SMO 2012 Junior Round 2

Let $O$ be the centre of a parallelogram $ABCD$ and $P$ be any point in the plane. Let $M,N$ be the midpoints of $AP,BP$, respectively, and $Q$ be the intersection of $MC$ and $ND$. Prove that $O,P$ and $Q$ are collinear.

Does there exist an integer $A$ such that each of the ten digits $0,1,\ldots,9$ appears exactly once as a digit in exactly one of the numbers $A,A^2,A^3$?

In $\triangle ABC$, the external bisectors of $\angle A$ and $\angle B$ meet at a point $D$. Prove that the circumcentre of $\triangle ABD$ and the points $C,D$ lie on the same straight line.

Determine the values of the positive integer $n$ for which the following system of equations has a solution in positive integers $x_1,x_2,\ldots,x_n$. Find all solutions for each such $n$.
$$x_1+x_2+\cdots+x_n=16\qquad(1)$$
$$\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_n}=1\qquad(2)$$

Suppose $S=\{a_1,a_2,\ldots,a_{15}\}$ is a set of $15$ distinct positive integers chosen from $2,3,\ldots,2012$ such that every two of them are coprime. Prove that $S$ contains a prime number. (Note: Two positive integers $m,n$ are coprime if their only common factor is $1$.)