# Further Mathematicians Rise to Mathematical Olympiad Challenge

On the 8th October, 20 further mathematicians took part in the gruelling two and a half hour Mathematical Olympiad for Girls, a competition which organisers try to discourage students from entering as it is really tough. The Mathematical Olympiad for Girls, run by the UK Mathematics Trust, is aimed at girls and young women across the UK and challenges girls to tackle advanced mathematical problems. Despite long written solutions not being needed this year, due to entering solutions online, the questions were no less challenging.

Our girls did really well overall but special mentions should go to Zareen (top 25%), Sydney (top 10%) and Diana and Emily (top 5%).

We are incredibly proud of their achievements.

**A sample question from the Mathematical Olympiad paper:**

Daniel and Alessia each have eight boxes labelled 1 to 8. To start, each of them has

balloons, all in their box 1.

(a) Alessia is playing a game with her balloons. She may make the following move:

• Choose a box containing at least two balloons. Pop a single balloon in box and

then move another balloon from box to box + 1.

Find the value of such that, after a finite number of moves, all that remains is a single

balloon in box 8. (2 marks)

(b) Daniel is playing a different game with his balloons. He may use any combination of

the following two moves:

• Pop two balloons in box and move a third balloon from box to box + 1.

• Pop a single balloon in each of boxes and + 1 and then move another balloon

from box + 1 to box + 2.

What is the smallest value of such that after a finite number of moves, all that remains

(i) is a single balloon in box 4?

(ii) is a single balloon in box 8? (8 marks)