SMO 2015 Junior Round 2

Let $n=\overline{30x070y03}$ be a 9-digit integer. Find all possible values of the pair $(x,y)$, so that $n$ is a multiple of $37$.

In a convex hexagon $ABCDEF$, $AB$ is parallel to $DE$, $BC$ is parallel to $EF$ and $CD$ is parallel to $FA$. Prove that the triangles $ACE$ and $BDF$ have the same area.

There are 30 children, $a_1,a_2,\ldots,a_{30}$, seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of 1000 candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.

Let $A$ be a set of numbers chosen from $1,2,\ldots,2015$ with the property that any two distinct numbers, say $x$ and $y$, in $A$ determine a unique isosceles triangle (which is non equilateral) whose sides are of length $x$ or $y$. What is the largest possible size of $A$?

Find all positive integers $k$ such that $k^k+1$ is divisible by $30$. Justify your answer.