CMO 2016

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 $v_{1}, \ldots, v_{10}$ :

$$v_{i}=1+\frac{6 v_{i}^{2}}{v_{1}^{2}+v_{2}^{2}+\cdots+v_{10}^{2}} (i=1, \ldots, 10) .$$

Find all 10-tuples $\left(v_{1}, v_{2}, \ldots, v_{10}\right)$ 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 $n$.

Lavaman versus the Flea. Let $A, B$, and $F$ be positive integers, and assume $A<B<2 A$. 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 $\frac{A}{n+1} \leq B-A<\frac{A}{n}$.

Let $\triangle A B C$ be an acute-angled triangle with altitudes $A D$ and $B E$ meeting at $H$. Let $M$ be the midpoint of segment $A B$, and suppose that the circumcircles of $\triangle D E M$ and $\triangle A B H$ meet at points $P$ and $Q$ with $P$ on the same side of $C H$ as $A$. Prove that the lines $E D$, $P H$, and $M Q$ all pass through a single point on the circumcircle of $\triangle A B C$.