AUSTRIAN-POLISH 1980

A sequence of integers $1 = x_1 < x_2 < x_3 <...$ satisfies $x_{n+1} \le 2n$ for all $n$ . Show that every positive integer $k$ can be written as $x_j -x_i$ for some $i, j$ .

Let $a_1,a_2,a_3,\dots$ be a sequence of real numbers satisfying the inequality $$|a_{k+m}-a_k-a_m| \leq 1 \text{for all} \ k,m \in \mathbb{Z}_{>0}.$$ Show that the following inequality holds for all positive integers $k,m$ $$\left| \frac{a_k}{k}-\frac{a_m}{m} \right| < \frac{1}{k}+\frac{1}{m}.$$

Through the endpoints $A$ and $B$ of a diameter $AB$ of a given circle, the tangents $\ell$ and $m$ have been drawn. Let $C\ne A$ be a point on $\ell$ and let $q_1,q_2$ be two rays from $C$ . Ray $q_i$ cuts the circle in $D_i$ and $E_i$ with $D_i$ between $C$ and $E_i, i = 1,2$ . Rays $AD_1,AD_2,AE_1,AE_2$ meet $m$ in the respective points $M_1,M_2,N_1,N_2$ . Prove that $M_1M_2 = N_1N_2$ .