OBM 2013

Let $\Gamma$ be a circle and $A$ a point outside $\Gamma$ . The tangent lines to $\Gamma$ through $A$ touch $\Gamma$ at $B$ and $C$ . Let $M$ be the midpoint of $AB$ . The segment $MC$ meets $\Gamma$ again at $D$ and the line $AD$ meets $\Gamma$ again at $E$ . Given that $AB=a$ , $BC=b$ , compute $CE$ in terms of $a$ and $b$ .

Arnaldo and Bernaldo play the following game: given a fixed finite set of positive integers $A$ known by both players, Arnaldo picks a number $a \in A$ but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer $b$ (not necessarily in $A$ ). Then Arnaldo tells the number of divisors of $ab$ . Show that Bernaldo can choose $b$ in a way that he can find out the number $a$ chosen by Arnaldo.

Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^*$ from the non-zero reals to the non-zero reals, such that $$f(x+y) \left(f(x) + f(y)\right) = f(xy)$$ for all non-zero reals $x, y$ such that $x+y \neq 0$ .

Find the largest $n$ for which there exists a sequence $(a_0, a_1, \ldots, a_n)$ of non-zero digits such that, for each $k$ , $1 \le k \le n$ , the $k$ -digit number $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$ divides the $(k+1)$ -digit number $\overline{a_{k} a_{k-1}a_{k-2} \ldots a_0}$ .

P.S.: This is basically the same problem as http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=548550.

Let $x$ be an irrational number between 0 and 1 and $x = 0.a_1a_2a_3\cdots$ its decimal representation. For each $k \ge 1$ , let $p(k)$ denote the number of distinct sequences $a_{j+1} a_{j+2} \cdots a_{j+k}$ of $k$ consecutive digits in the decimal representation of $x$ . Prove that $p(k) \ge k+1$ for every positive integer $k$ .

The incircle of triangle $ABC$ touches sides $BC, CA$ and $AB$ at points $D, E$ and $F$ , respectively. Let $P$ be the intersection of lines $AD$ and $BE$ . The reflections of $P$ with respect to $EF, FD$ and $DE$ are $X,Y$ and $Z$ , respectively. Prove that lines $AX, BY$ and $CZ$ are concurrent at a point on line $IO$ , where $I$ and $O$ are the incenter and circumcenter of triangle $ABC$ .