# [Solutions] United Kingdom Mathematical Olympiad For Girls 2016

1. The diagram shows a figure consisting of six line segments and a circle, each containing three points. Each point is labelled with a real number. The sum of the three numbers on each line segment or circle is $T$. Prove that each number is equal to $\dfrac{1}{3} T$.
2. The diagram shows two circles $C_{1}$ and $C_{2}$ with diameters $P A$ and $A Q$. The circles meet at the points $A$ and $B$, and the line $P A$ is a tangent to $C_{2}$ at $A$. Prove that $$\frac{P B}{B Q}=\frac{\text { area } C_{1}}{\text { area } C_{2}}.$$
3. Punam puts counters onto some of the cells of a $5 \times 5$ board. She can put more than one counter on each cell, and she can leave some cells empty. She tells Quinn how many counters there are in each row and column. These ten numbers are all different. Can Quinn always work out which cells, if any, are empty?
4. a) In the trapezium $A B C D$, the and $D C$ are parallel. The point midpoint of $B C$, and $N$ is the of $D A$. Prove that $2 M N=A B+C D$.
b) The diagram shows part of a tiling of the plane by squares and equilateral triangles. Each tile has edges of length $2$. The points $X$ and $Y$ are at the centres of square tiles. What is the distance $X Y$ ?
5. Alia, Bella and Catherine are multiplying fractions, aiming to obtain integers. Each of them can multiply as many fractions as she likes (including just one), and can use the same fraction more than once.
• Alia's fractions are of the form $\dfrac{n+1}{n}$, where $n$ is a positive integer.
• Bella's fractions are of the form $\dfrac{6 p-5}{3 p+6}$, where $p$ is a positive integer.
• Catherine's fractions are of the form $\dfrac{4 q-1}{2 q+1}$, where $q$ is a positive integer.
Which integers can each of them obtain?
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