Problems

Consider Pascal’s triangle: it starts with \(1\), then each entry in the triangle is the sum of the two numbers above it. Prove that the diagonals of Pascal’s triangle add up to Fibonacci numbers.

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 \[F_{k+(s-1)k} = F_{k-1}F_{(s-1)k} + F_kF_{(s-1)k+1}.\]

Let \(m\) and \(n\) be positive integers. What positive integers can be written as \(m+n+\gcd(m,n)+\text{lcm}(m,n)\), for some \(m\) and \(n\)?

Denote by \(\gcd(m,n)\) the greatest common divisor of numbers \(m,n\), namely the largest possible \(d\) which divides both \(n\) and \(m\). Prove for any \(m,n\) that \[\gcd(F_n,F_m) = F_{\gcd(m,n)}.\]

Let \(n\ge r\) be positive integers. What is \(F_n^2-F_{n-r}F_{n+r}\) in terms of \(F_r\)?

On the questioners’ planet (where everyone can only ask questions. Cricks can only ask questions to which the answer is yes, and Goops can only ask questions to which the answer is no), you meet 4 alien mathematicians.

They’re called Alexander Grothendieck, Bernhard Riemann, Claire Voisin and Daniel Kan (you may like to shorten their names to \(A\), \(B\), \(C\) and \(D\)).
Alexander asks the following question “Am I the kind who could ask whether Bernhard could ask whether Claire could ask whether Daniel is a Goop?"

Amongst the final three (that is, Bernhard, Claire and Daniel), are there an even or an odd number of Goops?

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.