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*}
2018 Turkey MO (2nd Round)
Day 1
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.
Day 2
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?