The integers $1, 2, 3, \ldots, 2016$ are written on a board. You can choose any two numbers on the board and replace them with their average. For example, you can replace $1$ and $2$ with $1.5$, or you can replace $1$ and $3$ with a second copy of $2$. After $2015$ replacements of this kind, the board will have only one number left on it. (a) Prove that there is a sequence of replacements that will make the final number equal to $2$. (b) Prove that there is a sequence of replacements that will make the final number equal to $1000$.

Consider the following system of $10$ equations in $10$ real variables $v_1, \ldots, v_{10}$: \[v_i = 1 + \frac{6v_i^2}{v_1^2 + v_2^2 + \cdots + v_{10}^2} \qquad (i = 1, \ldots, 10).\]Find all $10$-tuples $(v_1, v_2, \ldots , v_{10})$ that are solutions of this system.

Find all polynomials $P(x)$ with integer coefficients such that $P(P(n) + n)$ is a prime number for infinitely many integers $n$.

Let $A, B$, and $F$ be positive integers, and assume $A < B < 2A$. A flea is at the number $0$ on the number line. The flea can move by jumping to the right by $A$ or by $B$. Before the flea starts jumping, Lavaman chooses finitely many intervals $\{m+1, m+2, \ldots, m+A\}$ consisting of $A$ consecutive positive integers, and places lava at all of the integers in the intervals. The intervals must be chosen so that: (i) any two distinct intervals are disjoint and not adjacent; (ii) there are at least $F$ positive integers with no lava between any two intervals; and (iii) no lava is placed at any integer less than $F$. Prove that the smallest $F$ for which the flea can jump over all the intervals and avoid all the lava, regardless of what Lavaman does, is $F = (n-1)A + B$, where $n$ is the positive integer such that $\frac{A}{n+1} \le B-A < \frac{A}{n}$.

Let $\triangle ABC$ be an acute-angled triangle with altitudes $AD$ and $BE$ meeting at $H$. Let $M$ be the midpoint of segment $AB$, and suppose that the circumcircles of $\triangle DEM$ and $\triangle ABH$ meet at points $P$ and $Q$ with $P$ on the same side of $CH$ as $A$. Prove that the lines $ED, PH,$ and $MQ$ all pass through a single point on the circumcircle of $\triangle ABC$.