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1+(3n+3)/(a²+b²+c²) perfect square if 3n+1 is perfect square

Source: Romanian District Olympiad 2017, Grade VII, Problem 1

October 9, 2018
number theory

Problem Statement

Let be a natural number n≥3 n\ge 3 with the property that 1+3n 1+3n is a perfect square. Show that there are three natural numbers a,b,c, a,b,c, such that the number 1+3n+3a2+b2+c2 1+\frac{3n+3}{a^2+b^2+c^2} is a perfect square.
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