Let $n$ be a given positive integer. Prove that there is no positive divisor $d$ of $2n^2$ such that $d^2n^2+d^3$ is a square of an integer.

Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$

Let $k$,$n$ be positive integers, $k,n>1$, $k<n$ and a $n \times n$ grid of unit squares is given. Ana and Maya take turns in coloring the grid in the following way: in each turn, a unit square is colored black in such a way that no two black cells have a common side or vertex. Find the smallest positive integer $n$ , such that they can obtain a configuration in which each row and column contains exactly $k$ black cells. Draw one example.

Let $ABC$ be a triangle such that $\measuredangle BAC = 60^{\circ}$. Let $D$ and $E$ be the feet of the perpendicular from $A$ to the bisectors of the external angles of $B$ and $C$ in triangle $ABC$, respectively. Let $O$ be the circumcenter of the triangle $ABC$. Prove that circumcircle of the triangle $BOC$ has exactly one point in common with the circumcircle of $ADE$.

Find all such functions $f:\mathbb{R} \to \mathbb{R}$ that for any real $x,y$ the following equation is true. $$f(f(x)+y)+1=f(x^2+y)+2f(x)+2y$$

Given a paper on which the numbers $1,2,3\dots ,14,15$ are written. Andy and Bobby are bored and perform the following operations, Andy chooses any two numbers (say $x$ and $y$) on the paper, erases them, and writes the sum of the numbers on the initial paper. Meanwhile, Bobby writes the value of $xy(x+y)$ in his book. They were so bored that they both performed the operation until only $1$ number remained. Then Bobby adds up all the numbers he wrote in his book, let’s call $k$ as the sum. $a$. Prove that $k$ is constant which means it does not matter how they perform the operation, $b$. Find the value of $k$.

Find all positive integers $n$ such that for all positive integers $m$, $1<m<n$, relatively prime to $n$, $m$ must be a prime number.