Problems

A simple polygon is a polygon that does not intersect itself and has no holes. Suppose we have a simple polygon \(S\) whose vertices consists of only integer coordinates.

The area turns out to be remarkably easy to calculate. Count up the number of points with integer coordinate inside the polygon and on the boundary; call them \(i\) and \(b\) respectively. The area is then \[A(S) = i+\frac{b}{2}-1.\]

In the picture above, \(i=3\) and \(b=11\), so \(A(S) = \frac{15}{2}\). Prove that this formula for the area \(A(S)\) is correct.

All of the rectangles in the figure below, which is drawn to scale, are similar to the big rectangle (that is, their sides are in the same ratio). Each number represents the area of the rectangle. What is the length \(AB\)?

Each square in a \(3\times3\) grid of squares is coloured red, white, blue, or green so that every \(2\times2\) square contains one square of each color. One such colouring is shown below. How many different colourings are possible?

For any positive integer \(k\), the factorial \(k!\) is defined as a product of all integers between 1 and \(k\) inclusive: \(k! = k \times (k-1) \times ... \times 1\). What’s the remainder when \(2025!+2024!+2023!+...+3!+2!+1!\) is divided by \(8\)?

In the picture below, we have a regular pentagon. The segments \(AB\) and \(CD\) have equal lengths. What is the angle \(\alpha\)?

We have two squares sharing the same centre, each with side length \(2\). Show that the area of overlap is at least \(3\).

A regular tetrahedron is a three dimensional shape with four faces. Each face of a regular tetrahedron is an equilateral triangle. Describe all rotational symmetries of a regular tetrahedron.

Two lines intersect at a point \(P\) at an angle of \(\alpha\). Show that a rotation in the plane around the point \(P\) through an angle \(2\alpha\) can be achieved by a reflection in one of the two lines followed by a reflection in the other line.

Show that a rotation about an axis on a sphere followed by a rotation about a different axis is again a rotation about some axis.

You have in your possession a rotation of the sphere about an axis \(l\) through an angle \(\alpha\neq k\times360^{\circ}\) 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 \(\theta'\) around an axis \(m'\). You can add to your possession each of the below:

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

2. the rotation \(r_2r_1\) obtained by doing \(r_1\) and then \(r_2\);

3. the rotation \(g^{-1}r_1g\), where \(g\) is any rotation of the sphere.

Can you get all the rotations of the sphere?