PL-OM 1984

Find the number of all real functions $f$ which map the sum of $n$ elements into the sum of their images, such that $f^{n-1}$ is a constant function and $f^{n-2}$ is not. Here $f^0(x) = x$ and $f^k = f \circ f^{k-1}$ for $k \ge 1$ .

Let $n$ be a positive integer. For all $i, j \in \{1,2,...,n\}$ define $a_{j,i} = 1$ if $j = i$ and $a_{j,i} = 0$ otherwise. Also, for $i = n+1,...,2n$ and $j = 1,...,n$ define $a_{j,i} = -\frac{1}{n}$ . Prove that for any permutation $p$ of the set $\{1,2,...,2n\}$ the following inequality holds: $\sum_{j=1}^{n}\left|\sum_{k=1}^{n} a_{j,p}(k)\right| \ge \frac{n}{2}$

Let $W$ be a regular octahedron and $O$ be its center. In a plane $P$ containing $O$ circles $k_1(O,r_1)$ and $k_2(O,r_2)$ are chosen so that $k_1 \subset P\cap W \subset k_2$ . Prove that $\frac{r_1}{r_2}\le \frac{\sqrt3}{2}$

A coin is tossed $n$ times, and the outcome is written in the form ( $a_1,a_2,...,a_n$ ), where $a_i = 1$ or $2$ depending on whether the result of the $i$ -th toss is the head or the tail, respectively. Set $b_j = a_1 +a_2 +...+a_j$ for $j = 1,2,...,n$ , and let $p(n)$ be the probability that the sequence $b_1,b_2,...,b_n$ contains the number $n$ . Express $p(n)$ in terms of $p(n-1)$ and $p(n-2)$ .

A regular hexagon of side $1$ is covered by six unit disks. Prove that none of the vertices of the hexagon is covered by two (or more) discs.

Cities $P_1,...,P_{1025}$ are connected to each other by airlines $A_1,...,A_{10}$ so that for any two distinct cities $P_k$ and $P_m$ there is an airline offering a direct flight between them. Prove that one of the airlines can offer a round trip with an odd number of flights.