[Solutions] Junior Balkan Mathematical Olympiad 2019

  1. Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number $$x^p + y^p + z^p - x - y - z$$ is a product of exactly three distinct prime numbers.
  2. Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that $$a^4 - 2019a = b^4 - 2019b = c.$$ Prove that $- \sqrt{c} < ab < 0$.
  3. Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.
  4. A $5 \times 100$ table is divided into $500$ unit square cells, where $n$ of them are coloured black and the rest are coloured white. Two unit square cells are called adjacent if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible value of $n$.

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