PAMO 2019

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows:

• $a_0 = 3$ , $a_1 = 2$ , and $a_2 = 12$ ; and
• $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$ .

Show that $a_n$ is always a strictly positive integer.

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$ ?

Let $ABC$ be a triangle, and $D$ , $E$ , $F$ points on the segments $BC$ , $CA$ , and $AB$ respectively such that $$\frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}.$$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle.

The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$ . The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$ . Show that $AO$ is perpendicular to $EF$ .

A square is divided into $N^2$ equal smaller non-overlapping squares, where $N \geq 3$ . We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself).

• Show that it is possible to find a broken line composed of $4$ segments for $N = 3$ .
• Find the minimum number of segments in this broken line for arbitrary $N$ .

Find the $2019$ th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$ .