Problem:
Find, with proof, all positive integers n for which 2n+12n+2011n is a perfect square.
Solution:
The answer is n=1. Clearly, n=1 is a solution because 2+12+2011=452. Next we show that there is no other solutions.
Assume that n≥2. If n is odd, then 2n+12n+2011n cannot be a perfect square because it is congruent to 3 modulo 4 . If n is even, we can complete our solution in two ways.
- 2n+12n+2011n cannot be a perfect square because it is congruent to 2 modulo 3 .
- 2n+12n+2011n cannot be a perfect square because it is in between two consecutive perfect squares. Indeed, say n=2k, then
(2011k)2<22k+122k+20112k=4k+144k+20112k<1+2â‹…2011k+20112k=(2011k+1)2
The problems on this page are the property of the MAA's American Mathematics Competitions