University of Toronto Math Academy
Qualifying Quiz
Instructions
We call it a quiz, but it’s really a challenge: a chance for you to show us how you approach
new problems and new concepts in mathematics. We care about more than just your final
answers—your reasoning, your explanations, and your presentation are just as important.
Some of the problems are intended to be hard. We do not expect you to solve every
problem completely. Send us your work, even if it is only partial solutions or conjectures.
There is no time limit on this quiz.
If you need clarification on a problem, please email outreach@math.toronto.edu. You
may not consult or get help from anyone else.
We also think that these problems are fun. So, enjoy, and good luck!
The problems
1. At the University of Toronto Math Academy, each student always tells the truth or
always lies. One day there are 21 students in class and every student says “Everybody
else is a liar”. What is the largest possible number of truthtellers? What is the largest
possible number of liars?
2. Which integer numbers can be written as the difference of two perfect squares?
Your answer should be a list of all the integer numbers that can be written as the
difference of two perfect squares. Once you have that, you still have to do two
things. First, prove that all the numbers in your list are, in fact, the difference of
two squares. Second, prove that no other integer numbers are the difference of two
perfect squares.
3. I have a collection of chameleons in my garden. Each chameleon is either red, green,
or black. The chameleons sometimes change colours: if two chameleons of different
colours run into each other, they both change to the third colour. This is the only
type of colour change that may happen. Yesterday I noticed that I had 4 green
chameleons, 5 red chameleons, and 6 black chameleons. Prove that my collection of
chameleons will never be monochromatic (that is, all the same colour).
4. Rey and Kylo Ren are playing a game. They have a big bag of M&Ms of various
colours and they take turns eating them. In their turn, they can choose to eat one
or two M&Ms, but they can only eat two if they are of different colours. Rey will
