IZhO 2014

Points $M, N, K$ lie on the sides $B C, C A, A B$ respectively, of a triangle $A B C$, and are different from its vertices. The triangle $M N K$ is called beautiful if the triangles $M N K$ and $A B C$ are similar (with the vertices respectively in this order). Show that if in the triangle $A B C$ there are two beautiful triangles with a common vertex, then $\triangle A B C$ is right-angled.

Do there exist? functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that a) $f$ is a surjective function; and b) $f(f(x))=(x-1) f(x)+2$ for all $x$ real.

There are given 100 distinct positive integers. We call a pair of integers among them good if the ratio of its elements is either 2 or 3 . What is the maximum number $g$ of good pairs that these 100 numbers can form? (A same number can be used in several pairs.)

Does it exist? a polynomial $P(x)$ with integer coefficients, such that $P(1+\sqrt{3})=2+\sqrt{3}$ and $P(3+\sqrt{5})=3+\sqrt{5}$.

Let $U=\{1,2,3, \ldots, 2014\}$. For all $a, b, c \in \mathbb{N}$ let $f(a, b, c)$ be the number of ordered sextuplets $\left(X_{1}, X_{2}, X_{3}, Y_{1}, Y_{2}, Y_{3}\right)$ of subsets of $U$, satisfying the following conditions

(i) $Y_{1} \subseteq X_{1} \subseteq U$ and $\left|X_{1}\right|=a$;

(ii) $Y_{2} \subseteq X_{2} \subseteq U \backslash Y_{1}$ and $\left|X_{2}\right|=b$;

(iii) $Y_{3} \subseteq X_{3} \subseteq U \backslash\left(Y_{1} \cup Y_{2}\right)$ and $\left|X_{3}\right|=c$.

Prove $f(\sigma(a), \sigma(b), \sigma(c))$ does not change, for permutations $\sigma$ of $a, b, c$.

Let $A B C D$ be a convex quadrilateral, partitioned by four lines as in the picture below, such that the meeting points of these lines all lie on the diagonals of $A B C D$. Prove that if the corner quadrilaterals 1, 2,3 and the center quadrilateral 4 are all tangential, then the corner quadrilateral 5 is also tangential.