2026 AIME I Problem 13

For each nonnegative integer r less that 502, define S_r=\sum_{m\geq 0}\binom{10,000}{502m+r}, where \binom{10,000}{n} is defined to be 0 when n>10,000. That is, S_r is the sum of all the binomial coefficients of the form \binom{10,000}{k} for which 0\leq k\leq 10,000 and k-r is a multiple of 502. Find the number of integers in the list S_0,S_1,S_2,\ldots ,S_{501} that are multiples of the prime number 503.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.