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2013 Japan Mathematical Olympiad Finals Problem 3

Source:

February 11, 2013
algebrainductionnumber theory proposednumber theory

Problem Statement

Let n≥2n\geq 2 be a positive integer. Find the minimum value of positive integer mm for which there exist positive integers a1, a2, ⋯ ,ana_1,\ a_2,\ \cdots, a_n such that :
∙ a1<a2<⋯<an=m\bullet\ a_1<a_2<\cdots <a_n=m
∙ a12+a222, a22+a322, ⋯ , an−12+an22\bullet \ \frac{a_1^2+a_2^2}{2},\ \frac{a_2^2+a_3^2}{2},\ \cdots,\ \frac{a_{n-1}^2+a_n^2}{2} are all square numbers.
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