KURSCHAK 2011

Let $a_1, a_2,...$ be an infinite sequence of positive integers such that for any $k,\ell\in \mathbb{Z_+}$ , $a_{k+\ell}$ is divisible by $\gcd(a_k,a_\ell)$ . Prove that for any integers $1\leqslant k\leqslant n$ , $a_na_{n-1}\dots a_{n-k+1}$ is divisible by $a_ka_{k-1}\dots a_1$ .

Let $n$ be a positive integer. Denote by $a(n)$ the ways of expression $n=x_1+x_2+\dots$ where $x_1\leqslant x_2 \leqslant\dots$ are positive integers and $x_i+1$ is a power of $2$ for each $i$ . Denote by $b(n)$ the ways of expression $n=y_1+y_2+\dots$ where $y_i$ is a positive integer and $2y_i\leqslant y_{i+1}$ for each $i$ . Prove that $a(n)=b(n)$ .

Given $2n$ points and $3n$ lines on the plane. Prove that there is a point $P$ on the plane such that the sum of the distances of $P$ to the $3n$ lines is less than the sum of the distances of $P$ to the $2n$ points.