Problems

Let $X$ be a finite set, and let $\mathcal{P}X$ be the power set of $X$ - that is, the set of subsets of $X$. For subsets $A$ and $B$ of $X$, define $A\ast B$ as the symmetric difference of $A$ and $B$ - that is, those elements that are in either $A$ or $B$, but not both. In formal set theory notation, this is $A\ast B=(A\cup B)\mathrm{\setminus }(A\cap B)$.

Prove that $(\mathcal{P}X,\ast )$ forms a group.

The lengths of three sides of a right-angled triangle are all integers.

Show that one of them is divisible by $5$.

How many integers are there $n>1$ such that $a^{25}-a$ is divisible by $n$ for every integer $a$.

Given a pile of five cards, is it true that reversing the order of the pile by counting the cards out one by one leaves no card in its original position?

Today we will discover some ideas related to non-isosceles triangles. This restriction comes from the fact that in isosceles triangles, a median and a height coincide.

Let $p$ be a prime number, $a$ be an integer, not divisible by $p$. Prove that $a^p-a$ is divisible by $p$.

Let $n$ be an integer. Denote by $\varphi (n)$ the number of integers from $1$ to $n-1$ coprime to $n$. Find $\varphi (n)$ in the following cases:

• $n$ is a prime number.

• $n=p^k$ for a prime $p$.

• $n=pq$ for two different primes $p$ and $q$.

Let $n$ be an integer and let $a$ be an integer coprime to $n$. Prove that $a^{\varphi (n)-1}-1$ is divisible by $n$.

With a pile of four cards, does reversing the order of the pile by counting the cards out one by one leaves no card in its original position?

You have in your hands a royal flush! That is, Ace, King, Queen, Jack and $10$ of spades. How many shuffles of your hand swap the Ace and Jack?