CMO 2010

For a positive integer $n$, an $n$-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to $n$ squares in the $n^{\text {th }}$ row, such that all the left-most squares in each row are aligned vertically. For example, the 5 -staircase is shown below.

Let $f(n)$ denote the minimum number of square tiles required to tile the $n$ staircase, where the side lengths of the square tiles can be any positive integer. For example, $f(2)=3$ and $f(4)=7$.

(a) Find all $n$ such that $f(n)=n$.

(b) Find all $n$ such that $f(n)=n+1$.

Let $A, B, P$ be three points on a circle. Prove that if $a$ and $b$ are the distances from $P$ to the tangents at $A$ and $B$ and $c$ is the distance from $P$ to the chord $A B$, then $c^{2}=a b$.

Three speed skaters have a friendly race on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does 1 lap a minute, the fastest one does 3.14 laps a minute, and the middle one does $L$ laps a minute for some $1<L<3.14$. The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting

Each vertex of a finite graph can be colored either black or white. Initially all vertices are black. We are allowed to pick a vertex $\mathrm{P}$ and change the color of $P$ and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a sequence of operations of this type?

Let $P(x)$ and $Q(x)$ be polynomials with integer coefficients. Let $a_{n}=n !+n$. Show that if $P\left(a_{n}\right) / Q\left(a_{n}\right)$ is an integer for every $n$, then $P(n) / Q(n)$ is an integer for every integer $n$ such that $Q(n) \neq 0$.