50th Mathematical Olympiad in Ukraine, Third Round (January 23, 2010)

Question

Prove that for all positive integers n equation a^n + 2010b^n = c^n+1 has infinitely many natural solutions a, b, c.

Accepted Answer

Consider a = b = 2011k^n+1, where k is a positive integer, then a^n + 2010b^n = 2011^n k^n(n+1) + 2010 2011^n k^n(n+1) = 2011^n+1 k^n(n+1). Since 2011 2011^n k^n(n+1) and 2011 2010, we have 2011^n+1 k^n(n+1) = (2011k^n)^n+1 = c^n+1 c = 2011k^n.