IMC 2006

Compare $\tan(\sin x)$ with $\sin(\tan x)$ , for $x\in \left]0,\frac{\pi}{2}\right[$ .

Let $v_{0}$ be the zero ector and let $v_{1},...,v_{n+1}\in\mathbb{R}^{n}$ such that the Euclidian norm $|v_{i}-v_{j}|$ is rational for all $0\le i,j\le n+1$ . Prove that $v_{1},...,v_{n+1}$ are linearly dependent over the rationals.

Let $a, b, c, d$ three strictly positive real numbers such that $$a^{2}+b^{2}+c^{2}=d^{2}+e^{2},$$ $$a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.$$ Compare $$a^{3}+b^{3}+c^{3}$$ with $$d^{3}+e^{3},$$