IR-MO 1997

Let $x, y, z$ be real numbers greater than $1$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$ . Prove that $$\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}\leq \sqrt{x+y+z}.$$

In an acute triangle $ABC$ let $AD$ and $BE$ be altitudes, and $AP$ and $BQ$ be bisectors. Let $I$ and $O$ be centers of incircle and circumcircle, respectively. Prove that the points $D, E$ , and $I$ are collinear if and only if the points $P, Q$ , and $O$ are collinear.

Let $\mathcal P$ be the set of all points in $\mathbb R^n$ with rational coordinates. For the points $A,B \in \mathcal l{P}$ , one can move from $A$ to $B$ if the distance $AB$ is $1$ . Prove that every point in $\mathcal l{ P}$ can be reached from any other point in $\mathcal{P}$ by a finite sequence of moves if and only if $n \geq 5$ .