id stringlengths 12 12 | generation_step int64 1 101 | problem stringlengths 57 421 | answer stringlengths 1 72 |
|---|---|---|---|
synco_000001 | 1 | How many ordered triples (x, y, z) of positive integers satisfy x + y + z = 15 and x ≤ y ≤ z? | 19 |
synco_000002 | 1 | Find the number of positive integer solutions (a, b) to the equation 3a + 5b = 100 where both a and b are even numbers. | 3 |
synco_000003 | 1 | Find the number of positive integer solutions (x, y) to the system of equations 3x + 2y = 12 and 5x + 4y = 20. | 0 |
synco_000004 | 1 | How many 3-digit numbers have all even digits and the sum of their digits divisible by 3? | 33 |
synco_000005 | 1 | Find the number of strictly increasing sequences of 5 positive integers (a1 < a2 < a3 < a4 < a5) where each term is at least 2 more than the previous term, and the sum of the sequence is exactly 50. | 377 |
synco_000006 | 1 | Find the number of sequences (a₁, a₂, a₃, a₄, a₅) of positive integers such that a₁ + a₂ + a₃ + a₄ + a₅ = 100 and each term is at least twice the previous term (a₂ ≥ 2a₁, a₃ ≥ 2a₂, etc.). | 361 |
synco_000007 | 1 | How many 4-element subsets of {1, 2, ..., 10} have the property that the sum of the smallest and largest elements equals the sum of the middle two elements? | 50 |
synco_000008 | 1 | How many sequences of four positive integers (a, b, c, d) exist such that each number is less than 10 and a + b = c + d? | 489 |
synco_000009 | 1 | How many integer sequences (a₁,a₂,a₃,a₄) exist where each element is between 1 and 5 inclusive, and the sum of the first two elements equals the sum of the last two elements? | 85 |
synco_000010 | 1 | A sequence is defined where each term after the first two follows these rules: if the term's position is odd (n=1,3,5,...), then a_n = a_{n-1} * a_{n-2}; if even (n=2,4,6,...), then a_n = a_{n-1} + a_{n-2}. Given a_1 = 2 and a_2 = 3, compute a_5. | 54 |
synco_000011 | 1 | How many positive integer solutions exist to the equation x + y + z = 15 where x ≤ y ≤ z and each variable is at least 2? | 12 |
synco_000012 | 1 | A fair die is rolled three times. What is the probability that the sum of the three rolls is exactly 10, given that the first roll is even and the second roll is a multiple of 3? | 1/9 |
synco_000013 | 1 | Find the number of positive integer solutions (a, b) to the equation 3a + 5b = 50 such that a < 2b. | 1 |
synco_000014 | 1 | A box contains red, blue, and green marbles. The total number of marbles is 100. The number of red marbles is twice the number of blue marbles, and the number of green marbles is 20 less than the number of blue marbles. If two marbles are drawn without replacement, what is the probability that the first is red and the ... | 2/33 |
synco_000015 | 1 | Find the number of integer solutions (x, y, z) to the equation x + y + z = 100 where x, y, z are positive integers, x ≤ y ≤ z, and x + y > z. | 208 |
synco_000016 | 1 | Find the number of positive integer solutions (a, b, c) to the equation 2a + 3b + 5c = 100 where a < b < c and all variables are distinct. | 6 |
synco_000017 | 1 | Find the number of positive integers x and y such that 2x + 3y = 100 and x + y is a multiple of 5. | 3 |
synco_000018 | 1 | How many 5-digit numbers with all even digits have a digit sum divisible by 5? | 500 |
synco_000019 | 1 | Find the number of positive integer solutions (x, y) to the equation 3x + 5y = 100. | 6 |
synco_000020 | 1 | A bag contains red and blue marbles. The number of red marbles is twice the number of blue marbles. Two marbles are drawn without replacement. The probability that both are red is 2/5. How many blue marbles are there? | 2 |
synco_000021 | 1 | How many ways are there to distribute 12 identical balls into 3 distinct boxes such that each box contains at least 2 balls and no box contains more than 5 balls? | 10 |
synco_000022 | 1 | Find the number of positive integer solutions (x, y) to the equation 3x + 5y = 100 where x > y. | 4 |
synco_000023 | 1 | Find the number of positive integer solutions (x, y) to the equation 2x + 3y = 100 where both x and y are even numbers. | 8 |
synco_000024 | 1 | Find the number of positive integer solutions (x, y) to the equation 3x + 5y = 100 where both x and y are even numbers. | 3 |
