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-0f5cd312e2baba17e6d2e15acf4978e534e17c2bbf51559a9dd70504e27fba9c | Your cat has $n$ kittens which are on average 1.5 kg and each weighs at least 1 kg. A number $M$ is called catty if there is some collection of the kittens with a total mass of $M$ kg. Let $d$ be the largest possible difference between any two adjacent catty numbers on a number line. Find the value of $d$. | [
"\\dfrac{3}{2}"
] | [
"Your cat has $n$ kittens which are on average 1.5 kg and each weighs at least 1 kg. A number $M$ is called catty if there is some collection of the kittens with a total mass of $M$ kg. Let $d$ be the largest possible difference between any two adjacent catty numbers on a number line. Find the value of $d$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88551f4beaf5de8d5fe16338871342ada46b2a599a91dbf2bba74082c0ccb739 | Your cat has $n$ kittens, each with a mass of at least $1$ kg and an average mass of $1.5$ kg. A number $M$ is called catty if there is some collection of the kittens with a total mass of $M$ kg. Determine the maximum possible difference between any two adjacent catty numbers on a number line. | [
"\\dfrac{3}{2}"
] | [
"Your cat has $n$ kittens, each with a mass of at least $1$ kg and an average mass of $1.5$ kg. A number $M$ is called catty if there is some collection of the kittens with a total mass of $M$ kg. Determine the maximum possible difference between any two adjacent catty numbers on a number line."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f90bcabde5ba865f85d57da3567473553b03e7fabbb3218461ba1a09ce5e56ec | Your friend Mary has 2 siblings, and you don't know whether they are boys or girls or both. There are 5 people in the family. You give a card to your friend to take to her house, and when she knocks on the door, a female person opens the door. What is the probability that it is your friend Mary? | [
"\\(\\frac{1}{3}\\)"
] | [
"Your friend Mary has 2 siblings, and you don't know whether they are boys or girls or both. There are 5 people in the family. You give a card to your friend to take to her house, and when she knocks on the door, a female person opens the door. What is the probability that it is your friend Mary?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 9
} | 27 | 0 | 0 |
omr-ws-f4563b913bf5bb5f8495df6a9f396e31975b3f056e8a30eb992807974aa04948 | Your friend has two children. You meet one of them and find out that he is a boy. What is the probability that the other child is a girl? | [
"\\(\\frac{1}{2}\\)"
] | [
"Your friend has two children. You meet one of them and find out that he is a boy. What is the probability that the other child is a girl?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-7b2480d2d7f852c77d1f6678400a63b26f2f13082dca8b7e9eb9d2439d2b73db | Your wardrobe contains two red socks, two green socks, two blue socks, and two yellow socks. It is currently dark right now, but you decide to pair up the socks randomly. What is the probability that none of the pairs are of the same color? | [
"\\dfrac{4}{7}"
] | [
"Your wardrobe contains two red socks, two green socks, two blue socks, and two yellow socks. It is currently dark right now, but you decide to pair up the socks randomly. What is the probability that none of the pairs are of the same color?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-ff6ca9bf3ffd12b4e41693fa041bc9cb0d197b5e662bedccc704a1b5700d02fb | You’re initially given $10$ factories at positions $1-10$ (each position has one factory). Every day, each factory at position $x$ generates another factory at position $x+7$. After $290$ days, how many factories do you have at position $2022$? | [
"41905"
] | [
"You’re initially given $10$ factories at positions $1-10$ (each position has one factory). Every day, each factory at position $x$ generates another factory at position $x+7$. After $290$ days, how many factories do you have at position $2022$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-a528051873fd04644340173a29bb170065e1f77a1fcc51a9f54c4a724b65fd1b | Yu has 12 coins, consisting of 5 pennies, 4 nickels, and 3 dimes. He tosses them all in the air. What is the probability that the total value of the coins that land heads-up is exactly 30 cents? Express your answer as a common fraction. | [
"\\dfrac{23}{2048}"
] | [
"Yu has 12 coins, consisting of 5 pennies, 4 nickels, and 3 dimes. He tosses them all in the air. What is the probability that the total value of the coins that land heads-up is exactly 30 cents? Express your answer as a common fraction."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-b317cd24936d76eb4b5ec126ca2b23bcb7930578e8643c4c06bef5a333d4a2b0 | Yunseo has a supercomputer, equipped with a function $F$ that takes in a polynomial $P(x)$ with integer coefficients, computes the polynomial $Q(x) = (P(x)-1)(P(x)-2)(P(x)-3)(P(x)-4)(P(x)-5)$, and outputs $Q(x)$. Thus, for example, if $P(x) = x+3$, then $F(P(x)) = (x+2)(x+1)(x)(x-1)(x-2) = x^5 - 5x^3 + 4x$. Yunseo, bei... | [
"5^{2017} - 2017"
] | [
"Yunseo has a supercomputer, equipped with a function $F$ that takes in a polynomial $P(x)$ with integer coefficients, computes the polynomial $Q(x) = (P(x)-1)(P(x)-2)(P(x)-3)(P(x)-4)(P(x)-5)$, and outputs $Q(x)$. Thus, for example, if $P(x) = x+3$, then $F(P(x)) = (x+2)(x+1)(x)(x-1)(x-2) = x^5 - 5x^3 + 4x$. Yunseo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-47957805a2036394ab8914bbc0940030f18799eba380cec37a92ae77eb2969ba | Yura put $2001$ coins of $1$, $2$, or $3$ kopeykas in a row. It turned out that between any two $1$-kopeyka coins there is at least one coin; between any two $2$-kopeyka coins there are at least two coins; and between any two $3$-kopeyka coins there are at least three coins. How many $3$-kopeyka coins could Yura put? | [
"500 or 501"
] | [
"Yura put $2001$ coins of $1$, $2$, or $3$ kopeykas in a row. It turned out that between any two $1$-kopeyka coins there is at least one coin; between any two $2$-kopeyka coins there are at least two coins; and between any two $3$-kopeyka coins there are at least three coins. How many $3$-kopeyka coins could Yura p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-251aa2c4176b3f4407678a39d809837a30b3ad194d9d8e4ebea09c8eb26b5c2b | Yuri and Vlad are playing a game on a $4 \times 100$ table. Yuri first chooses 73 squares of size $2 \times 2$ (which can intersect but cannot be equal). Vlad then colors the cells of the table using 4 colors such that each row, each column, and each $2 \times 2$ square chosen by Yuri contains all 4 colors. Vlad pays 2... | [
"48"
] | [
"Yuri and Vlad are playing a game on a $4 \\times 100$ table. Yuri first chooses 73 squares of size $2 \\times 2$ (which can intersect but cannot be equal). Vlad then colors the cells of the table using 4 colors such that each row, each column, and each $2 \\times 2$ square chosen by Yuri contains all 4 colors. Vla... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-c5d2df70b037779a5d2bf55e52060f671edeeadc82d1be86e908ce496133160a | Zach has chosen five numbers from the set $\{1, 2, 3, 4, 5, 6, 7\}$. If he told Claudia what the product of the chosen numbers was, that would not be enough information for Claudia to figure out whether the sum of the chosen numbers was even or odd. What is the product of the chosen numbers? | [
"420"
] | [
"Zach has chosen five numbers from the set $\\{1, 2, 3, 4, 5, 6, 7\\}$. If he told Claudia what the product of the chosen numbers was, that would not be enough information for Claudia to figure out whether the sum of the chosen numbers was even or odd. What is the product of the chosen numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f66f1d18f0fe9acd2035d243b1aa216ca677049862e0a10486cf44e34f322080 | Zach needs to create a billboard that is 20 feet high. Each letter is 2 feet high, and there is a ½-foot space between the bottom of one line of words and the top of the next line of words. Additionally, there is a 1 ½-foot border at both the top and bottom of the billboard. How many lines of words can Zach fit onto th... | [
"7"
] | [
"Zach needs to create a billboard that is 20 feet high. Each letter is 2 feet high, and there is a ½-foot space between the bottom of one line of words and the top of the next line of words. Additionally, there is a 1 ½-foot border at both the top and bottom of the billboard. How many lines of words can Zach fit on... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9394ebc76f34fe31ce60a33e91f0bf4159a9c5e7692ec8f3e4064910c49b7a81 | Zach threw the frisbee to Clive, who is standing 3 spaces to Zach’s left, and the frisbee would go half as far as if he threw it to Henry, who is standing 9 spaces to Zach’s left. How far would the frisbee go if Clive were to throw it to Henry? | [
"6"
] | [
"Zach threw the frisbee to Clive, who is standing 3 spaces to Zach’s left, and the frisbee would go half as far as if he threw it to Henry, who is standing 9 spaces to Zach’s left. How far would the frisbee go if Clive were to throw it to Henry?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-d01af19aeccf94be9d0d44a6ad6decf0e13111ae4bb65916ffe0535e53a8489d | Zach writes the numbers 1 through 64 in binary on a piece of paper without leading zeroes. How many times will he have written the two-digit sequence "10"? | [
"81"
] | [
