Given that $ \{a_n\}$ is a sequence in which all the terms are integers, and $ a_2$ is odd. For any natural number $ n$, $ n(a_{n + 1} - a_n + 3) = a_{n + 1} + a_n + 3$. Furthermore, $ a_{2009}$ is divisible by $ 2010$. Find the smallest integer $ n > 1$ such that $ a_n$ is divisible by $ 2010$. P.S.: I saw EVEN instead of ODD. Got only half of the points.

There are $ n$ points on the plane, no three of which are collinear. Each pair of points is joined by a red, yellow or green line. For any three points, the sides of the triangle they form consist of exactly two colours. Show that $ n<13$.

Let $ \triangle ABC$ be a right-angled triangle with $ \angle C=90^\circ$. $ CD$ is the altitude from $ C$ to $ AB$, with $ D$ on $ AB$. $ \omega$ is the circumcircle of $ \triangle BCD$. $ \omega_1$ is a circle situated in $ \triangle ACD$, it is tangent to the segments $ AD$ and $ AC$ at $ M$ and $ N$ respectively, and is also tangent to circle $ \omega$. (i) Show that $ BD\cdot CN+BC\cdot DM=CD\cdot BM$. (ii) Show that $ BM=BC$.

Find all non-negative integers $ m$ and $ n$ that satisfy the equation: \[ 107^{56}(m^2-1)-2m+5=3\binom{113^{114}}{n}\] (If $ n$ and $ r$ are non-negative integers satisfying $ r\le n$, then $ \binom{n}{r}=\frac{n}{r!(n-r)!}$ and $ \binom{n}{r}=0$ if $ r>n$.)