1. Let $n$ and $k$ be two positive integers such that $1\leq n \leq k$. Prove that, if $d^k+k$ is a prime number for each positive divisor $d$ of $n$, then $n+k$ is a prime number.
  2. Given $x_1,x_2,...,x_n$ real numbers, prove that there exists a real number $y$, such that $$\{y-x_1\}+\{y-x_2\}+...+\{y-x_n\} \leq \frac{n-1}{2}$$
  3. Let $I$ be the incenter of the scalene $\Delta ABC$ such that $AB<AC$, and let $I'$ be the reflection of point $I$ in line $BC$. The angle bisector $AI$ meets $BC$ at $D$ and circumcircle of $\Delta ABC$ at $E$. The line $EI'$ meets the circumcircle at $F$. Prove that
    a) $\dfrac{AI}{IE}=\dfrac{ID}{DE}$.
    b) $IA=IF$.
  4. The sides of an equilateral triangle are divided into n equal parts by $n-1$ points on each side. Through these points one draws parallel lines to the sides of the triangle. Thus, the initial triangle is divides into $n^2$ equal equilateral triangles. In every vertex of such a triangle there is a beetle. The beetles start crawling simultaneously, with equal speed, along the sides of the small triangles. When they reach a vertex, the beetles change the direction of their movement by $60^{\circ}$ or by $120^{\circ}$.
    a) Prove that if $n \geq 7$ then the beetles can move indefinitely on the sides of the small triangles without two beetles ever meeting in a vertex of a small triangle.
    b) Determine all the values of $n \geq 1$ for which the beetles can move along the sides of the small triangles without meeting in their vertices.

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