synco_000025 | 1 | How many 5-digit sequences (with leading zeros allowed) are there such that no two consecutive digits are the same and the sum of the digits is even? | 32805 |
synco_000026 | 1 | How many distinct arrangements are there of the letters in the word 'COMBINATORICS' such that no two vowels are adjacent? | 76204800 |
synco_000027 | 1 | How many 5-digit sequences composed of even digits (0, 2, 4, 6, 8) have a sum divisible by 5? | 625 |
synco_000028 | 1 | Find all positive integer solutions (x, y) to the equation 2x + 3y = 100, where x and y are both even numbers. How many such solutions exist? | 8 |
synco_000029 | 1 | A standard deck of 52 cards is shuffled. What is the probability that a randomly dealt 5-card hand contains exactly two aces and at least one king? | \frac{1013}{108290} |
synco_000030 | 1 | Find all positive integer solutions (x, y) to the equation 3x + 4y = 25 where x > y. How many such solutions exist? | 1 |
synco_000031 | 1 | How many binary strings of length 10 contain exactly four 1s and no two 1s are adjacent? | 35 |
synco_000032 | 2 | A library has 7 distinct math books and 5 distinct science books. How many ways can you select 6 books such that the number of math books is at least 3 and the number of science books is not equal to the number of math books? | 462 |
synco_000033 | 2 | A sequence is defined by a₁ = x, a₂ = y, and aₙ = aₙ₋₁ + aₙ₋₂ for n ≥ 3. Given that a₅ = 8, find the value of x + y, assuming x and y are positive integers. | 3 |
synco_000034 | 2 | How many positive integer solutions are there to the equation x + y + z = 10 such that x < y < z? | 4 |
synco_000035 | 2 | Find the number of ordered triples (x, y, z) of positive integers, each divisible by 5, such that x + y + z = 100 and x < y < z. | 24 |
synco_000036 | 2 | From a shelf containing 4 novels, 3 biographies, and 3 cookbooks, how many ways can you select 6 books such that there are at least 2 novels and at least 1 biography? | 178 |
synco_000037 | 2 | A fair 6-sided die is rolled repeatedly until the cumulative sum of all rolls is at least 10. What is the probability that the process stops exactly on the third roll? | 11/24 |
synco_000038 | 2 | How many ordered triplets (x, y, z) of positive integers satisfy x + y + z = 10 with x < y < z? | 4 |
synco_000039 | 2 | How many 5-digit numbers have non-decreasing digits (each digit ≥ the previous) and are divisible by 5? | 70 |
synco_000040 | 2 | How many positive integers less than 1000 have exactly two even digits and one odd digit in their decimal representation? | 325 |
synco_000041 | 2 | How many positive integer solutions (x, y) are there to the equation 2x + 5y = 25, where x is even and y is odd? | 1 |
synco_000042 | 2 | Find the number of positive integer solutions (x, y) to the equation 3x + 4y = 120 where both x and y are multiples of 2. | 4 |
synco_000043 | 2 | What is the probability that a randomly chosen 5-digit sequence (allowing leading zeros) has no two adjacent digits the same and starts with an even digit? | \frac{6561}{20000} |
synco_000044 | 2 | How many ways can 6 people be seated around a circular table if two of them refuse to sit next to each other? | 72 |
synco_000045 | 2 | A committee of 6 people is to be formed from 8 men and 7 women. The committee must include at least 2 women and at least 1 man, and the number of women must be more than the number of men. How many different committees are possible? | 1148 |
synco_000046 | 2 | Find the number of integer solutions (x, y, z) to the equation 2x + 3y + 5z = 100 where x, y, z are non-negative integers and x ≤ y ≤ z. | 21 |
synco_000047 | 2 | Define a sequence where a₁ = 1 and each subsequent term is the sum of all previous terms plus 2 times the term's position. Find the 5th term of this sequence. | 54 |
synco_000048 | 2 | How many 6-character passwords can be formed using digits 0-9 and letters A-Z (uppercase/lowercase) if: (1) the first character is a letter, (2) at least one even digit appears, (3) at least one vowel (A, E, I, O, U) appears, and (4) no two identical characters are adjacent? | 9440339750 |
