Find all pairs of nonnegative integers $ (x, y)$ satisfying $ (14y)^x + y^{x+y} = 2007$.

Let $ABCD$ be a convex quadrilateral inscribed in a circle with $M$ and $N$ the midpoints of the diagonals $AC$ and $BD$ respectively. Suppose that $AC$ bisects $\angle BMD$. Prove that $BD$ bisects $\angle ANC$.

Let $ a_1, a_2,\ldots ,a_8$ be $8$ distinct points on the circumference of a circle such that no three chords, each joining a pair of the points, are concurrent. Every $4$ of the $8$ points form a quadrilateral which is called a quad. If two chords, each joining a pair of the $8$ points, intersect, the point of intersection is called a bullet. Suppose some of the bullets are coloured red. For each pair $(i j)$, with $ 1 \le i < j \le 8$, let $r(i,j)$ be the number of quads, each containing $ a_i, a_j$ as vertices, whose diagonals intersect at a red bullet. Determine the smallest positive integer $n$ such that it is possible to colour $n$ of the bullets red so that $r(i,j)$ is a constant for all pairs $(i,j)$.

Two circles $ (O_1)$ and $ (O_2)$ touch externally at the point $C$ and internally at the points $A$ and $B$ respectively with another circle $(O)$. Suppose that the common tangent of $ (O_1)$ and $ (O_2)$ at $C$ meets $(O)$ at $P$ such that $PA=PB$. Prove that $PO$ is perpendicular to $AB$.

Prove the inequality \[\sum_{i<j} \frac{a_ia_j}{a_i + a_j} \le \frac{n}{2(a_1 + a_2 +\cdots + a_n)}\sum_{i<j} a_ia_j\] for all positive real numbers $ a_1, a_2,\ldots , a_n$.

Let $A,B,C$ be $3$ points on the plane with integral coordinates. Prove that there exists a point $P$ with integral coordinates distinct from $A,B$ and $C$ such that the interiors of the segments $PA,PB$ and $PC$ do not contain points with integral coordinates.