OME 2012

Determine if the number $\lambda_n=\sqrt{3n^2+2n+2}$ is irrational for all non-negative integers $n$ .

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

Let $x$ and $n$ be integers such that $1\le x\le n$ . We have $x+1$ separate boxes and $n-x$ identical balls. Define $f(n,x)$ as the number of ways that the $n-x$ balls can be distributed into the $x+1$ boxes. Let $p$ be a prime number. Find the integers $n$ greater than $1$ such that the prime number $p$ is a divisor of $f(n,x)$ for all $x\in\{1,2,\ldots ,n-1\}$ .

Find all positive integers $n$ and $k$ such that $(n+1)^n=2n^k+3n+1$ .

A sequence $(a_n)_{n\ge 1}$ of integers is defined by the recurrence $$a_1=1,\ a_2=5,\ a_n=\frac{a_{n-1}^2+4}{a_{n-2}}\ \text{for}\ n\ge 2.$$ Prove that all terms of the sequence are integers and find an explicit formula for $a_n$ .

Let $ABC$ be an acute-angled triangle. Let $\omega$ be the inscribed circle with centre $I$ , $\Omega$ be the circumscribed circle with centre $O$ and $M$ be the midpoint of the altitude $AH$ where $H$ lies on $BC$ . The circle $\omega$ be tangent to the side $BC$ at the point $D$ . The line $MD$ cuts $\omega$ at a second point $P$ and the perpendicular from $I$ to $MD$ cuts $BC$ at $N$ . The lines $NR$ and $NS$ are tangent to the circle $\Omega$ at $R$ and $S$ respectively. Prove that the points $R,P,D$ and $S$ lie on the same circle.