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a_n = A_n + B, M^2 smallest square, M < A +\sqrt{B}

Source: 2010 Dutch IMO TST1 P2

January 10, 2020

arithmetic sequencealgebra

Problem Statement

Let

A

and

B

be positive integers. Define the arithmetic sequence

a_0, a_1, a_2, ...

a_n = A_n + B

. Suppose that there exists an

n\ge 0

such that

a_n

is a square. Let

M

be a positive integer such that

M^2

is the smallest square in the sequence. Prove that

M < A +\sqrt{B}