Problems

You have in your possession a rotation of the sphere about an axis $l$ through an angle $\alpha \ne k\times {360}^{\circ }$ for any integer $k$.

Consider the following funny rules. Suppose you have a rotation $r_1$ through an angle $\theta$ around an axis $m$ and a rotation $r_2$ through an angle ${\theta }^{\prime }$ around an axis $m^{\prime }$. You can add to your possession each of the below:

1. the rotation $r_1^{-1}$ through $-\theta$ around $m$;

2. the rotation $r_2{r}_1$ obtained by doing $r_1$ and then $r_2$;

3. the rotation $g^{-1}{r}_{1}g$, where $g$ is any rotation of the sphere.

Can you get all the rotations of the sphere?

Let us colour each side of a hexagon using one of yellow, blue or green. Any two configurations that can be rotated or reflected onto each other will be the same colouring for us. How many colourings are there?

Today we will focus on applications of the Pythagorean theorem in geometry and number theory. This famous and ancient theorem states that in a right-angled triangle, the area of a square on a hypothenuse (the longest side) is the sum of the areas of the squares on the other two sides. $$a^2+b^2=c^2$$

There are over a 100 proofs of Pythagorean theorem, a quite simple one is visible below:

Four right triangles in this picture are identical (congruent): $\mathrm{△}A{A}^{\prime }D,\mathrm{△}EF{A}^{\prime },\mathrm{△}GFH$ and $\mathrm{△}CHD$. By moving the triangles around you can see that the large red square has the same area as the sum of areas of the other two squares (violet and green).

Today’s session is not only about geometry, we will also learn something about the equation $a^2+b^2=c^2$, where all the numbers $a,b,c$ are integers.

Let $u$ and $v$ be two positive integers, with $u>v$. Prove that a triangle with side lengths $u^2-v^2$, $2uv$ and $u^2+v^2$ is right-angled.

We call a triple of natural numbers (also known as positive integers) $(a,b,c)$ satisfying $a^2+b^2=c^2$ a Pythagorean triple. If, further, $a$, $b$ and $c$ are relatively prime, then we say that $(a,b,c)$ is a primitive Pythagorean triple.

Show that every primitive Pythagorean triple can be written in the form $(u^2-v^2,2uv,u^2+v^2)$ for some coprime positive integers $u>v$.

What symmetries does a regular hexagon have, and how many?

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 $5$.

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 particular restriction comes from the fact that in isosceles triangles a median and a height coincide.