Problems

In the diagram below, there are nine discs - each is black on one side, and white on the other side. Two have black face-up right now. Your task is to remove all of the discs by making a series of the following moves. Each move includes choosing a black disc, flipping over its neighbours\(^*\) and removing that black disc. Discs are ‘neighbours’ if they’re adjacent at the beginning - removing a disc creates a gap, so that at later stages, a disc may have two, one or even zero neighbours left. \[\circ\circ\circ\bullet\circ\circ\circ\circ\bullet\] Show that this task is impossible.

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 \(n>5040\).

Let \(a\) and \(b\) be two different \(9\)-digit numbers. It is known that each one of them contains all of the digits \(1,2,...9\). Find the maximal value of \(\gcd(a,b)\).

Take a regular dodecahedron as in the image. It has \(12\) regular pentagons as its faces, \(30\) edges, and \(20\) vertices. We can cut it with planes in various ways and the cut will be a polygon on a plane. Find out how many ways there are to cut a dodecahedron with a plane so that the polygon obtained is a regular hexagon.

For an odd number \(N\) denote by \(A\) the minimal positive difference between prime divisors of \(N\), denote by \(B\) the minimal positive difference between composite divisors of \(N\). Usually we have \(A<B\), but can we have \(A>B\)? (Disregard numbers such as \(15\) where one of \(A\) or \(B\) is not defined)

Let \(n\) be an integer bigger than \(1\), and \(p\) a prime number. Suppose that \(n\) divides \(p-1\) and \(p\) divides \(n^3-1\). Prove that \(4p-3\) is a square number.

Let \(n\) be a composite number. Arrange the factors of \(n\) greater than \(1\) in a circle. When can this be done such that neighbours in the circle are never coprime?

Let \(x\), \(y\), \(z\) and \(w\) be non-negative integers. Find all solutions to \(2^x3^y-5^z7^w=1\).

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?