# [Solutions] The Mathematical Danube Competition 2016 (Juniors)

1. Let $S=x_{1} x_{2}+x_{3} x_{4}+\ldots+x_{2015} x_{2016}$, where $x_{1}, x_{2} \ldots, x_{2006} \in \{\sqrt{3}-\sqrt{2}, \sqrt{3}+\sqrt{2}\} .$ Is the equality $S=2016$ possible?
2. Determine the poxitive integers $n>1$ such that, for any divisar $d$ of $n$, the numbers $d^{2} d+1$ and $d^{2}-d+1$ are prime.
3. Let $A B C$ ' be a triangle with $A B < A C$, $I$ its incenter, and $M$ the midpoint of the side $B C$. If $I A=I M$, determine the smallest possible value of the angle $AIM$.
4. A unit square is removed from the corner of the $n \times n$ grid where $n \geq 2$. Prowe that the remainder can be covered by copies of the $L$-shapes consisting of $3$ or $5$ unit squares depicted in the figure, Every aquare must be covered once and the $L$-shapes must not go over the bounds of the grid.
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