DE-MO 2006

Find all $n\in \mathbb Z^+$ , so that $$z_n = \underbrace{ 101\dots101}_{2n+1 \text{ digits} }$$ is prime.

Five points are on the surface of of a sphere of radius $1$ . Let $a_{\text{min}}$ denote the smallest distance (measured along a straight line in space) between any two of these points. What is the maximum value for $a_{\text{min}}$ , taken over all arrangements of the five points?

Let $D$ be a point inside a triangle $ABC$ such that $|AC| -|AD| \geq 1$ and $|BC|- |BD| \geq 1.$ Prove that for any point $E$ on the segment $AB$ , we have $|EC| -|ED| \geq 1.$

Let $x \neq 0$ be a real number satisfying $ax^2+bx+c=0$ with $a,b,c \in \mathbb{Z}$ obeying $|a|+|b|+|c| > 1$ . Then prove $$|x| \geq \frac{1}{|a|+|b|+|c|-1}.$$

Let a circle through $B$ and $C$ of a triangle $ABC$ intersect $AB$ and $AC$ in $Y$ and $Z$ , respectively. Let $P$ be the intersection of $BZ$ and $CY$ , and let $X$ be the intersection of $AP$ and $BC$ . Let $M$ be the point that is distinct from $X$ and on the intersection of the circumcircle of the triangle $XYZ$ with $BC$ . Prove that $M$ is the midpoint of $BC$