[Solutions] Nordic Mathematical Contest 2018

  1. 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.
  2. 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 $p_2$.
  3. 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.
  4. 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?

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