synco_000049 | 2 | In a class of 30 students, 18 are girls and 12 are boys. If 5 students are selected at random, what is the probability that exactly 2 are boys, given that at least 3 are girls? | \frac{44}{81} |
synco_000050 | 2 | Find the number of ordered triples (x, y, z) of positive integers such that 2x + 3y + 4z = 20 and x, y, z are all distinct. | 3 |
synco_000051 | 2 | Define a sequence where each term is the sum of the previous two terms, starting with 1 and 1. However, if a term is divisible by 5, it is replaced by 0. What is the 10th term in this modified sequence? | 0 |
synco_000052 | 2 | How many sequences of 5 positive integers (a, b, c, d, e) satisfy the following conditions: (1) the sum of the integers is 25, and (2) each integer is at least 3 and no more than 10? | 926 |
synco_000053 | 2 | Find the number of positive integer solutions (x, y) to the equation 3x + 5y = 100, where x + y is an even number. | 6 |
synco_000054 | 2 | How many 5-digit numbers with distinct digits have the property that the sum of the first three digits equals the sum of the last two digits? | 1252 |
synco_000055 | 2 | How many 5-digit numbers composed solely of digits 1, 2, and 3 have a digit sum divisible by 3? | 81 |
synco_000056 | 2 | A 4-digit lock code has the first digit fixed as 3 (an odd digit). What is the probability that the entire code contains at least one even digit and at least one odd digit? | 7/8 |
synco_000057 | 2 | A committee of 5 is to be formed from 10 men and 10 women. How many such committees have at least 2 women and no more than 3 men? | 13152 |
synco_000058 | 2 | Find all positive integer solutions (x, y) to 2x + 5y = 100 where x + y is divisible by 4. How many such solutions exist? | 2 |
synco_000059 | 2 | Three distinct even numbers are randomly selected from 1 to 20. What is the probability that their sum is divisible by 3? | 7/20 |
synco_000060 | 2 | Determine the number of sequences of length 5 where each element is 0, 1, or 2, the sum of the elements is exactly 3, and no two adjacent elements are both 2. | 30 |
synco_000061 | 2 | Consider a sequence defined by a₁ = 1, a₂ = 2, and for n ≥ 3, aₙ is the sum of all previous terms. Find the value of a₁₀. | 384 |
synco_000062 | 2 | How many 5-digit sequences (allowing leading zeros) have digits summing to a multiple of 3? | 33334 |
synco_000063 | 3 | How many ways can 100 identical candies be distributed to 5 children if each child receives at least 10 candies and no more than 30 candies? | 116601 |
synco_000064 | 3 | Find the number of positive integer solutions (x, y) to the equation 5x + 3y = 100. | 6 |
synco_000065 | 3 | Find the number of sequences of length 4 with elements from {1, 2, 3} such that the sum of the first two elements equals the product of the last two. | 11 |
synco_000066 | 3 | How many sequences of length 4 can be formed using digits 0-9 such that the sum of the first two digits equals the sum of the last two digits? | 670 |
synco_000067 | 3 | Consider the sequence defined by a₁ = 1, a₂ = 2, and for n ≥ 3, aₙ = aₙ₋₁ + aₙ₋₂ + 1. What is the value of a₁₀? | 143 |
synco_000068 | 3 | Find the smallest positive integer divisible by 3, 5, and 7, and when divided by 11 leaves a remainder of 2. | 420 |
synco_000069 | 3 | Find the number of positive integer solutions to 5x + 7y = 100 where both x and y are even. | 1 |
synco_000070 | 3 | Find the smallest positive integer N such that N is divisible by 4, 5, and 6, and when divided by 7 leaves a remainder of 3. | 360 |
synco_000071 | 3 | How many 5-character passwords can be formed using uppercase letters (A-Z) and digits (0-9) such that exactly two vowels (A, E, I, O, U) are present, no two vowels are adjacent, and at least one digit is included? | 3079500 |
synco_000072 | 3 | Find the number of positive integer solutions (x, y) to the equation 2x + 3y = 25 where x and y are both greater than 2. | 2 |
synco_000073 | 3 | How many 3-digit numbers with strictly increasing digits have a digit sum that is even? | 44 |
synco_000074 | 3 | Find the number of positive integer solutions (x, y) to the equation 5x + 7y = 100 where x ≤ y. | 1 |
