Let $s(n)$ be the number of positive integer divisors of $n$. Find the all positive values of $k$ that is providing $k=s(a)=s(b)=s(2a+3b)$.
2018 Turkey Junior National Olympiad
We are placing rooks on a $n \cdot n$ chess table that providing this condition: Every two rooks will threaten an empty square at least. What is the most number of rooks?
In an acute $ABC$ triangle which has a circumcircle center called $O$, there is a line that perpendiculars to $AO$ line cuts $[AB]$ and $[AC]$ respectively on $D$ and $E$ points. There is a point called $K$ that is different from $AO$ and $BC$'s junction point on $[BC]$. $AK$ line cuts the circumcircle of $ADE$ on $L$ that is different from $A$. $M$ is the symmetry point of $A$ according to $DE$ line. Prove that $K$,$L$,$M$,$O$ are circular.
For all $x,y,z$ positive real numbers, find the all $c$ positive real numbers that providing $$\frac{x^3y+y^3z+z^3x}{x+y+z}+\frac{4c}{xyz}\ge2c+2$$