1. 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.
2. 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 $x$.
3. 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?
4. The functions $f_0, f_1, f_2, ...$ are defined on the reals by $$f_0(x) = 8,\quad f_{n+1}(x) = \sqrt{x^2 + 6f_n(x)},\,\forall x.$$ For all $n$ solve the equation $f_n(x) = 2x$.
5. 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 $$\sum\limits_{i=1}^n SB_{2i-1} = \sum\limits_{i=1}^n SB_{2i}.$$
6. Show that $(k^3)!$ is divisible by $(k!)^{k^2+k+1}$.
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