Let $p,q$ be prime numbers such that their sum isn't divisible by $3$. Find the all $(p,q,r,n)$ positive integer quadruples satisfy: $$p+q=r(p-q)^n$$ Proposed by Şahin Emrah

Let $ABCD$ be a convex quadrilateral and let $\omega$ be a circle tangent to the lines $AB$ and $BC$ at points $A$ and $C$, respectively. $\omega$ intersects the line segments $AD$ and $CD$ again at $E$ and $F$, respectively, which are both different from $D$. Let $G$ be the point of intersection of the lines $AF$ and $CE$. Given $\angle ACB=\angle GDC+\angle ACE$, prove that the line $AD$ is tangent to th circumcircle of the triangle $AGB$.

In a country consisting of $2015$ cities, between any two cities there is exactly one direct round flight operated by some air company. Find the minimal possible number of air companies if direct flights between any three cities are operated by three different air companies.

Prove that $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \ge \dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+2(a+b+c)$$for the all $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2+2abc \le 1$.

A quadratic number is a real root of the equations $ax^2 + bx + c = 0$ where $|a|,|b|,|c|\in\{1,2,\ldots,10\}$. Find the smallest positive integer $n$ for which at least one of the intervals$$\left(n-\dfrac{1}{3}, n\right)\quad \text{and}\quad\left(n, n+\dfrac{1}{3}\right)$$does not contain any quadratic number.

Find the greatest possible integer value of the side length of an equilateral triangle whose vertices belong to the interior region of a square with side length $100$.

For the all $(m,n,k)$ positive integer triples such that $|m^k-n!| \le n$ find the maximum value of $\frac{n}{m}$ Proposed by Melih Üçer

A coloring of all plane points with coordinates belonging to the set $S=\{0,1,\ldots,99\}$ into red and white colors is said to be critical if for each $i,j\in S$ at least one of the four points $(i,j),(i + 1,j),(i,j + 1)$ and $(i + 1, j + 1)$ $(99 + 1\equiv0)$ is colored red. Find the maximal possible number of red points in a critical coloring which loses its property after recoloring of any red point into white.