CH-SMO 2010

Three coins lie on integer points on the number line. A move consists of choosing and moving two coins, the first one $1$ unit to the right and the second one $1$ unit to the left. Under which initial conditions is it possible to move all coins to one single point?

Let $\triangle{ABC}$ be a triangle with $AB\not=AC$ . The incircle with centre $I$ touches $BC$ , $CA$ , $AB$ at $D$ , $E$ , $F$ , respectively. Furthermore let $M$ the midpoint of $EF$ and $AD$ intersect the incircle at $P\not=D$ . Show that $PMID$ ist cyclic.

For $n\in\mathbb{N}$ , determine the number of natural solutions $(a,b)$ such that $$(4a-b)(4b-a)=2010^n$$ holds.

Let $x$ , $y$ , $z \in\mathbb{R}^+$ satisfying $xyz = 1$ . Prove that $$\frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\text{.}$$

Some sides and diagonals of a regular $n$ -gon form a connected path that visits each vertex exactly once. A parallel pair of edges is a pair of two different parallel edges of the path. Prove that (a) if $n$ is even, there is at least one parallel pair. (b) if $n$ is odd, there can't be one single parallel pair.

Find all functions $f: \mathbb{R}\mapsto\mathbb{R}$ such that for all $x$ , $y$ $\in\mathbb{R}$ , $$f(f(x))+f(f(y))=2y+f(x-y)$$ holds.

Let $m$ , $n$ be natural numbers such that $m+n+1$ is prime and divides $2(m^2+n^2)-1$ . Prove that $m=n$ .

In a village with at least one inhabitant, there are several associations. Each inhabitant is a member of at least $k$ associations, and any two associations have at most one common member. Prove that at least $k$ associations have the same number of members.

Let $k$ and $k'$ two concentric circles centered at $O$ , with $k'$ being larger than $k$ . A line through $O$ intersects $k$ at $A$ and $k'$ at $B$ such that $O$ seperates $A$ and $B$ . Another line through $O$ intersects $k$ at $E$ and $k'$ at $F$ such that $E$ separates $O$ and $F$ . Show that the circumcircle of $\triangle{OAE}$ and the circles with diametres $AB$ and $EF$ have a common point.

Let $n\geqslant 3$ and $P$ a convex $n$ -gon. Show that $P$ can be, by $n - 3$ non-intersecting diagonals, partitioned in triangles such that the circumcircle of each triangle contains the whole area of $P$ . Under which conditions is there exactly one such triangulation?