Problems

You have a row of coins and you can perform these three operations as many times as you like:

1. Remove three adjacent heads

2. Remove two adjacent tails

3. If there’s a head between two tails, then you can remove the head and swap the two tails to heads.

You apply these operations until you can’t make any more moves. Show that you will always get the same configuration at the end, no matter the order.

Let \(a\) be a positive integer, and let \(p\) be a prime number. Prove that \(a^p - a\) is a multiple of \(p\).

We have a square of side length 1. At each vertex of the square, we draw a circle of radius 1. What is the area bounded by all four circles?

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 \(A(S) = \frac{15}{2}\).

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 on the right below. How many different colourings are possible?

For any positive integer \(k\), the factorial \(k!\) is defined as a product of all integersbetween 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 \(8\)?

Find all functions \(f\) from the real numbers to the real numbers such that \(xy=f(x)f(y)-f(x+y)\) for all real numbers x and y.

There are two imposters and seven crewmates on the rocket ‘Plus’. How many ways are there for the nine people to split into three groups of three, such that each group has at least two crewmates? The two imposters and seven crewmates are all distinguishable from each other, but we’re not concerned with the order of the three groups.

For example: \(\{I1,C1,C2\}\), \(\{I2,C3,C4\}\) and \(\{C5,C6,C7\}\) is the same as
\(\{C3,C4,I2\}\), \(\{C5,C6,C7\}\) and \(\{I1,C2,C1\}\) but different from
\(\{I2,C1,C2\}\), \(\{I1,C3,C4\}\) and \(\{C5,C6,C7\}\).

Let \(n\) be a natural number, and let \(d(n)\) be the number of factors of \(n\). For example, the factors of \(6\) are \(1,2,3,6\), so \(d(6)=4\). Find all \(n\) such that \(d(n)+d(n+1)=5\).