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IMC 2010 Day 1 Problem 5

Suppose that $a,b,c$ are real numbers in the interval $[-1,1]$ such that $1+2abc \geq a^2+b^2+c^2$. Prove that $1+2(abc)^n\geq a^{2n}+b^{2n}+c^{2n}$ for all positive integers $n$.
IMC 2010 Day 1 Problem 5

Official Solution:
The conditions of the problem imply that $\displaystyle \begin{pmatrix} 1 & a & b \\ a&1& c \\ b&c&1 \end{pmatrix}$ is positive definite.