messages listlengths 2 2 | source stringclasses 5
values | _key stringlengths 16 16 | gold stringclasses 284
values |
|---|---|---|---|
[
{
"role": "user",
"content": "The integer 273 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 35.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | d3d82e7c895de134 | 58 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 12 and x * y = 35, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | cf17651e7da0738d | 74 |
[
{
"role": "user",
"content": "The integer 2717 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 24.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999.... | aime_bench/procedural_aime/factor_chain | 64df9d018348863d | 67 |
[
{
"role": "user",
"content": "The integer 165 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 15.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | 820fd80d1940ec22 | 34 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 13 and x * y = 36, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | 7cf90a7b8e2cf147 | 97 |
[
{
"role": "user",
"content": "The integer 255 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 29.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | b5041c2b7440322a | 54 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 10 = 2 and x mod 7 = 0. The answer is a positive integer less than 70.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the... | aime_bench/procedural_aime/lcm_residue | bf973dfff1192667 | 42 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 126, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | d55e32b3165bb051 | 108 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 120, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | 4c8d7a21759bebe4 | 66 |
[
{
"role": "user",
"content": "The integer 195 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 41.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | 7d0aeda66fa2335b | 62 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 10 = 5 and x mod 11 = 2. The answer is a positive integer less than 110.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve t... | aime_bench/procedural_aime/lcm_residue | f38e11bd4ebf4ab4 | 35 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 8 = 5 and x mod 7 = 6. The answer is a positive integer less than 56.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the ... | aime_bench/procedural_aime/lcm_residue | 506689b3e378181a | 13 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 432, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | 9bda7f18605d35b0 | 108 |
[
{
"role": "user",
"content": "The integer 2261 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 5.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | 3527444f2c36baa3 | 48 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 8 = 4 and x mod 11 = 9. The answer is a positive integer less than 88.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the... | aime_bench/procedural_aime/lcm_residue | 8c2edc5187e8c4b9 | 20 |
[
{
"role": "user",
"content": "Compute the value of ((8^9) * 2) mod 103, where ^ denotes integer exponentiation. The final answer is a non-negative integer less than 103.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between... | aime_bench/procedural_aime/chained_modular | 73cb1f531be20b1a | 49 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 396, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | fdba7229e9fda415 | 108 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 16 and x * y = 64, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | 98483ff93b915f1e | 128 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 22 and x * y = 121, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the pro... | aime_bench/procedural_aime/diophantine | 72e1c7000ac07d80 | 242 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 15 and x * y = 54, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | a9a68c0e9c1733e1 | 117 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 169, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | c267cd6ba5b33bd5 | 52 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 355, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | 2b21de8affdb379f | 94 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 60, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step and... | aime_bench/procedural_aime/iterated_digit | c2562b28ab7cdbde | 87 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 16 and x * y = 63, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | c12ff18fa9eca84f | 130 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 6 = 1 and x mod 17 = 2. The answer is a positive integer less than 102.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve th... | aime_bench/procedural_aime/lcm_residue | de490b5e00c64257 | 19 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 6 = 4 and x mod 19 = 14. The answer is a positive integer less than 114.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve t... | aime_bench/procedural_aime/lcm_residue | d8fcc8b10933e991 | 52 |
[
{
"role": "user",
"content": "The integer 665 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 1.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. S... | aime_bench/procedural_aime/factor_chain | 497c97a375ef80da | 32 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 12 = 8 and x mod 19 = 10. The answer is a positive integer less than 228.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve ... | aime_bench/procedural_aime/lcm_residue | b8529bb21c1a1d5a | 200 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 14 and x * y = 48, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | 309db9c182b64459 | 100 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 14 and x * y = 40, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | 85bc54aa6d5b7483 | 116 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 24 and x * y = 143, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the pro... | aime_bench/procedural_aime/diophantine | 2b82678ce5ac9f49 | 290 |
[
{
"role": "user",
"content": "The integer 2717 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 8.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | 3341927eb3206a6a | 51 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 227, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | fe08e2fbbb09b60f | 101 |
[
{
"role": "user",
"content": "The integer 357 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 46.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | ad80a9a721853a57 | 73 |
[
{
"role": "user",
"content": "The integer 561 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 22.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | 0e0fd15a2f06b2e7 | 53 |
[
{
"role": "user",
