Problems

We meet a group of people, all of whom are either knights or liars. Knights always tell the truth and liars always lie. Prove that it’s impossible for someone to say “I’m a liar".

We’re told that Leonhard and Carl are knights or liars (the two of them could be the same or one of each). They have the following conversation.

Leonhard says “If $49$ is a prime number, then I am a knight."

Carl says “Leonhard is a liar".
Prove that Carl is a liar.

Let $p$ be a prime number greater than $3$. Prove that $p^2-1$ is divisible by $12$.

Split the numbers from $1$ to $9$ into three triplets such that the sum of the three numbers in each triplet is prime. For example, if you split them into $124$, $356$ and $789$, then the triplet $124$ is correct, since $1+2+4=7$ is prime. But the other two triples are incorrect, since $3+5+6=14$ and $7+8+9=24$ are not prime.

A family is going on a big holiday, visiting Austria, Bulgaria, Cyprus, Denmark and Estonia. They want to go to Estonia before Bulgaria. How many ways can they visit the five countries, subject to this constraint?

Let $p$, $q$ and $r$ be distinct primes at least $5$. Can $p^2+q^2+r^2$ be prime? If yes, then give an example. If no, then prove it.

How many subsets of $\{1,2,...,n\}$ (that is, the integers from $1$ to $n$) have an even product? For the purposes of this question, take the product of the numbers in the empty set to be $1$.

Let $A$, $B$, $C$ and $D$ be four points labelled clockwise on the circumference of a circle. The diagonals $AC$ and $BD$ intersect at the centre $O$ of the circle. What can be deduced about the quadrilateral $ABCD$?

Consider the 7 different tetrominoes. Is it possible to cover a $4\times 7$ rectangle with exactly one copy of each of the tetrominoes? If it is possible, provide an example layout. If it is not possible, prove that it’s impossible.

We allow rotation of the tetrominoes, but not reflection. This means that we consider $S$ and $Z$ as different, as well as $L$ and $J$.

In the following grid, how many different ways are there of getting from the bottom left triangle to the bottom right triangle? You must only go from between triangles that share an edge and you can visit each triangle at most once. (You don’t have to visit all of the triangles.)