"Zach writes the numbers 1 through 64 in binary on a piece of paper without leading zeroes. How many times will he have written the two-digit sequence \"10\"?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8d5e541bc48e140c2f7a651c4b421911dd0028075ee3c86e3a3695be08b8cafa | Zachary has a superball which rebounds to 7/8 of its previous height. On the third bounce, the ball lands on a piece of gum and only rebounds to 1/8 of its previous height. Find the total distance the superball has traveled, in feet, if it starts at a height of 4 feet and is caught at its highest point after the third ... | [
"17.5078125"
] | [
"Zachary has a superball which rebounds to 7/8 of its previous height. On the third bounce, the ball lands on a piece of gum and only rebounds to 1/8 of its previous height. Find the total distance the superball has traveled, in feet, if it starts at a height of 4 feet and is caught at its highest point after the t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-51970cacba188a2bf7a8de1d76b313c82abcb6bc7d2ec63479de56eb894769d6 | Zachary lives on a street with houses numbered in order from 1 to n, with n between 25 and 75. The sum of the house numbers less than his equals the sum of the house numbers greater than his. What is Zachary's house number? | [
"35"
] | [
"Zachary lives on a street with houses numbered in order from 1 to n, with n between 25 and 75. The sum of the house numbers less than his equals the sum of the house numbers greater than his. What is Zachary's house number?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-bfb8b1127a99850732110a6701969229059c64400307a4d5c12df89ccee81f14 | Zachary paid for a $\$1$ burger with 32 coins and received no change. Each coin was either a penny or a nickel. What was the number of nickels Zachary used? | [
"17"
] | [
"Zachary paid for a $\\$1$ burger with 32 coins and received no change. Each coin was either a penny or a nickel. What was the number of nickels Zachary used?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e454e39b6a21518b52417e3b5e4dd87f7438989463818de62f5c32b3558f4a05 | Zachary wants to write down all \( n \)-digit positive integers using the digits 1, 2, and 3, such that no two 1's are next to each other. How many such integers can Zachary write? | [
"\\frac{(3 + 2\\sqrt{3})(1 + \\sqrt{3})^n + (3 - 2\\sqrt{3})(1 - \\sqrt{3})^n}{6}"
] | [
"Zachary wants to write down all \\( n \\)-digit positive integers using the digits 1, 2, and 3, such that no two 1's are next to each other. How many such integers can Zachary write?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-20341d7be4aa2c534d19db98ecf5007ebf0f545041fef7e14c2fc4d00c9b8b0e | Zadam Heng shoots free throws until he has made 5 shots. He wins if this takes him 9 or fewer attempts. The probability that Zadam makes any given attempt is \(\frac{1}{2}\). What is the probability that Zadam Heng wins the bet? | [
"\\(\\frac{1}{2}\\)"
] | [
"Zadam Heng shoots free throws until he has made 5 shots. He wins if this takes him 9 or fewer attempts. The probability that Zadam makes any given attempt is \\(\\frac{1}{2}\\). What is the probability that Zadam Heng wins the bet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-839cce36969cb2a47124c6173183b2b75a0063eecbd73945eab6f40c7fcd25e5 | Zahar has a stack of $n \geq 1$ notebooks, numbered from $1$ to $n$. He repeats the following operation: he randomly selects a notebook whose number $k$ does not match its position in the stack and moves it to the $k$th position, without altering the positions of the other notebooks. If no such notebook exists, he stop... | [
"4"
] | [
"Zahar has a stack of $n \\geq 1$ notebooks, numbered from $1$ to $n$. He repeats the following operation: he randomly selects a notebook whose number $k$ does not match its position in the stack and moves it to the $k$th position, without altering the positions of the other notebooks. If no such notebook exists, h... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-1e2273dcc62a88cc0003594b45316d8e1f7d0c20c343ca301afed53ba4954b97 | Zaki wants to complete the following sequence of numbers such that the difference between every pair of adjacent numbers is $1$, $4$, or $7$: $$1 \quad \_ \quad \_ \quad \_ \quad \_ \quad \_ \quad \_ \quad \_ \quad 1.$$ How many ways are there to fill in the blanks with the numbers $2$ to $8$, each appearing exactly on... | [
"10"
] | [
"Zaki wants to complete the following sequence of numbers such that the difference between every pair of adjacent numbers is $1$, $4$, or $7$:\n$$1 \\quad \\_ \\quad \\_ \\quad \\_ \\quad \\_ \\quad \\_ \\quad \\_ \\quad \\_ \\quad 1.$$\nHow many ways are there to fill in the blanks with the numbers $2$ to $8$, eac... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 6,
"tir": 13
} | 22 | 0 | 0 |
omr-ws-544a47687c56f3e4010a647e568e59c4e2de108632738ba8d926ba855e38c295 | Zan has created this iterative rule for generating sequences of whole numbers: 1) If a number is 25 or less, double the number. 2) If a number is greater than 25, subtract 12 from it. Let $F$ be the first number in a sequence generated by the rule above. $F$ is a "sweet number" if 16 is not a term in the sequence that ... | [
"16"
] | [
"Zan has created this iterative rule for generating sequences of whole numbers:\n\n1) If a number is 25 or less, double the number.\n\n2) If a number is greater than 25, subtract 12 from it.\n\nLet $F$ be the first number in a sequence generated by the rule above. $F$ is a \"sweet number\" if 16 is not a term in th... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-8dd3496d4ce6925ea27ef5810930bef4b711b99c90464efa59a2b31f30893c5a | Zara and Ada are playing a game. Ada begins by picking an integer from $1$ to $2011$ (inclusive). On each turn, Zara tries to guess Ada's number. Ada then tells Zara whether her guess is too high, too low, or correct. If Zara's guess is not correct, Ada adds or subtracts $1$ from her number (always constructing a new n... | [
"1008"
] | [
"Zara and Ada are playing a game. Ada begins by picking an integer from $1$ to $2011$ (inclusive). On each turn, Zara tries to guess Ada's number. Ada then tells Zara whether her guess is too high, too low, or correct. If Zara's guess is not correct, Ada adds or subtracts $1$ from her number (always constructing a ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff4f14a188e0316379b2edfdfc93a1caa7ed80d0425d1bd8f9dc78e656bb419b | Zara has a collection of 4 marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this? | [
"12"
] | [
"Zara has a collection of 4 marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-bf259e82a7749db3de3d26fc6e8f79c826cfbf6750649c9f2e19c150b48904e0 | Zeb, an alien, wants to approximate how long it will take for his money to triple at a $3\%$ interest rate. Find an approximation for this duration. | [
"38.039 years"
] | [
"Zeb, an alien, wants to approximate how long it will take for his money to triple at a $3\\%$ interest rate. Find an approximation for this duration."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-1b05943ce83cad71226bc31acc9298d39c96bbf3f2f8b216e267ed365a4bd0f7 | Zerglings were recently discovered to use the same $24$-hour cycle that we use. However, instead of making $12$-hour analog clocks like humans, Zerglings make $24$-hour analog clocks. On these special analog clocks, how many times during $1$ Zergling day will the hour and minute hands be exactly opposite each other? | [
"23"
] | [
"Zerglings were recently discovered to use the same $24$-hour cycle that we use. However, instead of making $12$-hour analog clocks like humans, Zerglings make $24$-hour analog clocks. On these special analog clocks, how many times during $1$ Zergling day will the hour and minute hands be exactly opposite each othe... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-241398e1137ed0d98b1a68dfb3ca20d497559c111cd08a0d3f85382bcb883b43 | Zeus and Hades take turns writing numbers on an infinite board. Starting with Zeus, who writes the number $1$, each player can either copy the number previously written or divide this number by $2$ and write it. After an infinite number of turns, the numbers written on the board form a sequence $(a_n)_{i\in\mathbb{Z}+}... | [
"4"
] | [
"Zeus and Hades take turns writing numbers on an infinite board. Starting with Zeus, who writes the number $1$, each player can either copy the number previously written or divide this number by $2$ and write it. After an infinite number of turns, the numbers written on the board form a sequence $(a_n)_{i\\in\\math... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-99e5b5408e0c47f2371fbf2de78fe0ad8f96dbade7338bbb284cf9ce59657e4f | Zeus and Poseidon play 1997 rounds of a game in which one of the players scores a point in each round. Zeus wins with a score of 1492 to 505. If all possible sequences of wins and losses with 1492 wins for Zeus are equally likely, what is the probability that the score was tied at some point during the game? | [
"\\dfrac{1010}{1997}"
] | [
"Zeus and Poseidon play 1997 rounds of a game in which one of the players scores a point in each round. Zeus wins with a score of 1492 to 505. If all possible sequences of wins and losses with 1492 wins for Zeus are equally likely, what is the probability that the score was tied at some point during the game?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ccd798c43eb81ef12ceea7eb5f87e489e5e5830594af571cf3031efaac7e208c | Zeus, Athena, and Poseidon arrive at Mount Olympus at a random time between $12:00$ pm and $12:00$ am and each stays for $3$ hours. If any two of them see each other during this period, it will be a good day. The probability of it being a good day is $\frac{m}{n}$ where $m$ and $n$ are coprime positive integers. Find $... | [
