Problems

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 \(u>v\).

The lengths of three sides of a right-angled triangle are all integers.

Show that one of them is divisible by \(5\).

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 \(\phi(n)\) 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 \(\phi(mn) = \phi(m)\phi(n)\).

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 \(\{C_5,C_6,C_7\}\).

Draw the plane tiling with regular hexagons.

Let \(\sigma(n)\) be the sum of the divisors of \(n\). For example, \(\sigma(12)=1+2+3+4+6+12=28\). We use \(\gamma\) to denote the Euler-Mascheroni constant - one way to define this is as \(\gamma:=\lim_{n\to\infty}(\sum_{k=1}^n\frac{1}{n}-\log n)\).

Prove that \(\sigma(n)<e^{\gamma}n\log\log n\) for all integers \(n>5040\).

A natural number \(N\) is called perfect if it equals the sum of its divisors, except for \(N\) itself. Prove that if \(2^r-1\) is prime, then \((2^r-1)2^{r-1}\) is a perfect number. Are there any odd perfect numbers?