OME 2018

Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares.

Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be balanced if the number of white tokens on its left plus the number of black tokens on its right is $n$ . Determine whether the number of balanced tokens is even or odd.

Let $ABC$ be an acute-angled triangle with circumcenter $O$ and let $M$ be a point on $AB$ . The circumcircle of $AMO$ intersects $AC$ a second time on $K$ and the circumcircle of $BOM$ intersects $BC$ a second time on $N$ .

Prove that $\left[MNK\right] \geq \frac{\left[ABC\right]}{4}$ and determine the equality case.

Points on a spherical surface with radius $4$ are colored in $4$ different colors. Prove that there exist two points with the same color such that the distance between them is either $4\sqrt{3}$ or $2\sqrt{6}$ .

(Distance is Euclidean, that is, the length of the straight segment between the points)

Let $a, b$ be coprime positive integers. A positive integer $n$ is said to be weak if there do not exist any nonnegative integers $x, y$ such that $ax+by=n$ . Prove that if $n$ is a weak integer and $n < \frac{ab}{6}$ , then there exists an integer $k \geq 2$ such that $kn$ is weak.