[Solutions] Nordic Mathematical Contest 2019

  1. The geometric mean of the non-negative numbers $a_1, a_2,\cdots, $ $a_n$ is defined as $\sqrt[n]{a_1a_2\cdots a_n}$. A set of different positive integers is called meaningful if for any finite nonempty subset the corresponding arithmetic and geometric means are both integers.
    a) Does there exist a meaningful set which consists of $2019$ numbers?.
    b) Does there exist an infinite meaningful set?.
  2. Let $a, b, c $ be the side lengths of a right angled triangle with $c > a, b$. Show that $$3<\frac{c^3-a^3-b^3}{c(c-a)(c-b)}\leq \sqrt{2}+2.$$
  3. The quadrilateral $ABCD$ satisfies $\angle ACD = 2\angle CAB$, $\angle ACB = 2\angle CAD $ and $CB = CD$. Show that $\angle CAB=\angle CAD$.
  4. Let $n$ be an integer with $n\geq 3$ and assume that $2n$ vertices of a regular $(4n + 1)-$gon are coloured. Show that there must exist three of the coloured vertices forming an isosceles triangle.
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