# [Solutions] United Kingdom Mathematical Olympiad For Girls 2017

1. All the digits $1$ to $9$ are to be placed in the circles in Figure 1, one in each, so that the total of the numbers in any line of four circles is the same. In the example shown in Figure 2, the total is equal to $20$. Prove that if the total $T$ is possible then the total $40-T$ is possible.
2. A positive integer is said to be jiggly if it has four digits, all non-zero, and no matter how you arrange those digits you always obtain a multiple of $12$. How many jiggly positive integers are there?
3. Four different points $A$, $B$, $C$ and $D$ lie on the curve with equation $y=x^{2}$. Prove that $A B C D$ is never a parallelogram.
4. Let $n$ be an odd integer greater than $3$ and let $M=n^{2}+2 n-7$. Prove that, for all such $n$, at least four different positive integers (excluding 1 and $M$) divide $M$ exactly.
5. Claire and Stuart play a game called Nifty Nines:
• they take turns to choose one number at a time, with Claire choosing first;
• numbers can only be chosen from the integers $1$ to $5$ inclusive;
• the game ends when $n$ numbers have been chosen (repetitions are permitted).
Stuart wins the game if the sum of the chosen numbers is a multiple of $9$, otherwise Claire wins. Find all values of $n$ for which Claire can ensure a win, whatever Stuart's choices were. You must prove that you have found them all.
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