[Solutions] United Kingdom Mathematical Olympiad For Girls 2019

1. At Mathsland Animal Shelter there are only cats and dogs. Unfortunately, one day $60$ of the animals managed to escape. Once a volunteer had realised, they counted the remaining animals. They noted that half of the cats and a third of the dogs had escaped.
   a) If the number of cats before the escape was $C$ and the number of dogs before the escape was $D$, write down an equation linking $C$ and $D$.
   b) If the total number of animals before the escape was $T$, write down an equation linking $C$, $D$ and $T$.
   c) Given that more cats than dogs escaped, find the largest possible value of $T$. You must justify why the value you have found is the largest.
2. Beth has a black counter and Wendy has a white counter. Beth and Wendy move their counters on the two boards below according to the starting positions and rules given. They always move their counters at the same time.
   a) At each turn, each player moves their counter either one square to the left or one square to the right. Prove that the black and white counters can never be in the same square at the same time. (Hint You may find it helpful to refer to the colours of the squares on the board in your explanation.)
   b) At each turn, each player moves their counter to a triangular cell which shares one edge with the cell that their counter is currently in. Can their counters ever be in the same cell at the same time? (Hint. If you think the two counters can never be in the same cell at the same time, you should give an argument that they cannot be in the same cell at the same time which works no matter which sequence of moves Beth and Wendy do. If you think the two counters can be in the same cell at the same time, you should give an example of a sequence of moves after which they are in the same cell at the same time.)
3. a) Seth wants to know how many positive whole numbers from one to one hundred are divisible by two or five. He thinks that the answer is 70 because there are fifty multiples of two and twenty multiples of five from one to one hundred. Explain why his answer is too large.
   b) Consider the list of $1800$ fractions $$\frac{1}{1800}, \frac{2}{1800}, \ldots, \frac{1799}{1800}, \frac{1800}{1800}.$$ How many are not in simplest form? Explain your reasoning. (Note. The fraction $\frac{900}{1800}$ is not in simplest form because it can be simplified to $\frac{1}{2}$.)
4. The diagram shows a rectangle placed inside a quarter circle of radius 1 , such that its vertices all lie on the perimeter of the quarter circle and one vertex coincides with the centre of the (whole) circle. Let the perimeter of such a rectangle be $P$.
   a) Show that $P=3$ is impossible.
   b) Find the largest possible value of $P$. You must fully justify why the value that you find is the largest.
   c) Instead a rectangle is placed inside a whole circle of radius $1, such that its vertices all lie on the circumference of the circle. If the perimeter of the rectangle is as large as possible, show that the rectangle must be a square and calculate its perimeter.
5. Let $n$ be a positive integer. Tracy writes a list of $10$ whole numbers between $1$ and $n$ (inclusive). Each number in the list is either equal to, one less than, or one more than the number before it. For example, when $n=7$, her list could be $5,5,6,7,6,6,5,5,6,6$ or $4,4,3,2,1,1,1,1,1,1$, her list could not be $1,3,3,4,5,5,6,7,7,7$ or $5,6,7,8,7,6,5,5,5,5$.
   a) Suppose that $n=3$. Stacey forms a list by copying Tracy's list, except that whenever Tracy writes a $1$, Stacey writes a $3$, and whenever Tracy writes a $3$, Stacey writes a $1$.
   • Which lists could Tracy write that would cause her list to be the same as Stacey's?
   • Explain why Tracy can write as many lists that start $2, 2, 1$ as start $2, 2, 3$.
   b) For which $n$ between $1$ and $10$ (inclusive) is the number of lists that Tracy could write odd?

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