"content": "The integer 561 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 8.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. S... | aime_bench/procedural_aime/factor_chain | ce25294f2e813d05 | 39 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 12 = 7 and x mod 13 = 2. The answer is a positive integer less than 156.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve t... | aime_bench/procedural_aime/lcm_residue | fc47d1365b88c85a | 67 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 402, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | 50ff8bd5bd673207 | 87 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 12 and x * y = 32, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | d753ab4d97729c1f | 80 |
[
{
"role": "user",
"content": "The integer 105 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 10.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | 14e3afe55ed733f8 | 25 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 15 and x * y = 50, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | b5a98f017d58160a | 125 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 107, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | a795dd55367b1440 | 17 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 6 = 2 and x mod 13 = 4. The answer is a positive integer less than 78.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the... | aime_bench/procedural_aime/lcm_residue | d5d494eb8b76fd54 | 56 |
[
{
"role": "user",
"content": "The integer 429 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 47.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | 5bb047375e4d7712 | 74 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 15 = 11 and x mod 13 = 10. The answer is a positive integer less than 195.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve... | aime_bench/procedural_aime/lcm_residue | c7763a843d2a68a7 | 101 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 15 = 3 and x mod 19 = 0. The answer is a positive integer less than 285.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve t... | aime_bench/procedural_aime/lcm_residue | 72a5187179c8f185 | 228 |
[
{
"role": "user",
"content": "Compute the value of ((6^12) * 2) mod 103, where ^ denotes integer exponentiation. The final answer is a non-negative integer less than 103.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer betwee... | aime_bench/procedural_aime/chained_modular | 23bbf292c6dcb054 | 18 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 330, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | 9856af266b688506 | 87 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 8 and x * y = 15, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the probl... | aime_bench/procedural_aime/diophantine | 1f988f4dcf7e1536 | 34 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 6 = 2 and x mod 7 = 2. The answer is a positive integer less than 42.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the ... | aime_bench/procedural_aime/lcm_residue | ba937b6bb37ed74b | 2 |
[
{
"role": "user",
"content": "The integer 255 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 49.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | 3e660fd0816b6ec1 | 74 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 6 = 3 and x mod 17 = 5. The answer is a positive integer less than 102.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve th... | aime_bench/procedural_aime/lcm_residue | e49a56e27b6520b7 | 39 |
[
{
"role": "user",
"content": "The integer 455 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 14.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | a7745cfaa08a6be8 | 39 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 194, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | 283a33511891a798 | 59 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 240, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | e939a3c86a27ca02 | 87 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 475, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | 7241e820b0c08426 | 52 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 4 and x * y = 4, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the proble... | aime_bench/procedural_aime/diophantine | 30708187b27302ab | 8 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 8 = 0 and x mod 9 = 4. The answer is a positive integer less than 72.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the ... | aime_bench/procedural_aime/lcm_residue | 5d05ea3c17a2df59 | 40 |
[
{
"role": "user",
"content": "Compute the value of ((9^8) * 7) mod 103, where ^ denotes integer exponentiation. The final answer is a non-negative integer less than 103.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between... | aime_bench/procedural_aime/chained_modular | 7c74e1df4d9ddd2c | 32 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 6 = 0 and x mod 7 = 2. The answer is a positive integer less than 42.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the ... | aime_bench/procedural_aime/lcm_residue | 1606d38040e11b51 | 30 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 6 = 0 and x mod 11 = 9. The answer is a positive integer less than 66.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the... | aime_bench/procedural_aime/lcm_residue | dae31a2654ac77b8 | 42 |
[
{
"role": "user",
"content": "Compute the value of ((6^9) * 9) mod 103, where ^ denotes integer exponentiation. The final answer is a non-negative integer less than 103.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between... | aime_bench/procedural_aime/chained_modular | b3bfffeaad17a7e3 | 39 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 69, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step and... | aime_bench/procedural_aime/iterated_digit | 2e682410b22a0344 | 87 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 23 and x * y = 130, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the pro... | aime_bench/procedural_aime/diophantine | 90d974d926ca7b93 | 269 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 18 and x * y = 80, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | d84fd5859d8c199f | 164 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 488, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | e9880a2ea51d4705 | 101 |
[
{
"role": "user",
"content": "The integer 969 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 37.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | 92a4e146f0f7dc75 | 76 |
[
{
"role": "user",
"content": "The integer 561 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 5.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. S... | aime_bench/procedural_aime/factor_chain | b3b4aa51af536fa6 | 36 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 388, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | 93c13d5ed7da8cc8 | 73 |
[
{
"role": "user",
"content": "The integer 2431 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 33.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999.... | aime_bench/procedural_aime/factor_chain | ba3237b4dc90c4a5 | 74 |
