Problems
Let \(p\) be a prime number greater than \(3\). Prove that \(p^2-1\) is divisible by \(12\).
You meet an alien, who you learn is thinking of a positive integer \(n\). They ask the following three questions.
“Am I the kind who could ask whether \(n\) is divisible by no primes other than \(2\) or \(3\)?"
“Am I the kind who could ask whether the sum of the divisors of \(n\) (including \(1\) and \(n\) themselves) is at least twice \(n\)?"
“Is \(n\) divisible by 3?"
Is this alien a Crick or a Goop?
Find the mistake in the sequence of equalities: \(-1=(-1)^{\frac{2}{2}}=((-1)^2)^{\frac{1}{2}}=1^{\frac{1}{2}}=1\).
Prove for any integers \(m,n\ge0\) that \(F_{m+n} = F_{m-1}F_n + F_mF_{n+1}\).
Corollary: if \(k\mid n\), then \(F_k\mid F_n\). This can be proven by induction if we write \(n=sk\) for a natural \(s\), then \[F_{k+(s-1)k} = F_{k-1}F_{(s-1)k} + F_kF_{(s-1)k+1}.\]
Suppose that \(p\) is a prime number. How many numbers are there less than \(p^2\) that are relatively prime to \(p^2\)?
How many cuboids are contained in an \(n\times n\times n\) cube? For example, we’ve got \(n^3\) cuboids of size \(1\times1\times1\), and obviously just \(1\) of size \(n\times n\times n\) (which is the whole cube itself). But we also have to count how many there of size \(1\times1\times2\), \(1\times2\times3\), and several more.
In the \(6\times7\) large rectangle shown below, how many rectangles are there in total formed by grid lines?
Simplify \(F_0-F_1+F_2-F_3+...-F_{2n-1}+F_{2n}\), where \(n\) is a positive integer.
Is \(\text{Nim}(7,11,15)\) a winning position or a losing position? If it is a winning position, what is the optimal move?
Imagine the Earth is a perfectly round solid ball. Let us drill from the North Pole, London and Beijing simultaneously and meet at the centre of Earth. A ball with three openings is formed. The surface of this ball is shown on the left of the picture below. Describe how to stretch this surface so that it looks like the surface of a donut with two holes as shown on the right.
