number is divisible by $3$, how many possibilities are there for the digit that was written?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bd057789dcb3dbf51d68ab60b02ec4f0c61535664c37572075da545936b308ff | A digital clock displays the time in standard format (starting at $12:00$, through times like $1:24$ and $3:30$, ending with $11:59$). Looking at the clock at a random time, what is the probability that the display contains a zero? | [
"\\(\\frac{5}{16}\\)"
] | [
"A digital clock displays the time in standard format (starting at $12:00$, through times like $1:24$ and $3:30$, ending with $11:59$). Looking at the clock at a random time, what is the probability that the display contains a zero?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 28
} | 48 | 0 | 0 |
omr-ws-78a2d4a935b92286fe3329277e185f6ee5c97138e79b48eb110a5c3d68ccc529 | A digital clock displays the time in the form $AB:CD$, but it is broken such that the colon and rightmost digit do not display, and the leftmost digit always shows a $6$. Therefore, anyone who looks at the clock mistakenly interprets the first visible digit as the hour and the other two visible digits as the minutes. I... | [
"405"
] | [
"A digital clock displays the time in the form $AB:CD$, but it is broken such that the colon and rightmost digit do not display, and the leftmost digit always shows a $6$. Therefore, anyone who looks at the clock mistakenly interprets the first visible digit as the hour and the other two visible digits as the minut... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3769304c53df58ae925bbaa7ea801a04637516354cc4acdb1a1206fe9e2bd61 | A digital clock marks the hours from 00:00 to 23:59. When this clock displays each digit 8 as if it were 2, for example, instead of showing 18:38, it shows 12:32. How many times in a day does the clock show an incorrect time? | [
"132"
] | [
"A digital clock marks the hours from 00:00 to 23:59. When this clock displays each digit 8 as if it were 2, for example, instead of showing 18:38, it shows 12:32. How many times in a day does the clock show an incorrect time?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c9ba828d66e3d1b641d837eae715d14d6d14aed6efff71fe3d8114b8d522d8c | A digital clock shows the time 4:56. How many minutes will pass until the clock next shows a time in which all of the digits are consecutive and in increasing order? | [
"458"
] | [
"A digital clock shows the time 4:56. How many minutes will pass until the clock next shows a time in which all of the digits are consecutive and in increasing order?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 5
} | 25 | 0 | 0 |
omr-ws-8e1ca055ebbe61f393271615a70b0156585d6ba8e6501d221703e8220589d1fd | A digital counter starts at 0000 and increments by 1, 2, 4, 8, ... (doubling each time a person enters the room) and decrements by 1 each time a person exits the room. What is the smallest number greater than 2007 that the counter can read? | [
"2036"
] | [
"A digital counter starts at 0000 and increments by 1, 2, 4, 8, ... (doubling each time a person enters the room) and decrements by 1 each time a person exits the room. What is the smallest number greater than 2007 that the counter can read?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 3
} | 4 | 0 | 0 |
omr-ws-35f74eee8fa8b04ffb76954306f4fb7862a69b30098eef3cc8af1bc3dc7c0fb6 | A digital music library contains 12 days of music and takes up 16,000 megabytes of disk space. On average, how many megabytes of disk space does one hour of music in this library take up? Express your answer to the nearest whole number. | [
"56"
] | [
"A digital music library contains 12 days of music and takes up 16,000 megabytes of disk space. On average, how many megabytes of disk space does one hour of music in this library take up? Express your answer to the nearest whole number."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c7282873db42400ee3ff0d5a7763abb8ce9ba28b6d2ff1f48bf3f323e06543fa | A digital number is defined as a positive integer that is equal to the sum of its digits each raised to the power of the number of digits. For example, $3126$ is a digital number because $3^4 + 1^4 + 2^4 + 6^4 = 3126$. Find the sum of all digital numbers less than $10,000$. | [
