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1 square is sum of 3 squares in 2 different ways

Source: JBMO 2008 Shortlist N8

October 14, 2017
JBMOnumber theory

Problem Statement

Let a,b,c,d,e,fa, b, c, d, e, f are nonzero digits such that the natural numbers abc‾,def‾\overline{abc}, \overline{def} and abcdef‾\overline{abcdef } are squares. a) Prove that abcdef‾\overline{abcdef} can be represented in two different ways as a sum of three squares of natural numbers. b) Give an example of such a number.
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