CZ-SK-MO 2019

Find all triplets $(x,y,z)\in\mathbb{R}^3$ such that \begin{align*} x^2-yz &= |y-z|+1, y^2-zx &= |z-x|+1, z^2-xy &= |x-y|+1. \end{align*}

Let be $ABCD$ a rectangle with $|AB|=a\ge b=|BC|$ . Find points $P,Q$ on the line $BD$ such that $|AP|=|PQ|=|QC|$ . Discuss the solvability with respect to the lengths $a,b$ .

Let $a,b,c,n$ be positive integers such that the following conditions hold (i) numbers $a,b,c,a+b+c$ are pairwise coprime, (ii) number $(a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca)$ is a perfect $n$ -th power. Prove, that the product $abc$ can be expressed as a difference of two perfect $n$ -th powers.

Let be $ABC$ an acute-angled triangle. Consider point $P$ lying on the opposite ray to the ray $BC$ such that $|AB|=|BP|$ . Similarly, consider point $Q$ on the opposite ray to the ray $CB$ such that $|AC|=|CQ|$ . Denote $J$ the excenter of $ABC$ with respect to $A$ and $D,E$ tangent points of this excircle with the lines $AB$ and $AC$ , respectively. Suppose that the opposite rays to $DP$ and $EQ$ intersect in $F\neq J$ . Prove that $AF\perp FJ$ .

Prove that there are infinitely many integers which cannot be expressed as $2^a+3^b-5^c$ for non-negative integers $a,b,c$ .

Assume we can fill a table $n\times n$ with all numbers $1,2,\ldots,n^2-1,n^2$ in such way that arithmetic means of numbers in every row and every column is an integer. Determine all such positive integers $n$ .