"\\(15\\)"
] | [
"Zeus, Athena, and Poseidon arrive at Mount Olympus at a random time between $12:00$ pm and $12:00$ am and each stays for $3$ hours. If any two of them see each other during this period, it will be a good day. The probability of it being a good day is $\\frac{m}{n}$ where $m$ and $n$ are coprime positive integers. ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 26,
"genselect": 3,
"tir": 3
} | 32 | 0 | 0 |
omr-ws-417a788386692131cf034f5866cd7e55165179aa9209610620be17a3716c441a | Zeus, Athena, and Poseidon arrive at Mount Olympus at a random time between $12:00$ pm and $12:00$ am and each stays for $3$ hours. If any two of them see each other during this period, it will be a good day. The probability of it being a good day is $\frac{m}{n}$ where $m$ and $n$ are coprime positive integers. Find $... | [
"7"
] | [
"Zeus, Athena, and Poseidon arrive at Mount Olympus at a random time between $12:00$ pm and $12:00$ am and each stays for $3$ hours. If any two of them see each other during this period, it will be a good day. The probability of it being a good day is $\\frac{m}{n}$ where $m$ and $n$ are coprime positive integers. ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b4d54f6a0e47e450ee5c573e3d7f1267313b17f5f5f5bbb4404d84b550ed7f0 | Zeus, Athena, and Poseidon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am and each stays for 3 hours. If any two of them see each other during this period, it will be a good day. The probability of it being a good day is $\frac{m}{n}$ where $m$ and $n$ are coprime positive integers. Find $m+n$. | [
"15"
] | [
"Zeus, Athena, and Poseidon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am and each stays for 3 hours. If any two of them see each other during this period, it will be a good day. The probability of it being a good day is $\\frac{m}{n}$ where $m$ and $n$ are coprime positive integers. Find $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 5,
"tir": 6
} | 41 | 0 | 0 |
omr-ws-325be4e9bed3c91e98a551b362a62f00498c8785e77301c48d12320f0a42b201 | Zeus, Athena, and Poseidon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am and stay for 3 hours. The probability that any two gods see each other during this period is \( \frac{m}{n} \) where \( m \) and \( n \) are coprime positive integers. Find \( m + n \). | [
"15"
] | [
"Zeus, Athena, and Poseidon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am and stay for 3 hours. The probability that any two gods see each other during this period is \\( \\frac{m}{n} \\) where \\( m \\) and \\( n \\) are coprime positive integers. Find \\( m + n \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 7,
"tir": 5
} | 31 | 0 | 0 |
omr-ws-54d071dbfc5340325b45ef18cc103b32ce6204fe313ebd44a1e89f06edaf0cb5 | Zeus, Athena, and Poseidon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am, and each stays for 3 hours. If any two of the gods see each other during this period, it will be a good day. The probability of it being a good day is $\frac{m}{n}$ where $m$ and $n$ are coprime positive integers. Find $m... | [
"47"
] | [
"Zeus, Athena, and Poseidon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am, and each stays for 3 hours. If any two of the gods see each other during this period, it will be a good day. The probability of it being a good day is $\\frac{m}{n}$ where $m$ and $n$ are coprime positive integers. F... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e5299ee98099a8f61dd3222dee358d661646a29b4eb9b254550d2efdc39a561 | Ziam has 6 different shirts, 5 different pairs of pants, 7 different pairs of shoes, and 4 different harrystyles. An outfit is comprised of one shirt, one pair of pants, one pair of shoes, and one harrystyle. If only 3 of Ziam’s outfits will make the space girls frown, what is the probability that a randomly chosen out... | [
"\\(\\frac{837}{840}\\)"
] | [
"Ziam has 6 different shirts, 5 different pairs of pants, 7 different pairs of shoes, and 4 different harrystyles. An outfit is comprised of one shirt, one pair of pants, one pair of shoes, and one harrystyle. If only 3 of Ziam’s outfits will make the space girls frown, what is the probability that a randomly chose... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-01038435c696d3ff18bff3033a4fc855c5b6ae5507319e6c67c30871ada614c0 | Zion has an average \( 2 \) out of \( 3 \) hit rate for \( 2 \)-pointers and \( 1 \) out of \( 3 \) hit rate for \( 3 \)-pointers. In a recent basketball match, Zion scored \( 18 \) points without missing a shot, and all the points came from \( 2 \)- or \( 3 \)-pointers. What is the probability that all his shots were ... | [
"\\dfrac{27}{803}"
] | [
"Zion has an average \\( 2 \\) out of \\( 3 \\) hit rate for \\( 2 \\)-pointers and \\( 1 \\) out of \\( 3 \\) hit rate for \\( 3 \\)-pointers. In a recent basketball match, Zion scored \\( 18 \\) points without missing a shot, and all the points came from \\( 2 \\)- or \\( 3 \\)-pointers. What is the probability t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-845990b531bd4fd0d80e7c10da2929ed10726c6c938316532cd5ebeeab614f83 | Zoe and Kylie each choose six distinct single digit positive integers and take the product of their numbers. If Zoe's product is $28$ greater than Kylie's product, what is the sum of Zoe's six numbers? | [
"30"
] | [
"Zoe and Kylie each choose six distinct single digit positive integers and take the product of their numbers. If Zoe's product is $28$ greater than Kylie's product, what is the sum of Zoe's six numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-20729136c62b164ae2ea689c595bfa2714798a573adebb94e3d5e9ad2fe7f22e | Zoey read 15 books, one at a time. The first book took her 1 day to read, the second book took her 2 days to read, the third book took her 3 days to read, and so on, with each book taking her 1 more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day ... | [
"Monday"
] | [
"Zoey read 15 books, one at a time. The first book took her 1 day to read, the second book took her 2 days to read, the third book took her 3 days to read, and so on, with each book taking her 1 more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 15
} | 63 | 0 | 0 |
omr-ws-91c9c0c8331cf44635da7ec87c68685f1c2def189b86bfcda58ad7c211de4e2b | Zortech thinks of a two-digit positive integer \( n \). He divulges the sum of its digits to Harazi and the number of its positive divisors to Namdung. Harazi says, "I don't know what the number is." Namdung replies, "Neither do I, but I know whether it's odd or even." Harazi then says, "In that case, I know what the n... | [
"64"
] | [
"Zortech thinks of a two-digit positive integer \\( n \\). He divulges the sum of its digits to Harazi and the number of its positive divisors to Namdung. Harazi says, \"I don't know what the number is.\" Namdung replies, \"Neither do I, but I know whether it's odd or even.\" Harazi then says, \"In that case, I kno... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3d0257e374a55650ff22b47ede0e7fb35dd9f1bdb92e97187cc5aafc5e2469fd | Zou and Chou are practicing their $100$ -meter sprints by running $6$ races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\frac23$ if they won the previous race but only $\frac13$ if they lost the previous race. The probability that Zou will win exactly $5... | [
"097"
] | [
"Zou and Chou are practicing their $100$ -meter sprints by running $6$ races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\\frac23$ if they won the previous race but only $\\frac13$ if they lost the previous race. The probability that Zou will win exa... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 44,
"tir": 13
} | 57 | 0 | 0 |
omr-ws-176c9242f85af635ca8b12b184d8743f91126be239e535d2719ccfafea365bf4 | Zou and Chou are practicing their 100-meter sprints by running 6 races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\frac{2}{3}$ if they won the previous race but only $\frac{1}{3}$ if they lost the previous race. The probability that Zou will win exactly... | [
"97"
] | [
"Zou and Chou are practicing their 100-meter sprints by running 6 races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\\frac{2}{3}$ if they won the previous race but only $\\frac{1}{3}$ if they lost the previous race. The probability that Zou will win ... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 109,
"genselect": 6,
"tir": 27
} | 142 | 0 | 0 |
omr-ws-8ef82d7befc75866bbce5cef0bd8029ed1981851e66e55b9c476af7df7e950d8 | Zou and Chou are practicing their 100-meter sprints by running 6 races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\frac{2}{3}$ if they won the previous race but only $\frac{1}{3}$ if they lost the previous race. The probability that Zou will win exactly... | [
"97"
] | [
"Zou and Chou are practicing their 100-meter sprints by running 6 races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\\frac{2}{3}$ if they won the previous race but only $\\frac{1}{3}$ if they lost the previous race. The probability that Zou will win ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4b67c6c351ed572fe64a4abba6bbe70b731d6af56da2263e7d3ac6038e4dad82 | Zé Roberto and Humberto are playing a game with 30 empty boxes in a queue. Each player takes a blue stone and places it in a box (a move). The winner is the player who, in their move, obtains three full consecutive boxes. If Zé Roberto is the first player, who has the winning strategy? | [
"Zé Roberto has the winning strategy."