"20662"
] | [
"A digital number is defined as a positive integer that is equal to the sum of its digits each raised to the power of the number of digits. For example, $3126$ is a digital number because $3^4 + 1^4 + 2^4 + 6^4 = 3126$. Find the sum of all digital numbers less than $10,000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-09323334c18dca084617d910d6ae131fa65251acdd563d0f919740f28895a576 | A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display? | [
"23"
] | [
"A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-00d579e492af79f691a1d5fd6677b7acdde4d365e937c9de2a1f77fa51c03f0e | A dilation of the plane sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A'(5,6)$. What distance does the origin $O(0,0)$ move under this transformation? | [
"\\(\\sqrt{13}\\)"
] | [
"A dilation of the plane sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A'(5,6)$. What distance does the origin $O(0,0)$ move under this transformation?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 32
} | 49 | 0 | 0 |
omr-ws-fec02ab11feead04998f871b3a1b78047634ce080f05a9d942b7cc45d7aed510 | A dilation of the plane with a positive scale factor transforms the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A'(5,6)$. What distance does the origin $O(0,0)$ move under this transformation? | [
"\\(\\sqrt{13}\\)"
] | [
"A dilation of the plane with a positive scale factor transforms the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A'(5,6)$. What distance does the origin $O(0,0)$ move under this transformation?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-9e164072d4e10cc28f5b13664f6d0f107c399b5c1e131fd77f44d22c4b9492ef | A direction is chosen on each edge of a tetrahedron. What is the greatest number of vectors obtained that add up to a zero vector? | [
"4"
] | [
"A direction is chosen on each edge of a tetrahedron. What is the greatest number of vectors obtained that add up to a zero vector?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 33,
"genselect": 2
} | 35 | 0 | 0 |
omr-ws-b1eadd28db6b4bc1165c4a5fa0f0ff411c7c7e6d98b913396194467216dff873 | A dirigible and an airplane are flying towards each other, having left their terminals at the same time. When they meet, the dirigible has made 100 km less than the airplane, and it arrives at the departure point of the airplane three hours after they pass each other. The airplane arrives at the airport of the dirigibl... | [
"The distance between the airports is equal to 500 km; the speed of the dirigible is 100 km/h and that of the airplane is 150 km/h"
] | [
"A dirigible and an airplane are flying towards each other, having left their terminals at the same time. When they meet, the dirigible has made 100 km less than the airplane, and it arrives at the departure point of the airplane three hours after they pass each other. The airplane arrives at the airport of the dir... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e27dbf19f4556d0085306849b3e9f9dc8c89f24cc3fd0df9693c2ee4e5f536d | A dirt bike track consists of two 90-meter straightaways and two semi-circles with a diameter of 40 meters. What is the total length of one lap around the track? | [
"\\( 180 + 40\\pi \\) m"
] | [
"A dirt bike track consists of two 90-meter straightaways and two semi-circles with a diameter of 40 meters. What is the total length of one lap around the track?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-30af825359e465da5e57067367a2d5694130e34071bd5c302afe178aba762bb1 | A disc is divided into 30 segments labelled by the numbers $50, 100, 150, \ldots, 1500$ in some order. Let $S$ be the maximum sum of the labels of any three successive segments. Calculate the minimum possible value of $S$. | [
"3000"
] | [
"A disc is divided into 30 segments labelled by the numbers $50, 100, 150, \\ldots, 1500$ in some order. Let $S$ be the maximum sum of the labels of any three successive segments. Calculate the minimum possible value of $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-62cefe33e6cb160f461262329b3d2bf1b80645c695e541800bcc6a5d6660e129 | A disc of radius $1$ unit is cut into quadrants (identical quarters), and the quadrants are placed in a square of side $1$ unit. What are the other possible areas of overlap? | [
