# [Solutions] United Kingdom Mathematical Olympiad For Girls 2013

1. The diagram shows three identical overlapping right-angled triangles, made of coloured glass, placed inside an equilateral triangle, one in each corner. The total area covered twice (dark grey) is equal to the area left uncovered (white). What fraction of the area of the equilateral triangle does one glass triangle cover?
2. In triangle $A B C$, the median from $A$ is the line $A M$, where $M$ is the midpoint of the side $B C$. In any triangle, the three medians intersect at the point called the centroid, which divides each median in the ratio $2: 1$. In the convex quadrilateral $A B C D$, the points $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ and $D^{\prime}$ are the centroids of the triangles $B C D$, $C D A$, $D A B$ and $A B C$, respectively.
a) By considering the triangle $M C D$, where $M$ is the midpoint of $A B$, prove that $C^{\prime} D^{\prime}$ is parallel to $D C$ and that $C^{\prime} D^{\prime}=\dfrac{1}{3} D C$.
b) Prove that the quadrilaterals $A B C D$ and $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ are similar.
3. a) Find all positive integers $a$ and $b$ for which $$a^{2}-b^{2}=18.$$ b) The diagram shows a sequence of points $P_{0}, P_{1}, P_{2}, P_{3}, P_{4}, \ldots$, which spirals out around the point $O$. For any point $P$ in the sequence, the line segment joining $P$ to the next point is perpendicular to $O P$ and has length $3$. The distance from $P_{0}$ to $O$ is $29$. What is the next value of $n$ for which the distance from $P_{n}$ to $O$ is an integer?
4. a) An ant can move from any square on an $8 \times 8$ chessboard to an adjacent square. (Two squares are adjacent if they share a side). The ant starts in the top left corner and visits each square exactly once. Prove that it is impossible for the ant to finish in the bottom right corner. (You may find it helpful to consider the chessboard colouring.)
b) A ladybird can move one square up, one square to the right, or one square diagonally down and left, as shown in the diagram, and cannot leave the board. Is it possible for the ladybird to start in the bottom left corner of an $8 \times 8$ board, visit every square exactly once, and return to the bottom left corner?
5. a) Find an integer solution of the equation $x^{3}+6 x-20=0$ and prove that the equation has no other real solutions.
b) Let $x$ be $\sqrt[3]{\sqrt{108}+10}-\sqrt[3]{\sqrt{108}-10}$. Prove that $x$ is equal to $2$.
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