OIM 2015

The number $125$ can be written as a sum of some pairwise coprime integers larger than $1$ . Determine the largest number of terms that the sum may have.

A line $r$ contains the points $A$ , $B$ , $C$ , $D$ in that order. Let $P$ be a point not in $r$ such that $\angle{APB} = \angle{CPD}$ . Prove that the angle bisector of $\angle{APD}$ intersects the line $r$ at a point $G$ such that: $\frac{1}{GA} + \frac{1}{GC} = \frac{1}{GB} + \frac{1}{GD}$

Let $\alpha$ and $\beta$ be the roots of $x^{2} - qx + 1$ , where $q$ is a rational number larger than $2$ . Let $s_1 = \alpha + \beta$ , $t_1 = 1$ , and for all integers $n \geq 2$ : $s_n = \alpha^n + \beta^n$ $t_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n$ Prove that, for all odd integers $n$ , $t_n$ is the square of a rational number.

Let $ABC$ be an acute triangle and let $D$ be the foot of the perpendicular from $A$ to side $BC$ . Let $P$ be a point on segment $AD$ . Lines $BP$ and $CP$ intersect sides $AC$ and $AB$ at $E$ and $F$ , respectively. Let $J$ and $K$ be the feet of the peroendiculars from $E$ and $F$ to $AD$ , respectively. Show that $\frac{FK}{KD}=\frac{EJ}{JD}$ .

Find all pairs of integers $(a,b)$ such that $(b^2+7(a-b))^2=a^{3}b$ .

Beto plays the following game with his computer: initially the computer randomly picks $30$ integers from $1$ to $2015$ , and Beto writes them on a chalkboard (there may be repeated numbers). On each turn, Beto chooses a positive integer $k$ and some if the numbers written on the chalkboard, and subtracts $k$ from each of the chosen numbers, with the condition that the resulting numbers remain non-negative. The objective of the game is to reduce all $30$ numbers to $0$ , in which case the game ends. Find the minimal number $n$ such that, regardless of which numbers the computer chooses, Beto can end the game in at most $n$ turns.