"1 + \\frac{\\pi}{3} - \\sqrt{3}"
] | [
"A disc of radius $1$ unit is cut into quadrants (identical quarters), and the quadrants are placed in a square of side $1$ unit. What are the other possible areas of overlap?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-827117de0c83a3cc12e1a808654dd9a810b311b63869ea745ad980d6637213df | A disc of radius $1$ unit is cut into quadrants (identical quarters), and the quadrants are placed in a square of side $1$ unit. What is the least possible area of overlap? | [
"\\pi - 1"
] | [
"A disc of radius $1$ unit is cut into quadrants (identical quarters), and the quadrants are placed in a square of side $1$ unit. What is the least possible area of overlap?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-225f0ee20bc4a6436fef76a3164007b3d341d9e4d7237c05c3c3979fd7053723 | A disc rotates back and forth with angular velocity \(\cos t\) rad/sec. An insect starting 10 cm from the center of the disc at time \(t=0\) crawls outward at a rate of \(5t\) cm/sec. Find the position, velocity, and speed of the insect after \(2\pi\) seconds. Give the components of position and velocity relative to a ... | [
"Position: \\( (0.1 + 0.1\\pi^2, 0) \\), Velocity: \\( (0.1\\pi, 0.1 + 0.1\\pi^2) \\), Speed: \\( \\sqrt{0.01\\pi^4 + 0.03\\pi^2 + 0.01} \\)"
] | [
"A disc rotates back and forth with angular velocity \\(\\cos t\\) rad/sec. An insect starting 10 cm from the center of the disc at time \\(t=0\\) crawls outward at a rate of \\(5t\\) cm/sec. Find the position, velocity, and speed of the insect after \\(2\\pi\\) seconds. Give the components of position and velocity... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-71e26d928b0f6259711b537212d754608b6f10aa5574e68994b6a475ecaa9e25 | A disc-shaped automatic cleaner moves on a flat floor. For every point on the boundary of the disc, there exists a straight line that has always contained this point during the motion. If the radius of the disc is $r$, find the maximum possible distance that the center of the disc can travel from its initial position, ... | [
"2r"
] | [
"A disc-shaped automatic cleaner moves on a flat floor. For every point on the boundary of the disc, there exists a straight line that has always contained this point during the motion. If the radius of the disc is $r$, find the maximum possible distance that the center of the disc can travel from its initial posit... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-41d27126c5e2a8a8c132f442f9b3d04120bddfc39ae455a498ce668f5880bd5b | A dishonest tradesman marks his goods at a 5% advance on the cost price and uses a fraudulent balance where the beam is horizontal when the weight in one scale is one-fifth more than the weight in the other. What is his actual gain percent? | [
"26%"
] | [
"A dishonest tradesman marks his goods at a 5% advance on the cost price and uses a fraudulent balance where the beam is horizontal when the weight in one scale is one-fifth more than the weight in the other. What is his actual gain percent?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-3ce92316cacc695b1013dcf5d44df028582e44a0e4d00d6daa5d835d380f7e7b | A disk in a record turntable makes 100 revolutions per minute and plays for 24 minutes and 30 seconds. The recorded line over the disk is a spiral with a diameter that decreases uniformly from 29 cm to 11.5 cm. Compute the length of the recorded line. | [
"155862.265526224 cm"
] | [
"A disk in a record turntable makes 100 revolutions per minute and plays for 24 minutes and 30 seconds. The recorded line over the disk is a spiral with a diameter that decreases uniformly from 29 cm to 11.5 cm. Compute the length of the recorded line."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 24
} | 44 | 0 | 0 |
omr-ws-33e9ebc75ee64e04c6c53fcbb1f294a59ae3316a679707bab48171d88c81c57a | A disk is divided into $2n$ equal sectors, with $n$ sectors colored black and $n$ sectors colored white. The white sectors are numbered from $1$ to $n$ clockwise, and the black sectors are numbered from $1$ to $n$ counterclockwise. Find the number of distinct half-disks consisting of $n$ consecutive sectors that contai... | [
