# [Solutions] The Mathematical Danube Competition 2017 (Juniors)

1. What is the smallest value that the sum of the digits of the number $3 n^{2}+n+1$, $n \in \mathbb{N}$, can take?
2. Let $n \geq 3$ be a positive integer. Consider an $n \times n$ square. In each cell of the square, one of the numbers from the set $M=\{1,2, \ldots, 2 n-1\}$ is to be written. One such filling is called "good' if, for every index $i, 1 \leq i \leq n$, row no. $i$ and column no. $i$, together, contain all the elements of $M$. a) Prove that there exists $n \geq 3$ for which a good filling exists. b) Prove that for $n=2017$ there is no good filling of the $n \times n$ square.
3. Consider an acute triangle $A B C$ in which $A_{1}, B_{1}$, and $C_{1}$ are the feet of the altitudes dropped from $A, B$, and $C$, respectively, and $H$ is the orthocenter. The perpendiculars dropped from $H$ onto $A_{1} C_{1}$ and $A_{1} B_{1}$ intersect lines $A B$ and $A C$ at $P$ and $Q$, respectively. Prove that the line perpendicular to $B_{1} C_{1}$ that passes through $A$ also contains the midpoint of the line segment $P Q$.
4. Determine the triples of positive integers $(x, y, z)$ such that $x^{4}+y^{4}=2 z^{2}$ and $x, y$ are co-prime.
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