In the triangle $ABC$, $AB>AC$, the extension of the altitude $AD$ with $D$ lying inside $BC$ intersects the circumcircle $\omega$ of the triangle $ABC$ at $P$. The circle through $P$ and tangent to $BC$ at $D$ intersects $\omega$ at $Q$ distinct from $P$ with $PQ=DQ$. Prove that $AD=BD-DC$.
Let $b_1,b_2,\ldots$ be a sequence of positive real numbers such that for each $n\ge1$,
$$b_{n+1}^2\ge\frac{b_1^2}{1^3}+\frac{b_2^2}{2^3}+\cdots+\frac{b_n^2}{n^3}.$$
Show that there is a positive integer $M$ such that
$$\sum_{n=1}^M\frac{b_{n+1}}{b_1+b_2+\cdots+b_n}>\frac{2013}{1013}.$$
In the following $6\times6$ array, one can choose any $k\times k$ subarray, with $1<k\le6$, 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 array are multiples of $3$?
$$\begin{bmatrix}2&0&1&0&2&0\\0&2&0&1&2&0\\1&0&2&0&2&0\\0&1&0&2&2&0\\1&1&1&1&2&0\\0&0&0&0&0&0\end{bmatrix}$$
Original source array:
Let $x,y$ be distinct real numbers such that $\dfrac{x^n-y^n}{x-y}$ is an integer for four consecutive positive integers $n$. Prove that $\dfrac{x^n-y^n}{x-y}$ is an integer for all positive integers $n$.
