# [Solutions] United Kingdom Mathematical Olympiad For Girls 2021

1. a) Find all whole numbers $x$ such that $$\left(x^{2}-7 x+11\right)^{\left(x^{2}-4 x+4\right)}=1.$$ b) Find all whole numbers $x$ such that $$\left(x^{2}-7 x+11\right)^{\left(x^{2}-4 x+4\right)}=-1 .$$
2. Consider a $4 \times 4$ grid numbered $1$ to $16$ left to right then top to bottom. Tile A or Tile B is placed onto the grid so that it covers three adjacent numbers.
a) If Tile A is placed onto the grid (the orientation of the tile may be changed), can the total of the uncovered numbers be a multiple of three?.
b) In how many different ways can Tile B be placed onto the grid (the orientation of the tile may be changed) so that the sum of the uncovered numbers is a multiple of three?
3. The diagram shows a quadrilateral $A B C D$, where $A B$ is $2cm$ and $\angle A B C$, $\angle A C D$ and $\angle D A B$ are right angles.
a) Let $E$ be the point on $D A$ such that $C E$ is perpendicular to $D A$. Prove that triangles $A B C$ and $D E C$ are similar.
b) Given that the area of quadrilateral $A B C D$ is $6cm^{2}$, find all possible values for the perimeter of quadrilateral $A B C D$.
4. Sam is playing a game. Her teacher gives her a positive whole number $A$, and then Sam chooses a positive whole number $S$. Sam then adds together all of the integers between $S$ and $S+A-1$ (inclusive) to obtain a total $T$. If $T$ is even, Sam wins the game. For example, if $A=4$, Sam can win by choosing $S=10$ because then $T=10+11+12+13=46$.
a) Show that if $A=4$, Sam will win the game no matter which number she chooses.
b) Show that if $A$ is a multiple of $4$, Sam will win the game no matter which number she chooses.
c) For which other values of $A$ can Sam choose an $S$ so that she wins? You must show how she can win for each of those values, and also explain why she cannot win for all the other values.
5. a) By considering their difference, or otherwise, find all possibilities for the common factors of $n$ and $n+3$. For $n \geq 2$, let $P(n)$ denote the largest prime factor of $n$.
b) If $a$ and $b$ are positive integers greater than 1, explain why $P(a b)$ must be equal to at least one of $P(a)$ or $P(b)$.
c) Find all positive integers $n$ such that $$P\left(n^{2}+2 n+1\right)=P\left(n^{2}+9 n+14\right).$$
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