# [Solutions] United Kingdom Mathematical Olympiad For Girls 2012

1. The numbers $a, b$ and $c$ are real. Prove that at least one of the three numbers $(a+b+c)^{2}-9 b c$, $(a+b+c)^{2}-9 c a$ and $(a+b+c)^{2}-9 a b$ is non-negative.
2. Let $S=\left\{a_{1}, a_{2}, \ldots, a_{n}\right\}$ where the $a_{i}$ are different positive integers. The sum of the elements of each non-empty proper subset of $S$ is not divisible by $n$. Show that the sum of all elements of $S$ is divisible by $n$. Note that a proper subset of $S$ consists of some, but not all, of the elements of $S$.
3. Find all positive integers $m$ and $n$ such that $m^{2}+8=3^{n}$.
4. Does there exist a positive integer $N$ which is a power of $2$, and a different positive integer $M$ obtained from $N$ by permuting its digits (in the usual base $10$ representation), such that $M$ is also a power of $2$? Note that we do not allow the base $10$ representation of a positive integer to begin with $0$.
5. Consider the triangle $A B C$. Squares $A L K B$ and $B N M C$ are attached to two of the sides, arranged in a "folded out" configuration (so the interiors of the triangle and the two squares do not overlap one another). The squares have centres $O_{1}$ and $O_{2}$ respectively. The point $D$ is such that $A B C D$ is a parallelogram. The point $Q$ is the midpoint of $K N$, and $P$ is the midpoint of $A C$.
a) Prove that triangles $A B D$ and $B K N$ are congruent.
b) Prove that $O_{1} Q O_{2} P$ is a square.
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