Problem 1
Find the smallest positive integer n such that no arithmetic progression of 1999 real terms contains exactly n integers.
Step 2 of 5: Describe a progression with n integers
Detailed analysis
If consecutive integer terms are separated by k progression steps, the normalized progression has kn-k+1 terms. We may add up to k-1 terms on either end without introducing another integer, so every length in the interval [kn-k+1, kn+k-1] occurs for some k.