"2"
] | [
"A disk is divided into $2n$ equal sectors, with $n$ sectors colored black and $n$ sectors colored white. The white sectors are numbered from $1$ to $n$ clockwise, and the black sectors are numbered from $1$ to $n$ counterclockwise. Find the number of distinct half-disks consisting of $n$ consecutive sectors that c... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-791fc8e3067767fab407d73da5d828f2b47f2312e466f694db1c41055cf0eac5 | A disk of radius $1$ rolls all the way around in the inside of a square of side length $s > 4$ and sweeps out a region of area $A$. A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$. The value of $s$ can be written as $a + \dfrac{b\pi}{c}$, where $a... | [
"9"
] | [
"A disk of radius $1$ rolls all the way around in the inside of a square of side length $s > 4$ and sweeps out a region of area $A$. A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$. The value of $s$ can be written as $a + \\dfrac{b\\pi}{c}$, w... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-651ae0746314d091f922da8ece234db457c36a10c1fe7f78a8e981ecfa2f00d1 | A disk of radius $1$ rolls all the way around the inside of a square of side length $s > 4$ and sweeps out a region of area $A$. A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$. The value of $s$ can be written as $a + \frac{b\pi}{c}$, where $a, b$... | [
"10"
] | [
"A disk of radius $1$ rolls all the way around the inside of a square of side length $s > 4$ and sweeps out a region of area $A$. A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$. The value of $s$ can be written as $a + \\frac{b\\pi}{c}$, where... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-d7e8ba1270d85e9562d2b951131a0153c4aa7071ed8b8cff2169a68a6cfcd14a | A disk of radius 1 cm has a small hole at a point halfway between the center and the circumference. The disk is moved once around the inside of a circle of radius 2 cm, keeping the disk in contact with the circle without slipping. A pen is put through the hole in the disk and draws a curve. What is the area enclosed by... | [
"\\dfrac{3\\pi}{4}"
] | [
"A disk of radius 1 cm has a small hole at a point halfway between the center and the circumference. The disk is moved once around the inside of a circle of radius 2 cm, keeping the disk in contact with the circle without slipping. A pen is put through the hole in the disk and draws a curve. What is the area enclos... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-be8435988159863be9383d7bef9d49aae74d750ccf664697e2c423b24f69f05c | A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. The smaller disk rolls along the outside of the larger disk until it has turned through an angle of $360^\circ$. If t... | [
"7"
] | [
"A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. The smaller disk rolls along the outside of the larger disk until it has turned through an angle of $360^\\circ$... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-35a48fa38bd65f6dad396408e56f275c68f987bcce99085dd03a779a5726831b | A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. The smaller disk rolls along the outside of the larger disk until it has turned through an angle of $360^\circ$. If t... | [
"058"
] | [
"A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. The smaller disk rolls along the outside of the larger disk until it has turned through an angle of $360^\\circ$... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 8,
"tir": 3
} | 48 | 0 | 0 |
omr-ws-5487d82854b8fc18a48fc2464e22ac4f05e60c01fe21e99e75621bb7f17123d7 | A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until t... | [
"058"
] | [
"A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk un... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 39,
"genselect": 7,
"tir": 4
} | 50 | 0 | 0 |
omr-ws-8f0e1722e809503e7f1356cfea0e7bcb79eb93dc90a01e95daf9d9a0cb3c84b1 | A diver is on a Ferris wheel with a radius of 50 feet, centered 65 feet above the ground. The Ferris wheel rotates counterclockwise at a constant rate, completing a full turn every 40 seconds. A cart starts 240 feet to the left of the base of the Ferris wheel and moves to the right at a constant speed of 15 feet per se... | [