[
{
"role": "user",
"content": "Compute the value of ((9^15) * 9) mod 103, where ^ denotes integer exponentiation. The final answer is a non-negative integer less than 103.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer betwee... | aime_bench/procedural_aime/chained_modular | 5f2c2a31d2433b7b | 23 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 350, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | 332d537c67efb94c | 17 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 18 and x * y = 81, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | 8fe21608f35c40a2 | 162 |
[
{
"role": "user",
"content": "The integer 627 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. S... | aime_bench/procedural_aime/factor_chain | 53c05ce2bfb972ac | 35 |
[
{
"role": "user",
"content": "Compute the value of ((2^18) * 4) mod 109, where ^ denotes integer exponentiation. The final answer is a non-negative integer less than 109.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer betwee... | aime_bench/procedural_aime/chained_modular | c0adeb8092f3535b | 105 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 12 = 3 and x mod 11 = 0. The answer is a positive integer less than 132.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve t... | aime_bench/procedural_aime/lcm_residue | 300c1176e2eb0510 | 99 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 9 and x * y = 18, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the probl... | aime_bench/procedural_aime/diophantine | 9770ea7692e5565b | 45 |
[
{
"role": "user",
"content": "Compute the value of ((7^9) * 7) mod 101, where ^ denotes integer exponentiation. The final answer is a non-negative integer less than 101.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between... | aime_bench/procedural_aime/chained_modular | 766027d746c38085 | 65 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 10 = 7 and x mod 7 = 0. The answer is a positive integer less than 70.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the... | aime_bench/procedural_aime/lcm_residue | 4a6c10293d0b2bc9 | 7 |
[
{
"role": "user",
"content": "The integer 2261 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 42.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999.... | aime_bench/procedural_aime/factor_chain | c70d2e92212135e9 | 85 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 404, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | 10a95e3a7ca10b3a | 17 |
[
{
"role": "user",
"content": "Compute the value of ((6^18) * 6) mod 103, where ^ denotes integer exponentiation. The final answer is a non-negative integer less than 103.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer betwee... | aime_bench/procedural_aime/chained_modular | e474b2d13706f77a | 44 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 12 and x * y = 20, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | 9da4db6ad92215b0 | 104 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 466, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | 04ecd10a99d12712 | 52 |
[
{
"role": "user",
"content": "The integer 2431 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 15.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999.... | aime_bench/procedural_aime/factor_chain | 72e67e14cfb66495 | 56 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 14 = 11 and x mod 13 = 7. The answer is a positive integer less than 182.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve ... | aime_bench/procedural_aime/lcm_residue | d93148fa7c3d3bd6 | 137 |
[
{
"role": "user",
"content": "Define f(n) = 7 * (sum of digits of n) + 3. Starting at n_0 = 459, compute n_3 = f(f(f(n_0))).\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the problem step by step an... | aime_bench/procedural_aime/iterated_digit | 26397135cfb6f66b | 108 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 8 and x * y = 12, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the probl... | aime_bench/procedural_aime/diophantine | 51b33a6b2af43bbe | 40 |
[
{
"role": "user",
"content": "The integer 195 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 18.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | ffffa82ebae8b0b6 | 39 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 15 = 3 and x mod 16 = 10. The answer is a positive integer less than 240.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve ... | aime_bench/procedural_aime/lcm_residue | fcd943847dbe0baa | 138 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 15 and x * y = 44, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | e21f0d8c1798cc94 | 137 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 17 and x * y = 52, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | fefee96e0aa27a6e | 185 |
[
{
"role": "user",
"content": "The integer 165 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 41.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | 2372288217c4effc | 60 |
[
{
"role": "user",
"content": "The integer 165 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 21.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | d1669c4f96e89c28 | 40 |
[
{
"role": "user",
"content": "Compute the value of ((8^12) * 5) mod 103, where ^ denotes integer exponentiation. The final answer is a non-negative integer less than 103.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer betwee... | aime_bench/procedural_aime/chained_modular | 7d6389a9d2debd19 | 96 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 14 = 2 and x mod 11 = 9. The answer is a positive integer less than 154.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve t... | aime_bench/procedural_aime/lcm_residue | b83ed3aee19865aa | 86 |
[
{
"role": "user",
"content": "Find the smallest positive integer x with x mod 15 = 12 and x mod 13 = 4. The answer is a positive integer less than 195.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve ... | aime_bench/procedural_aime/lcm_residue | c33c93156e20f716 | 147 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 19 and x * y = 90, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the prob... | aime_bench/procedural_aime/diophantine | 7fd64ce225340dd4 | 181 |
[
{
"role": "user",
"content": "Two positive integers x and y satisfy x + y = 23 and x * y = 132, with 2 <= x <= y. Compute the integer x^2 + y^2.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. Solve the pro... | aime_bench/procedural_aime/diophantine | 62712609cf8a72e0 | 265 |
[
{
"role": "user",
"content": "The integer 455 factors uniquely as a product of three distinct primes. Let s be the sum of those three primes. Compute s + 43.\n\nSolve carefully and end with '#### N' where N is the final integer answer.\n\nThis is an AIME problem. The answer is an integer between 0 and 999. ... | aime_bench/procedural_aime/factor_chain | d712f1b17d2a5ba3 | 68 |
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