] | [
"Zé Roberto and Humberto are playing a game with 30 empty boxes in a queue. Each player takes a blue stone and places it in a box (a move). The winner is the player who, in their move, obtains three full consecutive boxes. If Zé Roberto is the first player, who has the winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 10
} | 32 | 0 | 0 |
omr-ws-a134d58623b283ef87b85534b970a96ae7d919cdfaf334d0f512642995d64af4 | \( 10 \) coins with probabilities of \( 1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{10} \) of coming up heads are flipped. What is the probability that an odd number of them come up heads? | [
"\\dfrac{1}{2}"
] | [
"\\( 10 \\) coins with probabilities of \\( 1, \\frac{1}{2}, \\frac{1}{3}, \\ldots, \\frac{1}{10} \\) of coming up heads are flipped. What is the probability that an odd number of them come up heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ea700f70609a8434e302a6e53930ac831b7ba5b5afffcfb6a73b52e100c23094 | \( 100! \) can be expressed as \( p_1^{e_1} p_2^{e_2} \cdots p_{25}^{e_{25}} \), where \( p_1, p_2, \ldots, p_{25} \) are the prime factors of \( 100! \). If \( N = e_1 + e_2 + \cdots + e_{25} \), find \( N \). | [
"239"
] | [
"\\( 100! \\) can be expressed as \\( p_1^{e_1} p_2^{e_2} \\cdots p_{25}^{e_{25}} \\), where \\( p_1, p_2, \\ldots, p_{25} \\) are the prime factors of \\( 100! \\). If \\( N = e_1 + e_2 + \\cdots + e_{25} \\), find \\( N \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ad27ba1977589f281fac48f84ce49f073fcd6b770289bde6bafbe80b4f6f237 | \( 11 \) different books are on a \( 3 \)-shelf bookcase. In how many different ways can the books be arranged such that at most one shelf is empty? | [
"75 * 11!"
] | [
"\\( 11 \\) different books are on a \\( 3 \\)-shelf bookcase. In how many different ways can the books be arranged such that at most one shelf is empty?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 27
} | 74 | 0 | 0 |
omr-ws-988360695c81d75ada2cefbcbbc23670ae7e1f62e0dba251b54fed5594bdfd88 | \( 2021 \) is the product of two consecutive prime numbers. How many years will pass between the year that this last happened and the next year that this will happen? | [
"728"
] | [
"\\( 2021 \\) is the product of two consecutive prime numbers. How many years will pass between the year that this last happened and the next year that this will happen?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 5,
"tir": 11
} | 19 | 0 | 0 |
omr-ws-40696606b45583ec7edd3ea3d496ea7802862912e82258c216ac32166e927e7f | \( 5^n \) is written on the blackboard. The sum of its digits is calculated. Then the sum of the digits of the result is calculated and so on until we have a single digit. If \( n = 2014 \), what is this digit? | [
"4"
] | [
"\\( 5^n \\) is written on the blackboard. The sum of its digits is calculated. Then the sum of the digits of the result is calculated and so on until we have a single digit. If \\( n = 2014 \\), what is this digit?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-12fc6365ec76f3f933330e1435264d9c23ad8c8edf8dbff18e1c9f6bf08030aa | \( 6 \) cats can eat \( 6 \) fish in \( 1 \) day. \( c \) cats can eat \( 91 \) fish in \( d \) days. Given that \( c \) and \( d \) are whole numbers, and \( c \) is more than \( 1 \) but less than \( 10 \), find \( c + d \). | [
"20"
] | [
"\\( 6 \\) cats can eat \\( 6 \\) fish in \\( 1 \\) day. \\( c \\) cats can eat \\( 91 \\) fish in \\( d \\) days. Given that \\( c \\) and \\( d \\) are whole numbers, and \\( c \\) is more than \\( 1 \\) but less than \\( 10 \\), find \\( c + d \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0c1d1f389316eb1260962c643cfe6cc98d23c40acd5ea8c5cfb1205de8d6eaec | \( A \) and \( B \) play the following game with a polynomial of degree at least 4: \[ x^{2n} + \hspace{5mm} x^{2n-1} + \hspace{5mm} x^{2n-2} + \cdots + \hspace{5mm} x + 1 = 0 \] \( A \) and \( B \) take turns to fill in one of the blanks with a real number until all the blanks are filled up. If the resulting polynomia... | [
"A"
] | [
"\\( A \\) and \\( B \\) play the following game with a polynomial of degree at least 4:\n\\[ x^{2n} + \\hspace{5mm} x^{2n-1} + \\hspace{5mm} x^{2n-2} + \\cdots + \\hspace{5mm} x + 1 = 0 \\]\n\\( A \\) and \\( B \\) take turns to fill in one of the blanks with a real number until all the blanks are filled up. If th... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-66caaa1e8b5135b5c0be6d7f88a894a023c19e7ff9db0d83820542e2bce18f7c | \( A, B, C \) are distinct numbers in arithmetic progression (AP) such that \( \sin^2 A, \sin^2 B, \sin^2 C \) are also in AP. Find the smallest positive value of \( B \) (in degrees). | [
"45"
] | [
"\\( A, B, C \\) are distinct numbers in arithmetic progression (AP) such that \\( \\sin^2 A, \\sin^2 B, \\sin^2 C \\) are also in AP. Find the smallest positive value of \\( B \\) (in degrees)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-357cd3919a2eb4605c49eaaa4c5810a406b744d072ac40bebec70e285d4e03d2 | \( A, B, C, \) and \( D \) are different digits \( 0-9 \). If \( \log_A BC5 = D \), how many different possible values are there for \( A, B, C, \) and \( D \)? What are they? | [
"2"
] | [
"\\( A, B, C, \\) and \\( D \\) are different digits \\( 0-9 \\). If \\( \\log_A BC5 = D \\), how many different possible values are there for \\( A, B, C, \\) and \\( D \\)? What are they?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-01184b672bcf215407e08490aa76bed0c4847bda4e9064c4fea897b987338e82 | \( AB \) is a diameter of the circle \( O \), the point \( C \) lies on the line \( AB \) produced. A line passing through \( C \) intersects the circle \( O \) at points \( D \) and \( E \). \( OF \) is a diameter of the circumcircle \( O_1 \) of \( \triangle BOD \). Join \( CF \) and produce it, cutting the circle \(... | [
"no_valid_answer_found"
] | [
"\\( AB \\) is a diameter of the circle \\( O \\), the point \\( C \\) lies on the line \\( AB \\) produced. A line passing through \\( C \\) intersects the circle \\( O \\) at points \\( D \\) and \\( E \\). \\( OF \\) is a diameter of the circumcircle \\( O_1 \\) of \\( \\triangle BOD \\). Join \\( CF \\) and pro... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-226277a67491a00504bdfbace82e8399d7f313bc21b1dbb5ba05b250727fe25c | \( AB \) is the diameter of a circle with radius 2 and a side of an equilateral triangle \( ABC \). The circle intersects \( BC \) at \( D \). Compute the length of \( AD \). | [
"\\( 2\\sqrt{3} \\)"
] | [
"\\( AB \\) is the diameter of a circle with radius 2 and a side of an equilateral triangle \\( ABC \\). The circle intersects \\( BC \\) at \\( D \\). Compute the length of \\( AD \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-0006caf0b83613f9fd0c8d1011b2c2a17a24378d76b7cdede4724c8f39c252bc | \( ABC \) is a scalene triangle. Points \( D, E \), and \( F \) are selected on sides \( BC, CA \), and \( AB \) respectively. The cevians \( AD, BE \), and \( CF \) concur at point \( P \). If the areas of triangles \( AFP, FBP \), and \( CEP \) are 126, 63, and 24 respectively, then determine the area of triangle \( ... | [
"639"
] | [
