VMO 2012

Let the real sequence $(x_n)$ be defined by $x_1=3$ and $x_n=\dfrac{n+2}{3n}(x_{n-1}+2)$ for every $n\ge2$. Prove that the sequence has a finite limit as $n\to+\infty$, and find that limit.

Let $(a_n)$ and $(b_n)$ be arithmetic progressions, and let $m\gt 2$ be an integer. Consider $P_k(x)=x^2+a_kx+b_k$, $k=1,2,\ldots,m$. Prove that if $P_1$ and $P_m$ have no real roots, then none of the remaining polynomials has a real root.

In the plane, let $ABCD$ be a convex quadrilateral inscribed in a circle with center $O$, with neither pair of opposite sides parallel. Let $M=AB\cap CD$ and $N=AD\cap BC$, where the intersections are of the supporting lines. Let $P,Q,S,T$ respectively be the intersections of the internal angle bisectors of the pairs $\angle MAN$ and $\angle MBN$; $\angle MBN$ and $\angle MCN$; $\angle MCN$ and $\angle MDN$; $\angle MDN$ and $\angle MAN$. Assume that $P,Q,S,T$ are pairwise distinct.
(1) Prove that $P,Q,S,T$ lie on one circle, and let $I$ be its center.
(2) Let $E=AC\cap BD$. Prove that $E,O,I$ are collinear.

Let $n$ be a positive integer. There are $n$ male students and $n$ female students standing in a row in an arbitrary order. Each student $X$ receives as many candies as there are ways to choose two students of the opposite sex to $X$, one on each side of $X$. Prove that the total number of candies received by all $2n$ students does not exceed $\dfrac13n(n^2-1)$.

A group consists of five girls, denoted by $G_1,G_2,G_3,G_4,G_5$, and twelve boys. Seventeen chairs are arranged in a row. Seat the group so that all the following conditions hold:
1. Every chair has exactly one occupant.
2. From left to right, the girls occur in the order $G_1,G_2,G_3,G_4,G_5$.
3. At least three boys sit between $G_1$ and $G_2$.
4. At least one and at most four boys sit between $G_4$ and $G_5$.
How many such arrangements are there? Two arrangements are different if some chair has different occupants in them.

Let $a,b$ be odd natural numbers such that $a$ divides $b^2+2$ and $b$ divides $a^2+2$. Prove that $a$ and $b$ are terms of the natural-number sequence $(v_n)$ defined by $v_1=v_2=1$ and $v_n=4v_{n-1}-v_{n-2}$ for every $n\ge3$.

Find all functions $f:\mathbb R\to\mathbb R$ satisfying all three conditions: (1) $f$ is surjective; (2) $f$ is increasing on $\mathbb R$; (3) $f(f(x))=f(x)+12x$ for every real $x$.