NMC 2018

Let $k$ be a positive integer and $P$ a point in the plane. We wish to draw lines, none passing through $P$ , in such a way that any ray starting from $P$ intersects at least $k$ of these lines. Determine the smallest number of lines needed.

A sequence of primes $p_1, p_2, \dots$ is given by two initial primes $p_1$ and $p_2$ , and $p_{n+2}$ being the greatest prime divisor of $p_n + p_{n+1} + 2018$ for all $n \ge 1$ . Prove that the sequence only contains finitely many primes for all possible values of $p_1$ and $p_2$ .

Let $ABC$ be a triangle with $AB < AC$ . Let $D$ and $E$ be on the lines $CA$ and $BA$ , respectively, such that $CD = AB$ , $BE = AC$ , and $A$ , $D$ and $E$ lie on the same side of $BC$ . Let $I$ be the incenter of triangle $ABC$ , and let $H$ be the orthocenter of triangle $BCI$ . Show that $D$ , $E$ , and $H$ are collinear.

Let $f = f(x,y,z)$ be a polynomial in three variables $x$ , $y$ , $z$ such that $f(w,w,w) = 0$ for all $w \in \mathbb{R}$ . Show that there exist three polynomials $A$ , $B$ , $C$ in these same three variables such that $A + B + C = 0$ and $$f(x,y,z) = A(x,y,z) \cdot (x-y) + B(x,y,z) \cdot (y-z) + C(x,y,z) \cdot (z-x).$$ Is there any polynomial $f$ for which these $A$ , $B$ , $C$ are uniquely determined?