OIM 2018

For each integer $n \ge 2$ , find all integer solutions of the following system of equations:

$$x_1 = (x_2 + x_3 + x_4 + ... + x_n)^{2018}$$ $$x_2 = (x_1 + x_3 + x_4 + ... + x_n)^{2018}$$ $$\vdots$$ $$x_n = (x_1 + x_2 + x_3 + ... + x_{n - 1})^{2018}$$

Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$ and $AB = AC$ . Let $M$ be the midpoint of $BC$ . A point $D \neq A$ is chosen on the semicircle with diameter $BC$ that contains $A$ . The circumcircle of triangle $DAM$ cuts lines $DB$ and $DC$ at $E$ and $F$ respectively. Show that $BE = CF$ .

In a plane we have $n$ lines, no two of which are parallel or perpendicular, and no three of which are concurrent. A cartesian system of coordinates is chosen for the plane with one of the lines as the $x$ -axis. A point $P$ is located at the origin of the coordinate system and starts moving along the positive $x$ -axis with constant velocity. Whenever $P$ reaches the intersection of two lines, it continues along the line it just reached in the direction that increases its $x$ -coordinate. Show that it is possible to choose the system of coordinates in such a way that $P$ visits points from all $n$ lines.

A set $X$ of positive integers is said to be iberic if $X$ is a subset of $\{2, 3, \dots, 2018\}$ , and whenever $m, n$ are both in $X$ , $\gcd(m, n)$ is also in $X$ . An iberic set is said to be olympic if it is not properly contained in any other iberic set. Find all olympic iberic sets that contain the number $33$ .

Let $n$ be a positive integer. For a permutation $a_1, a_2, \dots, a_n$ of the numbers $1, 2, \dots, n$ we define $$b_k = \min_{1 \leq i \leq k} a_i + \max_{1 \leq j \leq k} a_j$$ We say that the permutation $a_1, a_2, \dots, a_n$ is guadiana if the sequence $b_1, b_2, \dots, b_n$ does not contain two consecutive equal terms. How many guadiana permutations exist?

Let $ABC$ be an acute triangle with $AC > AB > BC$ . The perpendicular bisectors of $AC$ and $AB$ cut line $BC$ at $D$ and $E$ respectively. Let $P$ and $Q$ be points on lines $AC$ and $AB$ respectively, both different from $A$ , such that $AB = BP$ and $AC = CQ$ , and let $K$ be the intersection of lines $EP$ and $DQ$ . Let $M$ be the midpoint of $BC$ . Show that $\angle DKA = \angle EKM$ .