NL-BXMO-TST 2017

Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine the minimum number of integers in a complete sequence of $n$ numbers.

Let define a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that : $i)$ $f(p)=1$ for all prime numbers $p$ . $ii)$ $f(xy)=xf(y)+yf(x)$ for all positive integers $x,y$ find the smallest $n \geq 2016$ such that $f(n)=n$

Let $ABC$ be a triangle with $\angle A = 90$ and let $D$ be the orthogonal projection of $A$ onto $BC$ . The midpoints of $AD$ and $AC$ are called $E$ and $F$ , respectively. Let $M$ be the circumcentre of $BEF$ . Prove that $AC$ and $BM$ are parallel.

A quadruple $(a; b; c; d)$ of positive integers with $a \leq b \leq c \leq d$ is called good if we can colour each integer red, blue, green or purple, in such a way that $i$ of each $a$ consecutive integers at least one is coloured red; $ii$ of each $b$ consecutive integers at least one is coloured blue; $iii$ of each $c$ consecutive integers at least one is coloured green; $iiii$ of each $d$ consecutive integers at least one is coloured purple. Determine all good quadruples with $a = 2.$

Determine all pairs of prime numbers $(p; q)$ such that $p^2 + 5pq + 4q^2$ is the square of an integer.