VN-TST 2005

Let $(I),(O)$ be the incircle, and, respectiely, circumcircle of $ABC$ . $(I)$ touches $BC,CA,AB$ in $D,E,F$ respectively. We are also given three circles $\omega_a,\omega_b,\omega_c$ , tangent to $(I),(O)$ in $D,K$ (for $\omega_a$ ), $E,M$ (for $\omega_b$ ), and $F,N$ (for $\omega_c$ ).

a) Show that $DK,EM,FN$ are concurrent in a point $P$ ;

b) Show that the orthocenter of $DEF$ lies on $OP$ .

Given $n$ chairs around a circle which are marked with numbers from 1 to $n$ .There are $k$ , $k \leq 4 \cdot n$ students sitting on those chairs .Two students are called neighbours if there is no student sitting between them. Between two neighbours students ,there are at less 3 chairs. Find the number of choices of $k$ chairs so that $k$ students can sit on those and the condition is satisfied.

Find all functions $f: \mathbb{Z} \mapsto \mathbb{Z}$ satisfying the condition: $f(x^3 +y^3 +z^3 )=f(x)^3+f(y)^3+f(z)^3.$

Let be given positive reals $a$ , $b$ , $c$ . Prove that: $\frac{a^{3}}{\left(a+b\right)^{3}}+\frac{b^{3}}{\left(b+c\right)^{3}}+\frac{c^{3}}{\left(c+a\right)^{3}}\geq \frac{3}{8}$ .

Let $p\in \mathbb P,p>3$ . Calcute:

a) $S=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{2k^2}{p}\right]-2 \cdot \left[\frac{k^2}{p}\right]$ if $p\equiv 1 \mod 4$

b) $T=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{k^2}{p}\right]$ if $p\equiv 1 \mod 8$

$n$ is called diamond 2005 if $n=\overline{...ab999...99999cd...}$ , e.g. $2005 \times 9$ . Let $\{a_n\}:a_n< C\cdot n,\{a_n\}$ is increasing. Prove that $\{a_n\}$ contain infinite diamond 2005.

Compare with this problem.