TR-MO 2018

Find all pairs $(x,y)$ of real numbers that satisfy, \begin{align*} x^2+y^2+x+y &= xy(x+y)-\frac{10}{27} |xy| & \leq \frac{25}{9}. \end{align*}

Let $P$ be a point in the interior of the triangle $ABC$ . The lines $AP$ , $BP$ , and $CP$ intersect the sides $BC$ , $CA$ , and $AB$ at $D,E$ , and $F$ , respectively. A point $Q$ is taken on the ray $[BE$ such that $E\in [BQ]$ and $m(\widehat{EDQ})=m(\widehat{BDF})$ . If $BE$ and $AD$ are perpendicular, and $|DQ|=2|BD|$ , prove that $m(\widehat{FDE})=60^\circ$ .

A sequence $a_1,a_2,\dots$ satisfy $$\sum_{i =1}^n a_{\lfloor \frac{n}{i}\rfloor }=n^{10},$$ for every $n\in\mathbb{N}$ . Let $c$ be a positive integer. Prove that, for every positive integer $n$ , $$\frac{c^{a_n}-c^{a_{n-1}}}{n}$$ is an integer.

In a triangle $ABC$ , the bisector of the angle $A$ intersects the excircle that is tangential to side $[BC]$ at two points $D$ and $E$ such that $D\in [AE]$ . Prove that, $$\frac{|AD|}{|AE|}\leq \frac{|BC|^2}{|DE|^2}.$$

Let $a_1,a_2,a_3,a_4$ be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let $i,j,k\in\{1,2,3,4\}$ with $i \neq j$ , $j\neq k$ , and $k\neq i$ . Determine the maximum number of triples $(i,j,k)$ for which $$({\rm gcd}(a_i,a_j))^2|a_k.$$

Initially, there are 2018 distinct boxes on a table. In the first stage, Yazan and Bozan, starting with Yazan, take turns make $2016$ moves each, such that, in each move, the person whose turn selects a pair of boxes that is not written on the board, and writes the pair on the board.

In the second stage, Bozan enumerates the $4032$ pairs with numbers from $1,2,\dots,4032$ , in whichever order he wants, and puts $k$ balls in each boxes written contained in the $k^{th}$ pair. Is there a strategy for Bozan that guarantees that the number of balls in each box are distinct?