Find all pairs of real numbers $(x, y)$ for which the inequality $y^2 + y + \sqrt{y - x^2 -xy} \le 3xy$ holds.

Let $\omega_1$ and $\omega_2$ be two circles , with respective centres $O_1$ and $O_2$ , that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\omega_2$ in $A$ and $C$ and the line $O_2A$ inetersects $\omega_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\omega_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.

A table consists of $17 \times 17$ squares. In each square one positive integer from $1$ to $17$ is written, every such number is written in exactly $17$ squares. Prove that there is a row or a column of the table that contains at least $5$ different numbers.

We denote by gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd$(4, 6, 8)=2$ and gcd $(12, 15)=3$.) Suppose that positive integers $a, b, c$ satisfy the following four conditions: $\bullet$ gcd $(a, b, c)=1$, $\bullet$ gcd $(a, b + c)>1$, $\bullet$ gcd $(b, c + a)>1$, $\bullet$ gcd $(c, a + b)>1$. a) Is it possible that $a + b + c = 2015$? b) Determine the minimum possible value that the sum $a+ b+ c$ can take.