Problems

Let \(n\) be an integer number, \(a\) be an integer, coprime with \(n\). Prove that \(a^{\phi(n)-1}-1\) is divisible by \(n\).

In the diagram below, I wish to write the numbers \(6, 11, 19, 23, 25, 27\) and \(29\) in the squares, but I want the sum of the numbers in the horizontal row to equal the sum of the numbers in the vertical column. What number should I put in the blue square with the question mark?

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 ‘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 textitsemiperimeter, \(s=\frac{a+b+c}{2}\). Then the area of this triangle is \[\sqrt{s(s-a)(s-b)(s-c)}.\] Prove this formula is correct.

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.

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 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 \(8\)?

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