"12"
] | [
"A diver is on a Ferris wheel with a radius of 50 feet, centered 65 feet above the ground. The Ferris wheel rotates counterclockwise at a constant rate, completing a full turn every 40 seconds. A cart starts 240 feet to the left of the base of the Ferris wheel and moves to the right at a constant speed of 15 feet p... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-ae19abd5ea3857656056c817a3843c1a03a7b984eb4f3a13400feb190ce660ce | A divided highway into the city passes under a number of bridges. The arch over each lane is in the form of a semi-ellipse, where the height is equal to the width. What is the lowest bridge under which a truck 6 feet wide and 12 feet high can theoretically pass? | [
"\\( 6\\sqrt{5} \\)"
] | [
"A divided highway into the city passes under a number of bridges. The arch over each lane is in the form of a semi-ellipse, where the height is equal to the width. What is the lowest bridge under which a truck 6 feet wide and 12 feet high can theoretically pass?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-93455282d8f958c9072206f739b7d437719276402c08d62a3e69592e600b814b | A divisor \( d \) of \( 3683488 \) is called a superdivisor if the sum of the digits of \( d \) also divides \( 3683488 \). Find all superdivisors of \( 3683488 \). | [
"\\{1, 2, 4, 8, 1951, 230218\\}"
] | [
"A divisor \\( d \\) of \\( 3683488 \\) is called a superdivisor if the sum of the digits of \\( d \\) also divides \\( 3683488 \\). Find all superdivisors of \\( 3683488 \\)."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 26
} | 26 | 0 | 0 |
omr-ws-711fb63095881373bcf639ec1d3c204a1e374184b215f43c9ef49270dc3e1fff | A divisor of a number is a proper divisor if it is not equal to the number. What is the sum of the proper divisors of $432$? | [
"808"
] | [
"A divisor of a number is a proper divisor if it is not equal to the number. What is the sum of the proper divisors of $432$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f3f7064db285ed4052c77dd65b8a1797a1873c81ddf2b8c6210568f5b3bde3bc | A doctor sends different stimuli \( S_i \) and receives different responses \( R_j \) where \( 1 \leq i, j \leq N \). If the doctor sends the stimulus \( S_i \), the probability that he receives the response \( R_i \) is \( p \) (where \( 0 < p < 1 \)), otherwise, he receives another response randomly from the other re... | [
"\\dfrac{p^2 (N - 1)}{p^2 (N - 1) + (1 - p)^2}"
] | [
"A doctor sends different stimuli \\( S_i \\) and receives different responses \\( R_j \\) where \\( 1 \\leq i, j \\leq N \\). If the doctor sends the stimulus \\( S_i \\), the probability that he receives the response \\( R_i \\) is \\( p \\) (where \\( 0 < p < 1 \\)), otherwise, he receives another response rando... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1c85e2644d98e3235aa5fea60e8b88f6567aba32e25120ac38d92691ca40b70d | A dodecagon inscribed in a circle has six sides of length $a$ and six sides of length $b$, in any order. Let $C$ be a vertex adjacent to a side $AC$ of length $a$ and a side $CB$ of length $b$. Calculate the angle $\angle ACB$. | [
"\\(30^\\circ\\)"
] | [
"A dodecagon inscribed in a circle has six sides of length $a$ and six sides of length $b$, in any order. Let $C$ be a vertex adjacent to a side $AC$ of length $a$ and a side $CB$ of length $b$. Calculate the angle $\\angle ACB$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 3
} | 4 | 0 | 0 |
omr-ws-425bd149dd0062f59bdf55eca7e872dfb5fce833c69f55c48a3d66556b07e7db | A dodecagon is inscribed in a circle with radius $r$. The polygon has six sides of length $8\sqrt{3}$ that alternate with six sides of length $15$. Find $r^2$. | [
"777"
] | [
"A dodecagon is inscribed in a circle with radius $r$. The polygon has six sides of length $8\\sqrt{3}$ that alternate with six sides of length $15$. Find $r^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-fb40e8c27e24b25f823a388a6136b61c7d27b9a8e73e50461007e083f33d3a97 | A dodecagon labelled with 12 months is rolled in a game. One "turn" of the game is to roll it until the face showing "May" appears, and the number of rolls is recorded. What is the probability of having 5 consecutive turns where the number of rolls is not greater than 10? | [
