2025 AIME II Problems/Problem 8

Problem

From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of cents, where is a positive integer. He uses the so-called greedy algorithm, successively choosing the coin of greatest value that does not cause the value of his collection to exceed For example, to get 42 cents, Silas will choose a 25-cent coin, then a 10-cent coin, then 7 1-cent coins. However, this collection of 9 coins uses more coins than necessary to get a total of 42 cents; indeed, choosing 4 10-cent coins and 2 1-cent coins achieves the same total value with only 6 coins. In general, the greedy algorithm succeeds for a given if no other collection of 1-cent, 10-cent, and 25-cent coins gives a total value of cents using strictly fewer coins than the collection given by the greedy algorithm. Find the number of values of between and inclusive for which the greedy algorithm succeeds.