synco_000075 | 3 | How many 6-character passwords can be formed using uppercase letters (A-Z) and digits (0-9) such that the password contains at least two vowels (A, E, I, O, U), at least one digit, and no two vowels are adjacent? | 192525000 |
synco_000076 | 3 | A sequence is defined by a₁ = 1, a₂ = 2, and for n ≥ 3, aₙ = aₙ₋₁ + aₙ₋₂ + n. Find the value of a₅. | 23 |
synco_000077 | 3 | Find all positive integer solutions (a, b) to the equation 2a + 3b = 100 where a + b is a perfect square. | (8, 28), (47, 2) |
synco_000078 | 3 | A password is created using digits 0-9, exactly 5 characters long. The first digit must be even, the last digit must be odd, and no two adjacent digits can be the same. How many such passwords are possible? | 16400 |
synco_000079 | 3 | How many ways can you select 3 distinct integers from {1, 2, ..., 10} such that their sum is divisible by 3 and no two numbers are consecutive? | 20 |
synco_000080 | 3 | A bag contains 6 red marbles, 4 blue marbles, and 3 green marbles. Three marbles are drawn at random without replacement. Given that at least one marble is blue, what is the probability that exactly two are red? | 30/101 |
synco_000081 | 3 | Determine the number of binary sequences of length 5 where no two consecutive 1s appear and the sum of the bits is even. | 7 |
synco_000082 | 3 | How many ways are there to distribute 7 identical balls into 3 distinct boxes such that each box contains a prime number of balls? | 3 |
synco_000083 | 3 | A function f(n) is defined as the sum of the first n positive integers. What is the smallest n for which f(n) is divisible by 100? | 24 |
synco_000084 | 3 | How many positive integer solutions exist to the equation x + y + z = 10 such that x < y < z? | 4 |
synco_000085 | 3 | From a standard 52-card deck, how many 5-card hands contain exactly two hearts and the remaining three cards all from the same non-heart suit (spades, clubs, or diamonds)? | 66924 |
synco_000086 | 3 | Find the number of positive integer solutions (x, y) to the equation 3x + 5y = 50 where x > y. | 2 |
synco_000087 | 3 | A bag contains 10 red marbles and 15 blue marbles. If 5 marbles are drawn at random without replacement, what is the probability that exactly 2 are red and 3 are blue, given that the first marble drawn is red? | \frac{195}{506} |
synco_000088 | 3 | Find the number of positive integer solutions (x, y, z) to the equation 2x + 3y + 5z = 100 where x ≤ y ≤ z. | 18 |
synco_000089 | 3 | Find all positive integers x, y, z such that x + y + z = 50, x < y < z, and x² + y² = z². Determine the sum of all possible values of x. | 0 |
synco_000090 | 3 | Find the number of positive integer solutions (x, y) to the equation 3x + 5y = 100 where x is a multiple of y. | 2 |
synco_000091 | 3 | Find the number of positive integer solutions (x, y) to the equation 3x + 5y = 100 where both x and y are even numbers. | 3 |
synco_000092 | 3 | A committee of 5 people is formed from 6 men and 5 women. What is the probability that the committee has at least 2 women, given that there are more men than women on the committee? | 200/281 |
synco_000093 | 3 | How many 6-character passwords can be formed using lowercase letters and digits, with exactly two vowels (a, e, i, o, u), no adjacent vowels, and at least one digit? | 182260000 |
synco_000094 | 3 | What is the probability that three distinct numbers selected from 1 to 100 have a product divisible by 100? | \frac{1333}{10780} |
synco_000095 | 4 | How many 5-digit numbers have all even digits and their digit sum divisible by 5? | 500 |
synco_000096 | 4 | Find the number of positive integer solutions (x, y, z) to the equation 2x + 3y + 5z = 100 where x < y < z. | 6 |
synco_000097 | 4 | How many 4-digit numbers with distinct digits have the first digit even, the last digit odd, and the middle two digits are both prime numbers? | 144 |
synco_000098 | 4 | Find the number of positive integer solutions (x, y) to the equation 2x + 3y = 100 where both x and y are even numbers. | 8 |
synco_000100 | 4 | How many 5-digit numbers have all even digits and are divisible by 3? | 834 |
synco_000101 | 4 | How many 6-digit numbers have all even digits and a digit sum divisible by 3? | 4167 |
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