RMO 2015!! #7

Number Theory Level 3

Find the sum of all positive integer $n$ for which the equation $x^n + (2 + x)^{n} + (2 - x)^{n} = 0$ has an integer as a solution.

0 2 1 3

1 solution

Dev Sharma
Sep 29, 2015

If $n$ is even, then term in LHS are non negative and cannot be 0. So there will be no integer solution.

If $n = 1$ then $x = -4$ is the solution.

If n is odd and at least 3, then x must be even, otherwise LHS is odd. But it wouldn't satisfy.

Can you pls explain the part of the last line of your solution that says "But it wouldn't satisfy"

rajdeep das - 4 years, 8 months ago