EMC 2016

Dag 1

Vraag 1

Is there a sequence $a_1, \ldots, a_{2016}$ of positive integers, such that every sum
$$a_r + a_{r+1} + \ldots + a_{s-1} + a_s$$
(with $1 \leq r \leq s \leq 2016$) is a composite number, but

$GCD (a_i, a_{i+1}) = 1$ for all $i=1,2,\ldots, 2015$;
$GCD (a_i, a_{i+1})= 1$ for all $i=1,2,\ldots, 2015$ and $GCD (a_i, a_{i+2})= 1$ for all $i=1,2,\ldots, 2014$?

$GCD(x,y)$ denotes the greatest common divisor of $x,y$.

Vraag 2

For two positive integers $a$ and $b$, Ivica and Marica play the following game: Given two piles of $a$ and $b$ cookies, on each turn a player takes $2n$ cookies from one of the piles, of which he eats $n$ and puts $n$ of them on the other pile. Number $n$ is arbitrary in every move. Players take turns alternatively, with Ivica going first. The player who cannot make a move, loses. Assuming both players play perfectly, determine all pairs of numbers $(a, b)$ for which Marica has a winning strategy.

Vraag 3

Determine all functions $f \colon \mathbb R \to \mathbb R$ such that equality
$$ f(x+y+yf(x)) = f(x) + f(y) + xf(y) $$
holds for all real numbers $x,y$.

Vraag 4

Let $C_1,C_2$ be circles intersecting in $X,Y$. Let $A,D$ be points on $C_1$ and $B,C$ on $C_2$ such that $A,X,C$ are collinear and $D,X,B$ are collinear. The tangent to circle $C_1$ at $D$ intersects $BC$ and the tangent to $C_2$ at $B$ in $P,R$ respectively. The tangent to $C_2$ at $C$ intersects $AD$ and tangent to $C_1$ at $A$, in $Q,S$ respectively. Let $W$ be the intersection of $AD$ with the tangent to $C_2$ at $B$ and $Z$ the intersection of $BC$ with the tangent to $C_1$ at $A$. Prove that the circumcircles of triangles $YWZ,RSY$ and $PQY$ have two points in common, or are tangent in the same point.