Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$

Each positive integer should be colored red or green in such a way that the following two conditions are met: - Let $n$ be any red number. The sum of any $n$ (not necessarily different) red numbers is red. - Let $m$ be any green number. The sum of any $m$ (not necessarily different) green numbers is green. Determine all such colorings.

The $ABCDE$ pentagon is inscribed in a circle and $AB = BC = CD$. Segments $AC$ and $BE$ intersect at $K$, and Segments $AD$ and $CE$ intersect at point$ L$. Prove that $AK = KL$.

Determine the largest two-digit number $d$ with the following property: for any six-digit number $\overline{aabbcc}$ number $d$ is a divisor of the number $\overline{aabbcc}$ if and only if the number $d$ is a divisor of the corresponding three-digit number $\overline{abc}$. Note The numbers $a \ne 0, b$ and $c$ need not be different.

Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Point $P$ lies on the segment $AB$, and points $S_1$ and $S_2$ are the centers of the circumcircles of $APC$ and $BPC$, respectively. Show that the midpoint of segment $S_1S_2$ lies on the perpendicular bisector of segment $CM$.

Decide whether there are infinitely many primes $p$ having a multiple in the form $n^2 + n + 1$ for some natural number $n$

Find all natural numbers $n$ such that the sum of the three largest divisors of $n$ is $1457$.

In a certain group there are $n \ge 5$ people, with every two people who do not know each other exactly having one mutual friend and no one knows everyone else. Prove $5$ of $n$ people, may sit at a circle around the table so that each of them sits between a) friends, b) strangers.

Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle ABC =\angle BCD > 90^o$. The circle circumscribed around the triangle $ABC$ intersects the sides $AD$ and $CD$ at points $K$ and $L$, respectively, different from any vertex of the quadrilateral $ABCD$ . Segments $AL$ and $CK$ intersect at point $P$. Prove that $\angle ADB =\angle PDC$.

Let $a, b, c$ be positive real numbers for which $ab + ac + bc \ge a + b + c$. Prove that $a + b + c \ge 3$.

There is a square $ABCD$ in the plane with $|AB|=a$. Determine the smallest possible radius value of three equal circles to cover a given square.