IMC 2010 · Problem 5

Suppose that $a, b, c$ are real numbers in the interval $[-1,1]$ such that

$$1 + 2abc \ge a^2 + b^2 + c^2.$$

Prove that

$$1 + 2(abc)^n \ge a^{2n} + b^{2n} + c^{2n}$$

for all positive integers $n$.