MathLabs

Problem 1

Find the smallest positive integer n such that no arithmetic progression of 1999 real terms contains exactly n integers.
Step 1 of 5: Integer terms are equally spaced
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ai+2j=ai+j+(ai+j−ai)a_{i+2j}=a_{i+j}+(a_{i+j}-a_i)
Detailed analysis

If terms aia_i and ai+ja_{i+j} are integers, then ai+2j=ai+j+(ai+j−ai)a_{i+2j}=a_{i+j}+(a_{i+j}-a_i) is also an integer. Thus all integer terms form an equally spaced subsequence. Scaling and translating lets us represent them by 1,2,...,n.