Problems

a) A mighty dragon has several rubies in his treasure. He is able to divide the rubies into groups of $3$, $5$ or $11$. How many rubies does he have, if we know that is fewer than $200$?
b) The same dragon also has some emeralds. He is $6$ emeralds short to be able to divide them into groups of $13$, one emerald short to be able to divide them into groups of $5$, but if he wants to divide them into groups of $8$, he is left with one emerald. How many emeralds does he have if we know it is fewer than $500$?

Let $n$ be a natural number. Show that the fraction $\frac{21n+4}{14n+3}$ is irreducible, i.e. it cannot be simplified.

Let $m$ and $n$ be two positive integers with $m<n$ such that $$gcd(m,n)+\text{lcm}(m,n)=m+n.$$ Show that $m$ divides $n$.

The numbers $x,a,b$ are natural. Show that $gcd(x^a-1,x^b-1)=x^{gcd(a,b)}-1$.

Let $p$ be a prime number bigger than $3$. Prove that $p^2-1$ is a multiple of 24.

Is it possible to draw $K_5$ without intersecting edges on a Möbius band? Recall that $K_5$ is the complete graph on $5$ vertices. That is, $5$ points with an edge between every pair of different points.

In chess, knights can move one square in one direction and two squares in a perpendicular direction. This is often seen as an ‘L’ shape on a regular chessboard. A closed knight’s tour is a path where the knight visits every square on the board exactly once, and can get to the first square from the last square.

This is a closed knight’s tour on a $6\times 6$ chessboard.

Can you draw a closed knight’s tour on a $3\times 3$ torus? That is, a $3\times 3$ square with both pairs of opposite sides identified in the same direction, like the diagram below.

You may remember the game Nim. We will now play a slightly modified version, called Thrim. In Thrim, there are two piles of stones (or any objects of your choosing), one of size $1$ and the other of size $5$.
Whoever takes the last stone wins. The players take it in turns to remove stones - they can only remove stones from one pile at a time, and they can remove at most $3$ stones at a time.
Does the player going first or the player going second have a winning strategy?

A circle is inscribed in a triangle (that is, the circle touches the sides of the triangle on the inside). Let the radius of the circle be $r$ and the perimeter of the triangle be $p$. Prove that the area of the triangle is $\frac{pr}{2}$.

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".