ÖMO 2011

Every brick has $5$ holes in a line. The holes can be filled with bolts (fitting in one hole) and braces (fitting into two neighboring holes). No hole may remain free. One puts $n$ of these bricks in a line to form a pattern from left to right. In this line no two braces and no three bolts may be adjacent. How many different such patterns can be produced with $n$ bricks?

We consider permutations $f$ of the set $\mathbb{N}$ of non-negative integers, i.e. bijective maps $f$ from $\mathbb{N}$ to $\mathbb{N}$ , with the following additional properties: $$f(f(x)) = x \text{and} \left| f(x)-x\right| \leqslant 3 \text{for all } x \in\mathbb{N}\text{.}$$ Further, for all integers $n > 42$ , $$\left.M(n)=\frac{1}{n+1}\sum_{j=0}^n \left|f(j)-j\right|<2,011\text{.}\right.$$ Show that there are infinitely many natural numbers $K$ such that $f$ maps the set $$\left\{ n\mid 0\leqslant n\leqslant K\right\}$$ onto itself.

We are given a non-isosceles triangle $ABC$ with incenter $I$ . Show that the circumcircle $k$ of the triangle $AIB$ does not touch the lines $CA$ and $CB$ . Let $P$ be the second point of intersection of $k$ with $CA$ and let $Q$ be the second point of intersection of $k$ with $CB$ . Show that the four points $A$ , $B$ , $P$ and $Q$ (not necessarily in this order) are the vertices of a trapezoid.

Determine all pairs $(a,b)$ of non-negative integers, such that $a^b+b$ divides $a^{2b}+2b$ . (Remark: $0^0=1$ .)

Let $k$ and $n$ be positive integers. Show that if $x_j$ ( $1\leqslant j\leqslant n$ ) are real numbers with $\sum_{j=1}^n\frac{1}{x_j^{2^k}+k}=\frac{1}{k}$ , then $$\sum_{j=1}^n\frac{1}{x_j^{2^{k+1}}+k+2}\leqslant\frac{1}{k+1}\text{.}$$

Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ touch each outside at point $Q$ . The other endpoints of the diameters through $Q$ are $P$ on $k_1$ and $R$ on $k_2$ . We choose two points $A$ and $B$ , one on each of the arcs $PQ$ of $k_1$ . ( $PBQA$ is a convex quadrangle.) Further, let $C$ be the second point of intersection of the line $AQ$ with $k_2$ and let $D$ be the second point of intersection of the line $BQ$ with $k_2$ . The lines $PB$ and $RC$ intersect in $U$ and the lines $PA$ and $RD$ intersect in $V$ . Show that there is a point $Z$ that lies on all of these lines $UV$ .