"\\( ABC \\) is a scalene triangle. Points \\( D, E \\), and \\( F \\) are selected on sides \\( BC, CA \\), and \\( AB \\) respectively. The cevians \\( AD, BE \\), and \\( CF \\) concur at point \\( P \\). If the areas of triangles \\( AFP, FBP \\), and \\( CEP \\) are 126, 63, and 24 respectively, then determine... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ebb6a37e91f7bee832bb648f8bc2b6f2b6a025b13846d7c2ba6a79ba92d866c | \( ABC \) is a triangle with \( \angle BAC = 10^\circ \), \( \angle ABC = 150^\circ \). Let \( X \) be a point on \( [AC] \) such that \( |AX| = |BC| \). Find \( \angle BXC \). | [
"20^\\circ"
] | [
"\\( ABC \\) is a triangle with \\( \\angle BAC = 10^\\circ \\), \\( \\angle ABC = 150^\\circ \\). Let \\( X \\) be a point on \\( [AC] \\) such that \\( |AX| = |BC| \\). Find \\( \\angle BXC \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-74cd8d543bbdab019d7c1be030ff48c2693995573a2dcda5b4cb1dc34bc35f7f | \( ABC \) is an isosceles triangle with \( AB = BC = 12 \). Points \( P \) and \( Q \) are on \( AB \) and \( AC \) such that \( AP = AQ = 8 \). \( PC \) and \( QB \) intersect at \( X \). The area of \( APXQ \) is 8. What is the area of \( ABC \)? | [
"36"
] | [
"\\( ABC \\) is an isosceles triangle with \\( AB = BC = 12 \\). Points \\( P \\) and \\( Q \\) are on \\( AB \\) and \\( AC \\) such that \\( AP = AQ = 8 \\). \\( PC \\) and \\( QB \\) intersect at \\( X \\). The area of \\( APXQ \\) is 8. What is the area of \\( ABC \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-a53384148d0d08a2186294d8ba892c54858f1da3ee66106363ada1cd75ee3388 | \( ABCD \) is a convex quadrilateral in which \( \overline{AB} \parallel \overline{CD} \). Let \( U \) denote the intersection of the extensions of \( \overline{AD} \) and \( \overline{BC} \). \( \Omega_1 \) is the circle tangent to line segment \( \overline{BC} \) which also passes through \( A \) and \( D \), and \( ... | [
"20"
] | [
"\\( ABCD \\) is a convex quadrilateral in which \\( \\overline{AB} \\parallel \\overline{CD} \\). Let \\( U \\) denote the intersection of the extensions of \\( \\overline{AD} \\) and \\( \\overline{BC} \\). \\( \\Omega_1 \\) is the circle tangent to line segment \\( \\overline{BC} \\) which also passes through \\... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-733ff962c51352ba8c9347e421d05ac33ee6dfa2bff2316cb2fa3caace78e78a | \( ABCD \) is a convex quadrilateral with \( AB = 2 \sqrt{3} \) and \( CD = 3 \). The diagonals intersect at \( M \) and \( \angle CMD = 30^\circ \). Find the radius of the circumcircle of \( ABCD \). | [
"\\sqrt{39}"
] | [
"\\( ABCD \\) is a convex quadrilateral with \\( AB = 2 \\sqrt{3} \\) and \\( CD = 3 \\). The diagonals intersect at \\( M \\) and \\( \\angle CMD = 30^\\circ \\). Find the radius of the circumcircle of \\( ABCD \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-3e7818b8b3677dd324e06a5576a5903a9bf0436ebef321e6aa71de1f4584ce22 | \( ABCD \) is a convex quadrilateral. Suppose that rays \( \overrightarrow{AB} \) and \( \overrightarrow{DC} \) intersect at \( X \), and rays \( \overrightarrow{AD} \) and \( \overrightarrow{BC} \) intersect at \( Y \). If it is known that: \[ AB = BC = CA = BX = DY = 1 \] then what is \( AD \)? | [
"\\sqrt[3]{2}"
] | [
"\\( ABCD \\) is a convex quadrilateral. Suppose that rays \\( \\overrightarrow{AB} \\) and \\( \\overrightarrow{DC} \\) intersect at \\( X \\), and rays \\( \\overrightarrow{AD} \\) and \\( \\overrightarrow{BC} \\) intersect at \\( Y \\). If it is known that:\n\\[ AB = BC = CA = BX = DY = 1 \\]\nthen what is \\( A... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-4beae7b3ab532a66b485870ac52f20ce08d381fd5606b32bec68429c005afdb8 | \( ABCD \) is a cyclic quadrilateral with \( AB = 8, BC = 4, CD = 1 \), and \( DA = 7 \). Let \( O \) and \( P \) denote the circumcenter and intersection of \( AC \) and \( BD \) respectively. The value of \( OP^2 \) can be expressed as \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime, positive intege... | [
"589"
] | [
"\\( ABCD \\) is a cyclic quadrilateral with \\( AB = 8, BC = 4, CD = 1 \\), and \\( DA = 7 \\). Let \\( O \\) and \\( P \\) denote the circumcenter and intersection of \\( AC \\) and \\( BD \\) respectively. The value of \\( OP^2 \\) can be expressed as \\( \\frac{m}{n} \\), where \\( m \\) and \\( n \\) are relat... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-928a2464d236b28d1f1f5c9f7f553eaea4d2dd14fc4d7b6a3e08109c9efeff8b | \( ABCD \) is a rectangular sheet of paper. \( E \) and \( F \) are points on \( AB \) and \( CD \) respectively such that \( BE < CF \). If \( BCEF \) is folded over \( EF \), \( C \) maps to \( C' \) on \( AD \) and \( B \) maps to \( B' \) such that \( \angle AB'C' \cong \angle B'EA \). If \( AB' = 5 \) and \( BE = ... | [
"580"
] | [
"\\( ABCD \\) is a rectangular sheet of paper. \\( E \\) and \\( F \\) are points on \\( AB \\) and \\( CD \\) respectively such that \\( BE < CF \\). If \\( BCEF \\) is folded over \\( EF \\), \\( C \\) maps to \\( C' \\) on \\( AD \\) and \\( B \\) maps to \\( B' \\) such that \\( \\angle AB'C' \\cong \\angle B'E... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-f5cede2bdf95c79fcf94626c81c75f8472bce4d3ff81195827bfe1d2e516e1e0 | \( ABCDE \) is a convex pentagon with \( BC = CD \), \( DE = EA \), \( \angle BCD = 90^\circ \), and \( \angle DEA = 90^\circ \). Given that \( \angle ACE = \alpha \) and \( \angle BEC = \beta \), calculate the sum of the angles of the triangle formed by segments \( AC \), \( CE \), and \( EB \). | [
"180^\\circ"
] | [
"\\( ABCDE \\) is a convex pentagon with \\( BC = CD \\), \\( DE = EA \\), \\( \\angle BCD = 90^\\circ \\), and \\( \\angle DEA = 90^\\circ \\). Given that \\( \\angle ACE = \\alpha \\) and \\( \\angle BEC = \\beta \\), calculate the sum of the angles of the triangle formed by segments \\( AC \\), \\( CE \\), and \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-858e7c0c1c5bf7a836967ddb67857df48e7aab33aab901d7458619d399b7eba0 | \( ABCDE \) is a cyclic pentagon with \( BC = CD = DE \). The diagonals \( AC \) and \( BE \) intersect at \( M \). \( N \) is the foot of the altitude from \( M \) to \( AB \). We have \( MA = 25, MD = 113 \), and \( MN = 15 \). \( [ABE] \) can be expressed as \( \frac{m}{n} \) where \( m \) and \( n \) are relatively... | [
"no_valid_answer_found"
] | [
"\\( ABCDE \\) is a cyclic pentagon with \\( BC = CD = DE \\). The diagonals \\( AC \\) and \\( BE \\) intersect at \\( M \\). \\( N \\) is the foot of the altitude from \\( M \\) to \\( AB \\). We have \\( MA = 25, MD = 113 \\), and \\( MN = 15 \\). \\( [ABE] \\) can be expressed as \\( \\frac{m}{n} \\) where \\( ... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e3f21d5a4c104e6a56a43a2eb392c0fa82de45351b9b9e396651e6cdc2a8218 | \( ABCDE \) is inscribed in a circle with \( AB = BC = CD = DE = 4 \) and \( AE = 1 \). Compute \( (1 - \cos \angle B)(1 - \cos \angle ACE) \). | [
"\\(\\frac{1}{64}\\)"
] | [
"\\( ABCDE \\) is inscribed in a circle with \\( AB = BC = CD = DE = 4 \\) and \\( AE = 1 \\). Compute \\( (1 - \\cos \\angle B)(1 - \\cos \\angle ACE) \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 26,
"genselect": 3,
"tir": 14
} | 43 | 0 | 0 |
omr-ws-e716f55270f3a549a0fef924c1bc60fe4506a838588fd531d200da4c7420509b | \( ABCDEF \) is a convex hexagon with \( AB = BC = CD = DE = EF = FA = 5 \) and \( AC = CE = EA = 6 \). Find the area of \( ABCDEF \). | [
