Problems

We ‘typically’ use the formula $\frac{1}{2}bh$ for the area of a triangle, where $b$ is the length of the base, and $h$ is the perpendicular height. Here’s another one, called Heron’s formula.

Call the sides of the triangle $a$, $b$ and $c$. The perimeter is $a+b+c$. We call half of this the semiperimeter, $s=\frac{a+b+c}{2}$. Then the area of this triangle is $$\sqrt{s(s-a)(s-b)(s-c)}.$$ Prove this formula is correct.

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?

Today we will be finding the areas of some geometric figures. Here is a brief reminder of how to calculate the area of common shapes.

In the picture below, the area of the rectangle is $|AB|\times |AD|$.

The area of a triangle is given by $\frac{1}{2}bh$, where $b$ is the length of a chosen base and $h$ is the height. In this case, the segment $AB$ is the base and $CD$ is the altitude corresponding to the base $AB$.

The area of a circle with radius $r$ is $\pi r^2$. The number $\pi$ is approximately 3.14159 to five decimal points.

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.

Let $\varphi (n)$ be the Euler’s function, namely the amount of numbers from $1$ to $n$, coprime with $n$. For two natural numbers $m,n$ such that $\mathbb{GCD}(m,n)=1$ prove that $\varphi (mn)=\varphi (m)\varphi (n)$.

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 $3\times 3$ grid of squares is coloured red, white, blue, or green so that every $2\times 2$ 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 $8$?

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: $\{I_1,C_1,C_2\}$, $\{I_2,C_3,C_4\}$ and $\{C_5,C_6,C_7\}$ is the same as $\{C_3,C_4,I_2\}$, $\{C_5,C_6,C_7\}$ and $\{I_1,C_2,C_1\}$ but different from $\{I_2,C_1,C_2\}$, $\{I_1,C_3,C_4\}$ and $\{C_5,C_6,C_7\}$.

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 $d(n)+d(n+1)=5$.