PL-OM 1992

Segments $AC$ and $BD$ meet at $P$ , and $|PA| = |PD|$ , $|PB| = |PC|$ . $O$ is the circumcenter of the triangle $PAB$ . Show that $OP$ and $CD$ are perpendicular.

Find all functions $f : \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ , where $\mathbb{Q}^{+}$ is the set of positive rationals, such that $f(x+1) = f(x) + 1$ and $f(x^3) = f(x)^3$ for all $x$ .

Show that for real numbers $x_1, x_2, ... , x_n$ we have: $$\sum\limits_{i=1}^n \sum\limits_{j=1}^n \dfrac{x_ix_j}{i+j} \geq 0$$ When do we have equality?

The functions $f_0, f_1, f_2, ...$ are defined on the reals by $f_0(x) = 8$ for all $x$ , $f_{n+1}(x) = \sqrt{x^2 + 6f_n(x)}$ . For all $n$ solve the equation $f_n(x) = 2x$ .

The base of a regular pyramid is a regular $2n$ -gon $A_1A_2...A_{2n}$ . A sphere passing through the top vertex $S$ of the pyramid cuts the edge $SA_i$ at $B_i$ (for $i = 1, 2, ... , 2n$ ). Show that $\sum\limits_{i=1}^n SB_{2i-1} = \sum\limits_{i=1}^n SB_{2i}$ .

Show that $(k^3)!$ is divisible by $(k!)^{k^2+k+1}$ .