If $x,y,z\in[-1,1]$ and $1+2xyz\geq x^2+y^2+z^2$, then can we infer $1+2(xyz)^n\geq x^{2n}+y^{2n}+z^{2n}$?

This problem was in IMC 2010.

Assuming $x,y,z\in [-1,1]$, suppose that $$1+2xyz\geqslant x^2 + y^2 + z^2$$ Can we infer from this that $$1+2(xyz)^n\geqslant x^{2n} + y^{2n} + z^{2n}$$ for any positive integer $n$?

Solution 1:

We can assume that $ab\geq0$ because

if $ab<0$, $ac<0$ and $bc<0$ we get $a^2b^2c^2<0$, which is contradiction.

The condition gives $(1-a^2)(1-b^2)\geq(c-ab)^2$ and we need to prove that $$\left(1-a^{2n}\right)\left(1-b^{2n}\right)\geq\left(c^n-a^nb^n\right)^2,$$ which is C-S: $$\left(c^n-a^nb^n\right)^2=(c-ab)^2\left(\sum\limits_{i=0}^{n-1}c^i(ab)^{n-1-i}\right)^2\leq(c-ab)^2\left(\sum\limits_{i=0}^{n-1}(ab)^{n-1-i}\right)^2\leq$$ $$\leq(1-a^2)(1-b^2)\sum_{i=0}^{n-1}a^{2i}\sum_{i=0}^{n-1}b^{2i}=\left(1-a^{2n}\right)\left(1-b^{2n}\right)$$ and we are done!