In a room, there are three-legged stools and four-legged chairs. When people sat down on all of these seats, there were 39 legs (human and stool/chair legs) in the room. How many stools are there in the room?
A pawn stands on one of the squares of an endless in both directions chequered strip of paper. It can be shifted by \(m\) squares to the right or by \(n\) squares to the left. For which \(m\) and \(n\) can it move to the next cell to the right?
There are 100 notes of two types: \(a\) and \(b\) pounds, and \(a \neq b \pmod {101}\). Prove that you can select several bills so that the amount received (in pounds) is divisible by 101.
For which \(n\) is the expression \(n^4+4^n\) prime?
Find all solutions to \(x^2+2=y^3\) in the natural numbers.
McDonald’s used to sell Chicken McNuggets in boxes of 6, 9 or 20 in the UK before they introduced the Happy Meal. What is the largest number of Chicken McNuggets that could not be bought? For example, you wouldn’t have been able to buy 8 Chicken McNuggets, but you could have bought \(21 = 6+6+9\) Chicken McNuggets.
Show that the equation \(x^4+y^4=z^4\) cannot satisfied by integers \(x,y,z\) if none of them are 0.
A regular polygon has integer side lengths and its perimeter is 60. How many sides can it have?
Find positive integers \(x,y,z\) such that \(28x+30y+31z = 365\).
Given a piece of paper, we are allowed to cut it into 8 or 12 pieces. Can we get exactly 60 pieces of paper starting with a single piece?