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Problem 1329

AIME late

Algebra Difficulty 5.5 Prove it All-Soviet-Union Mathematical Olympiad · Soviet Union

An infinite arithmetic progression contains a square. Prove it contains infinitely many squares.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:

Let the square be $a^{2}$ and the difference $d$ , so that all numbers of the form $a^{2} + nd$ belong to the arithmetic progression (for $n$ a natural number). Take $n$ to be $2a + dr^{2}$ , then $a^{2} + nd = (a + dr)^{2}$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.