"63"
] | [
"\\( ABCDEF \\) is a convex hexagon with \\( AB = BC = CD = DE = EF = FA = 5 \\) and \\( AC = CE = EA = 6 \\). Find the area of \\( ABCDEF \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cc03e2d4907f70f7ef5653825cf160e69e8a0fae1bb4bae9e65cf68560e7bfe9 | \( ABCDEF \) is a regular hexagon with side length 1, with point \( G \) randomly located on the hexagon's boundary or in its interior. What is the maximum value of \( AG^2 + BG^2 + CG^2 + DG^2 + EG^2 + FG^2 \)? | [
"12"
] | [
"\\( ABCDEF \\) is a regular hexagon with side length 1, with point \\( G \\) randomly located on the hexagon's boundary or in its interior. What is the maximum value of \\( AG^2 + BG^2 + CG^2 + DG^2 + EG^2 + FG^2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0efdd7ce1f6472493d5ff948b0d4f5a5afeb7ae935bf1534a48e3ad829d58c7d | \( ABCDEFG \) is a regular heptagon inscribed in a unit circle centered at \( O \). \( l \) is the line tangent to the circumcircle of \( ABCDEFG \) at \( A \), and \( P \) is a point on \( l \) such that triangle \( AOP \) is isosceles. Let \( p \) denote the value of \( AP \cdot BP \cdot CP \cdot DP \cdot EP \cdot FP... | [
"113"
] | [
"\\( ABCDEFG \\) is a regular heptagon inscribed in a unit circle centered at \\( O \\). \\( l \\) is the line tangent to the circumcircle of \\( ABCDEFG \\) at \\( A \\), and \\( P \\) is a point on \\( l \\) such that triangle \\( AOP \\) is isosceles. Let \\( p \\) denote the value of \\( AP \\cdot BP \\cdot CP ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-12c907b9346b4089e171d8c7e710c3101ef32e5085ff4e0807f5053010470533 | \( M \) is a \( 3 \times 2 \) matrix, \( N \) is a \( 2 \times 3 \) matrix. What is \( MN \)? | [
"3 \\times 3"
] | [
"\\( M \\) is a \\( 3 \\times 2 \\) matrix, \\( N \\) is a \\( 2 \\times 3 \\) matrix. What is \\( MN \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ad8d220dd13d2b8dc87cdc1d0b3cac42f9f7bf45f1333216fa414cba0d4f123 | \( MATH \) and \( TEAM \) are two parallelograms. If the lengths of \( MH \) and \( AE \) are \( 13 \) and \( 15 \), and the distance from \( AM \) to \( T \) is \( 12 \), find the perimeter of \( AMHE \). | [
"42"
] | [
"\\( MATH \\) and \\( TEAM \\) are two parallelograms. If the lengths of \\( MH \\) and \\( AE \\) are \\( 13 \\) and \\( 15 \\), and the distance from \\( AM \\) to \\( T \\) is \\( 12 \\), find the perimeter of \\( AMHE \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-e92eab324dd9fee0a3620c2b460d20b030315a6b10920d87165a7c4d8583d91d | \( N \) players enter a ping pong tournament, where \( N = (S - 7)/102 \). After each round, half of the remaining players (rounded down if odd) are eliminated. After 3 rounds of play, in how many ways can we select the remaining players? | [
"0"
] | [
"\\( N \\) players enter a ping pong tournament, where \\( N = (S - 7)/102 \\). After each round, half of the remaining players (rounded down if odd) are eliminated. After 3 rounds of play, in how many ways can we select the remaining players?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-23ee82340d0e483f69fd0b1c84e5ea9cc69fb67dbe3a430860c2b4ca8380f486 | \( O \) is the circumcenter of an acute-angled triangle \( \Delta ABC \). \( T \) is the circumcenter of \( \Delta AOC \). \( M \) is the midpoint of \( AC \). On the sides \( AB \) and \( BC \), consider points \( D \) and \( E \) such that \( \angle BDM = \angle BEM = \angle ABC \). If \( BT \) intersects \( DE \) at... | [
"2"
] | [
"\\( O \\) is the circumcenter of an acute-angled triangle \\( \\Delta ABC \\). \\( T \\) is the circumcenter of \\( \\Delta AOC \\). \\( M \\) is the midpoint of \\( AC \\). On the sides \\( AB \\) and \\( BC \\), consider points \\( D \\) and \\( E \\) such that \\( \\angle BDM = \\angle BEM = \\angle ABC \\). If... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88db02ddfc4cc5fb97009969c3f2ff6b70df84082fcebd3b29d4723600da05f3 | \( P \) and \( Q \) are two points on the parabola \( y^2 = 4px \), the coordinates of \( P \) are \( (a, b) \), and the slope of \( PQ \) is \( m \). Find the \( y \)-coordinate of the midpoint of \( PQ \). The final answer should be independent of both \( a \) and \( b \). | [
"\\(\\frac{2p}{m}\\)"
] | [
"\\( P \\) and \\( Q \\) are two points on the parabola \\( y^2 = 4px \\), the coordinates of \\( P \\) are \\( (a, b) \\), and the slope of \\( PQ \\) is \\( m \\). Find the \\( y \\)-coordinate of the midpoint of \\( PQ \\). The final answer should be independent of both \\( a \\) and \\( b \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-2fccf9369b100feb13e766467ac06d1c3f2624e608d928e5ab6056370e26b8ff | \( P \) is a point on the plane of the square \( ABCD \) such that each of \( PAB \), \( PBC \), \( PCD \), and \( PDA \) is an isosceles (nondegenerate) triangle. How many possible positions are there for such a point \( P \)? | [
"9"
] | [
"\\( P \\) is a point on the plane of the square \\( ABCD \\) such that each of \\( PAB \\), \\( PBC \\), \\( PCD \\), and \\( PDA \\) is an isosceles (nondegenerate) triangle. How many possible positions are there for such a point \\( P \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-fc4ede829368212aeb8be46c9ee0063e6f2de848d50c7194935d1137cb2450c4 | \( P \) is the probability that if you flip a fair coin, 20 heads will occur before 19 tails. If \( P = \frac{m}{n} \) where \( m \) and \( n \) are relatively prime positive integers, find the remainder when \( m+n \) is divided by 1000. | [
"149"
] | [
"\\( P \\) is the probability that if you flip a fair coin, 20 heads will occur before 19 tails. If \\( P = \\frac{m}{n} \\) where \\( m \\) and \\( n \\) are relatively prime positive integers, find the remainder when \\( m+n \\) is divided by 1000."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e752647de7642f4d214359da12204b9d15c03dae36b01fea3e0705260c48c260 | \( P(x) \) is the polynomial of minimal degree that satisfies \( P(k) = \frac{1}{k(k+1)} \) for \( k = 1, 2, 3, \dots, 10 \). The value of \( P(11) \) can be written as \( -\frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Determine \( m+n \). | [
"71"
] | [
"\\( P(x) \\) is the polynomial of minimal degree that satisfies \\( P(k) = \\frac{1}{k(k+1)} \\) for \\( k = 1, 2, 3, \\dots, 10 \\). The value of \\( P(11) \\) can be written as \\( -\\frac{m}{n} \\), where \\( m \\) and \\( n \\) are relatively prime positive integers. Determine \\( m+n \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f18443910a2b8c8f087efe053366bf6731f3119cba98494019196ea64c6d1ede | \( R \) is a \( 6 \times 8 \) rectangle. \( R' \) is another rectangle with one vertex on each side of \( R \). \( R' \) can be rotated slightly and still remain within \( R \). Find the smallest perimeter that \( R' \) can have. | [
"20"
] | [
"\\( R \\) is a \\( 6 \\times 8 \\) rectangle. \\( R' \\) is another rectangle with one vertex on each side of \\( R \\). \\( R' \\) can be rotated slightly and still remain within \\( R \\). Find the smallest perimeter that \\( R' \\) can have."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-168120f0814f5a09000bb8ce57f58e63e9383dcd4ba69d0570656a3d9b509040 | \( R \) is a solution to \( x + \frac{1}{x} = \frac{\sin 210^\circ}{\sin 285^\circ} \). Suppose \( \frac{1}{R^{2006}} + R^{2006} = A \). Find \( \lfloor A^{10} \rfloor \). | [
