In the Triangle ABC AB>AC, the extension of the altitude AD with D lying inside BC intersects the circum-circle of the Triangle ABC at P. The circle through P and tangent to BC at D intersects the circum-circle of Triangle ABC at Q distinct from P with PQ=DQ. Prove that AD=BD-DC

Find all pairs of integers $(m,n)$ such that $m^3-n^3=2mn +8$

Let $b_1,b_2,... $ be a sequence of positive real numbers such that for each $ n\ge 1$, $$b_{n+1}^2 \ge \frac{b_1^2}{1^3}+\frac{b_2^2}{2^3}+...+\frac{b_n^2}{n^3}$$Show that there is a positive integer $M$ such that $$\sum_{n=1}^M \frac{b_{n+1}}{b_1+b_2+...+b_n} > \frac{2013}{1013}$$

In the following $6\times 6$ matrix, one can choose any $k\times k$ submatrix, with $1<k\leq6 $ and add $1$ to all its entries. Is it possible to perform the operation a finite number of times so that all the entries in the $6\times 6$ matrix are multiples of $3$? $ \begin{pmatrix} 2 & 0 & 1 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 2 & 0 \\ 1 & 0 & 2 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 & 2 & 0 \\ 1 & 1 & 1 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} $ Note: A $p\times q$ submatrix of a $m\times n$ matrix (with $p\leq m$, $q\leq n$) is a $p\times q$ matrix formed by taking a block of the entries of this size from the original matrix.

Let $x,y$ be distinct real numbers such that $\frac{x^n-y^n}{x-y}$ is an integer for $4$ consecutive positive integer $n$. Prove that $\frac{x^n-y^n}{x-y}$ is an integer for all positive integers $n$.