"\\(\\left( 1 - \\left( \\frac{11}{12} \\right)^{10} \\right)^5\\)"
] | [
"A dodecagon labelled with 12 months is rolled in a game. One \"turn\" of the game is to roll it until the face showing \"May\" appears, and the number of rolls is recorded. What is the probability of having 5 consecutive turns where the number of rolls is not greater than 10?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-b19f6b3b3637c3449a55ac3334702214f109116ad3c6aeba78cad16bd1ef44b4 | A dodecahedron (a polyhedron with $12$ faces, each a regular pentagon) is projected orthogonally onto a plane parallel to one of its faces to form a polygon. Find the measure (in degrees) of the largest interior angle of this polygon. | [
"144"
] | [
"A dodecahedron (a polyhedron with $12$ faces, each a regular pentagon) is projected orthogonally onto a plane parallel to one of its faces to form a polygon. Find the measure (in degrees) of the largest interior angle of this polygon."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ca43cd266586c54e935e1ca11d89118c5ce48bae661637e5edf0d1cd9ed75d25 | A dodecahedron has 12 pentagonal faces and 20 vertices. How many ways are there to color each vertex red, green, blue, or yellow, where two colorings are considered the same if you can rotate one to become the other? | [
"18325477888"
] | [
"A dodecahedron has 12 pentagonal faces and 20 vertices. How many ways are there to color each vertex red, green, blue, or yellow, where two colorings are considered the same if you can rotate one to become the other?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d6158d7ecf13a12ee3066c30f30e1f1e08a98db935d72cb4f542b4e07dfd7b5a | A dodecahedron is drawn in a three-dimensional coordinate system. A nonnegative integer is written on each face, and the sum of the integers is 11. How many faces of the dodecahedron are guaranteed to have the same integer written on them? | [
"5"
] | [
"A dodecahedron is drawn in a three-dimensional coordinate system. A nonnegative integer is written on each face, and the sum of the integers is 11. How many faces of the dodecahedron are guaranteed to have the same integer written on them?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-e2bcc40f43148f86ea3ce70ba3b1cbf47eb3fb1a23a73e5fac8784fe6b4ad29e | A dodo bird can jump in two ways: type I jumps 1 meter to the East and 3 meters to the North, and type II jumps 2 meters to the West and 4 meters to the South. Determine the number of type I and type II jumps the dodo bird must make to reach a point 19 meters to the East and 95 meters to the North of its starting posit... | [
"76"
] | [
"A dodo bird can jump in two ways: type I jumps 1 meter to the East and 3 meters to the North, and type II jumps 2 meters to the West and 4 meters to the South. Determine the number of type I and type II jumps the dodo bird must make to reach a point 19 meters to the East and 95 meters to the North of its starting ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d9c5d656a3261c4ac328563d8e87f54df6ab9719a03359e8c270269769990eb9 | A dog buries a bone at $(0,1)$. He is at the end of a 1-unit long leash, and his master walks down the positive $x$-axis, dragging the dog along. The dog pulls the leash taut. Provide parametric equations for the dog's path in terms of $\theta$, where $\theta$ is the angle made by the positive $x$-axis and the vector c... | [
"\\( x = \\cos \\theta + \\ln \\left( \\tan \\left( \\frac{\\theta}{2} \\right) \\right) \\), \\( y = \\sin \\theta \\)"
] | [
"A dog buries a bone at $(0,1)$. He is at the end of a 1-unit long leash, and his master walks down the positive $x$-axis, dragging the dog along. The dog pulls the leash taut. Provide parametric equations for the dog's path in terms of $\\theta$, where $\\theta$ is the angle made by the positive $x$-axis and the v... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 7,
"tir": 3
} | 17 | 0 | 0 |
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