"243"
] | [
"\\( R \\) is a solution to \\( x + \\frac{1}{x} = \\frac{\\sin 210^\\circ}{\\sin 285^\\circ} \\). Suppose \\( \\frac{1}{R^{2006}} + R^{2006} = A \\). Find \\( \\lfloor A^{10} \\rfloor \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ae2bb7ba179a9f50a9ed8ac3584328678ec4a69b184c7a0ba6e2f4a037674e33 | \( S \) is a collection of integers \( n \) with \( 1 \le n \le 50 \) so that each integer in \( S \) is composite and relatively prime to every other integer in \( S \). What is the largest possible number of integers in \( S \)? | [
"4"
] | [
"\\( S \\) is a collection of integers \\( n \\) with \\( 1 \\le n \\le 50 \\) so that each integer in \\( S \\) is composite and relatively prime to every other integer in \\( S \\). What is the largest possible number of integers in \\( S \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a7198735c993397fe84c9a4d20ec504ff9aba86af34a550b73a79a27f8b51843 | \( \alpha, \beta \), and \( \gamma \) are the roots of \( x(x-200)(4x+1) = 1 \). Let \( \omega = \tan^{-1}(\alpha) + \tan^{-1}(\beta) + \tan^{-1}(\gamma) \). The value of \( \tan(\omega) \) can be written as \( \frac{m}{n} \) where \( m \) and \( n \) are relatively prime positive integers. Determine the value of \( m ... | [
"167"
] | [
"\\( \\alpha, \\beta \\), and \\( \\gamma \\) are the roots of \\( x(x-200)(4x+1) = 1 \\). Let \\( \\omega = \\tan^{-1}(\\alpha) + \\tan^{-1}(\\beta) + \\tan^{-1}(\\gamma) \\). The value of \\( \\tan(\\omega) \\) can be written as \\( \\frac{m}{n} \\) where \\( m \\) and \\( n \\) are relatively prime positive inte... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9a9dd29c287cabe5c698649e72ce78b09fd19618d29920df4ada11d121db0bae | \( \angle BAC = 80^\circ \), \( |AB| = |AC| \), \( K \in [AB] \), \( L \in [AB] \), \( |AB|^2 = |AK| \cdot |AL| \), \( |BL| = |BC| \), \( \angle KCB = ? \) | [
"25^\\circ"
] | [
"\\( \\angle BAC = 80^\\circ \\), \\( |AB| = |AC| \\), \\( K \\in [AB] \\), \\( L \\in [AB] \\), \\( |AB|^2 = |AK| \\cdot |AL| \\), \\( |BL| = |BC| \\), \\( \\angle KCB = ? \\)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-4714d3d081790c010ba89017cd1e2d8ed8116080310d901742d08800bb2bb388 | \( \arcsin(a) = \arctan(x) \) for certain values of \( a \). Find \( a \) in terms of \( x \). | [
"\\dfrac{x}{\\sqrt{x^2 + 1}}"
] | [
"\\( \\arcsin(a) = \\arctan(x) \\) for certain values of \\( a \\). Find \\( a \\) in terms of \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-edcf0a3b7955fd24c39493458198bcfc721da749b9b56c9ba662fd9b7e43f449 | \( \sqrt{18 + \sqrt{308}} = \sqrt{a} + \sqrt{b} \) for some integers \( a \) and \( b \). Find \( a + b \). | [
"18"
] | [
"\\( \\sqrt{18 + \\sqrt{308}} = \\sqrt{a} + \\sqrt{b} \\) for some integers \\( a \\) and \\( b \\). Find \\( a + b \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-398692abfba84afbb2e1239b2c3c2c0e06eceae4491ed4919001ace7506e07d3 | \( \tan(x) = \frac{1 + \sin(40^\circ) - \cos(40^\circ)}{1 + \sin(40^\circ) + \cos(40^\circ)} \). Find all values of \( x \) if \( 0 \leq x < 360^\circ \) and all angles are measured in degrees. | [
"200^\\circ"
] | [
"\\( \\tan(x) = \\frac{1 + \\sin(40^\\circ) - \\cos(40^\\circ)}{1 + \\sin(40^\\circ) + \\cos(40^\\circ)} \\). Find all values of \\( x \\) if \\( 0 \\leq x < 360^\\circ \\) and all angles are measured in degrees."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-1deab52a81ab775649af588bc9e538fc26644154bf85d0212cae0c86adc4795b | \( \triangle ABC \) is isosceles with \( \angle C = 90^\circ \). A point \( P \) lies inside the triangle such that \( AP = 33 \), \( CP = 28\sqrt{2} \), and \( BP = 65 \). Let \( A \) be the area of \( \triangle ABC \). Find the remainder when \( 2A \) is divided by 1000. | [
"505"
] | [
"\\( \\triangle ABC \\) is isosceles with \\( \\angle C = 90^\\circ \\). A point \\( P \\) lies inside the triangle such that \\( AP = 33 \\), \\( CP = 28\\sqrt{2} \\), and \\( BP = 65 \\). Let \\( A \\) be the area of \\( \\triangle ABC \\). Find the remainder when \\( 2A \\) is divided by 1000."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-6e2b48c4b1e917eec81b660369aa38c4255825ef2fc648a05c72a5c015eae0f5 | \( \triangle DEF \) is inscribed inside \( \triangle ABC \) such that \( D, E, F \) lie on \( BC, AC, AB \), respectively. The circumcircles of \( \triangle DEC, \triangle BFD, \triangle AFE \) have centers \( O_1, O_2, O_3 \), respectively. Also, \( AB = 23, BC = 25, AC = 24 \), and \( \stackrel{\frown}{BF} = \stackre... | [
"14"
] | [
"\\( \\triangle DEF \\) is inscribed inside \\( \\triangle ABC \\) such that \\( D, E, F \\) lie on \\( BC, AC, AB \\), respectively. The circumcircles of \\( \\triangle DEC, \\triangle BFD, \\triangle AFE \\) have centers \\( O_1, O_2, O_3 \\), respectively. Also, \\( AB = 23, BC = 25, AC = 24 \\), and \\( \\stack... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-9a08f4b765b282c3e9f30b7b8a297592e0e6d039379a6e612b7d57c2208a1fcd | \( \triangle DEF \) is inscribed inside \( \triangle ABC \) such that \( D, E, F \) lie on \( BC, AC, AB \), respectively. The circumcircles of \( \triangle DEC, \triangle BFD, \triangle AFE \) have centers \( O_1, O_2, O_3 \), respectively. Also, \( AB = 23, BC = 25, AC = 24 \), and \( \stackrel{\frown}{BF} = \stackre... | [
"14"
] | [
"\\( \\triangle DEF \\) is inscribed inside \\( \\triangle ABC \\) such that \\( D, E, F \\) lie on \\( BC, AC, AB \\), respectively. The circumcircles of \\( \\triangle DEC, \\triangle BFD, \\triangle AFE \\) have centers \\( O_1, O_2, O_3 \\), respectively. Also, \\( AB = 23, BC = 25, AC = 24 \\), and \\( \\stack... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-baae54ad009622b854ecff06b19644bb4d5205e9df2658493b333f9243944adb | \( \triangle VA_0A_1 \) is isosceles with base \( \overline{A_1A_0} \). Construct \( A_2 \) on segment \( \overline{A_0V} \) such that \( \overline{A_0A_1} = \overline{A_1A_2} = b \). Construct \( A_3 \) on \( \overline{A_1V} \) such that \( b = \overline{A_2A_3} \). Continue this pattern: construct \( \overline{A_{2n}... | [
"270"
] | [
"\\( \\triangle VA_0A_1 \\) is isosceles with base \\( \\overline{A_1A_0} \\). Construct \\( A_2 \\) on segment \\( \\overline{A_0V} \\) such that \\( \\overline{A_0A_1} = \\overline{A_1A_2} = b \\). Construct \\( A_3 \\) on \\( \\overline{A_1V} \\) such that \\( b = \\overline{A_2A_3} \\). Continue this pattern: c... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f3351d622cc791169513115371f017009a51c1d340e54a4c07f16edf5f970409 | \( a \) and \( b \) are two positive integers both less than 2010, and \( a \neq b \). Find the number of ordered pairs \( (a, b) \) such that \( a^2 + b^2 \) is divisible by 5. Find \( a + b \) such that \( a^2 + b^2 \) is maximum. | [
"4017"
] | [
"\\( a \\) and \\( b \\) are two positive integers both less than 2010, and \\( a \\neq b \\). Find the number of ordered pairs \\( (a, b) \\) such that \\( a^2 + b^2 \\) is divisible by 5. Find \\( a + b \\) such that \\( a^2 + b^2 \\) is maximum."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-b9136d82d0bc4d5204a7c6497ef4e882af45b422c8f7ed774ef51a3866929747 | \( a, b, c \) are positive numbers such that \( a^2 + b^2 + c^2 = 2abc + 1 \). Find the maximum value of \( (a - 2bc)(b - 2ca)(c - 2ab) \). | [
"\\dfrac{1}{8}"
] | [
"\\( a, b, c \\) are positive numbers such that \\( a^2 + b^2 + c^2 = 2abc + 1 \\). Find the maximum value of \\( (a - 2bc)(b - 2ca)(c - 2ab) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3117365bc0348a1ded5ff3a28c3f68862959183eacbb55f6ef4a1bf9b22b3355 | \( f \) is a function from \(\mathbb{Z}^{+}\) to \(\mathbb{Z}^{+}\), where \(\mathbb{Z}^{+}\) is the set of non-negative integers, which has the following properties: a) \( f(n+1) > f(n) \) for each \( n \in \mathbb{Z}^{+} \), b) \( f(n + f(m)) = f(n) + m + 1 \) for all \( m, n \in \mathbb{Z}^{+} \). Find all possible ... | [
"\\( 2002 \\)"
] | [
"\\( f \\) is a function from \\(\\mathbb{Z}^{+}\\) to \\(\\mathbb{Z}^{+}\\), where \\(\\mathbb{Z}^{+}\\) is the set of non-negative integers, which has the following properties:\na) \\( f(n+1) > f(n) \\) for each \\( n \\in \\mathbb{Z}^{+} \\),\nb) \\( f(n + f(m)) = f(n) + m + 1 \\) for all \\( m, n \\in \\mathbb{... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-a6334a9caf9a358e935f5d18021401de918f86d5e2937da31b1d1ac1b6422b7b | \( n \) Pokémon are arranged in a circle, each holding 1 badge. The Pokémon pass badges in the following manner: the first Pokémon passes 1 badge to the second, the second passes 2 badges to the third, the third passes 1 badge to the fourth, and so on. The game continues until the last Pokémon passes his badge to the f... | [
"2^k"
] | [
"\\( n \\) Pokémon are arranged in a circle, each holding 1 badge. The Pokémon pass badges in the following manner: the first Pokémon passes 1 badge to the second, the second passes 2 badges to the third, the third passes 1 badge to the fourth, and so on. The game continues until the last Pokémon passes his badge t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-3c3648aa379d2c215a88ad32dcccb784593b66df09061226d7fe668af5fc91cd | \( n \) boys are sitting in a circle, each with an even number of sweets. Each boy gives half of his sweets to the boy on his left. The teacher then gives any boy who ends up with an odd number of sweets an extra one. After 10 rounds of this process, calculate the sum of the sweets each boy has if initially, the boys h... | [
"2n^2"
] | [
"\\( n \\) boys are sitting in a circle, each with an even number of sweets. Each boy gives half of his sweets to the boy on his left. The teacher then gives any boy who ends up with an odd number of sweets an extra one. After 10 rounds of this process, calculate the sum of the sweets each boy has if initially, the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ffef2db7358bd71149336491399d078e12a5dcdfcceb35469d19a101d2c7b5af | \( n \) boys are sitting in a circle, each with an even number of sweets. Each boy then gives half of his sweets to the boy on his left. The teacher then gives any boy who ends up with an odd number of sweets an extra one. After \( k \) rounds of this process, every boy has the same number of sweets. Find the minimum v... | [
"8"
] | [
"\\( n \\) boys are sitting in a circle, each with an even number of sweets. Each boy then gives half of his sweets to the boy on his left. The teacher then gives any boy who ends up with an odd number of sweets an extra one. After \\( k \\) rounds of this process, every boy has the same number of sweets. Find the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6c233abaf0d4e7d9b93bd59970555ec291476c9f3f4c0d202816d04343c8e682 | \( n \) bugs are situated at each vertex of a unit \( n \)-gon. Each bug begins to chase its counterclockwise neighbor at a speed of 1 unit per minute. How long will it take for the bugs to crash into each other? | [
"\\(\\frac{\\csc^{2}(\\frac{180}{n})}{2}\\)"
] | [
"\\( n \\) bugs are situated at each vertex of a unit \\( n \\)-gon. Each bug begins to chase its counterclockwise neighbor at a speed of 1 unit per minute. How long will it take for the bugs to crash into each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 3,
"tir": 7
} | 20 | 0 | 0 |
omr-ws-678c1375038c267d067485b365a1b4bbb25268b2621ab634960d0545b110fdfa | \( n \) points are marked on the plane and all lines passing through at least two marked points. It turned out that after each marked point, exactly one third of the lines drawn pass through the point. For what \( n \) is this possible? | [
"6"
] | [
"\\( n \\) points are marked on the plane and all lines passing through at least two marked points. It turned out that after each marked point, exactly one third of the lines drawn pass through the point. For what \\( n \\) is this possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d0af27e4c286129e8642d3c331502e352e6586768e0b4dc1631348ec170600a8 | \( p \) is a prime number. Find all integers \( k \) for which \( \sqrt{k^2 - pk} \) is a natural number. | [
"\\( k = \\left(\\frac{p + 1}{2}\\right)^2 \\) or \\( k = -\\left(\\frac{p - 1}{2}\\right)^2 \\)"
] | [
"\\( p \\) is a prime number. Find all integers \\( k \\) for which \\( \\sqrt{k^2 - pk} \\) is a natural number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 8
} | 27 | 0 | 0 |
omr-ws-7077a26980f613ad4e67ba94fca1b8db159bc1bfee78285a18fc0299a69eabcd | \( p \) is a prime number. Solve in integers \( p(x + y) = xy \). | [
"\\((x, y) = (p + 1, p^2 + p), (2p, 2p), (p^2 + p, p + 1)\\)"
] | [
"\\( p \\) is a prime number. Solve in integers \\( p(x + y) = xy \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba63a95a0e962c72037564f1cbb78cb3185a1ef7cf4090baec60a231b35dc477 | \( p \) is an odd prime. The integer \( k \) is in the range \( 1 \leq k \leq p-1 \). Let \( a_k \) be the number of divisors of \( kp + 1 \) that are greater than or equal to \( k \) and less than \( p \). Find the value of \( a_1 + a_2 + \cdots + a_{p-1} \). | [
"p-1"
] | [
"\\( p \\) is an odd prime. The integer \\( k \\) is in the range \\( 1 \\leq k \\leq p-1 \\). Let \\( a_k \\) be the number of divisors of \\( kp + 1 \\) that are greater than or equal to \\( k \\) and less than \\( p \\). Find the value of \\( a_1 + a_2 + \\cdots + a_{p-1} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1705db3d4388a3d7285f74fb2c2408feb8bcd103659c8e5ee00ae6a8a4af46e7 | \( p \) passengers get on a train with \( n \) wagons. Find the probability of having at least one passenger in each wagon. | [
"\\dfrac{1}{n^p} \\sum_{k=0}^{n} (-1)^k \\binom{n}{k} (n - k)^p"
] | [
"\\( p \\) passengers get on a train with \\( n \\) wagons. Find the probability of having at least one passenger in each wagon."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8601f9806132f5c25187278e4acd24019be4ca2380a398ebeea18fb5e1b8e6f0 | \( p \), \( q \), and \( r \) are three non-zero integers such that \( p + q + r = 26 \) and \( \frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{360}{pqr} = 1 \). Compute \( pqr \). | [
"576"
] | [
"\\( p \\), \\( q \\), and \\( r \\) are three non-zero integers such that \\( p + q + r = 26 \\) and \\( \\frac{1}{p} + \\frac{1}{q} + \\frac{1}{r} + \\frac{360}{pqr} = 1 \\). Compute \\( pqr \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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