Problems

Find the minimal natural number \(n>1\) such that \(n^6 - 2n^5 - n^4 + 4n^3 - n^2 - 2n +1\) is divisible by \(2025\).

In chess, knights can move one square in one direction and two squares in a perpendicular direction. This is often seen as an ‘L’ shape on a regular chessboard. A closed knight’s tour is a path where the knight visits every square on the board exactly once, and can get to the first square from the last square.

This is a closed knight’s tour on a \(6\times6\) chessboard.

Can you draw a closed knight’s tour on a \(4\times4\) torus? That is, a \(4\times4\) square with both pairs of opposite sides identified in the same direction, like the diagram below.

Prove that one can tile the whole plane without spaces and overlaps, using any non self-intersecting quadrilaterals of the same shape. Note: quadrilaterals might not be convex.

Let \(n\) be a positive integer. We denote by \(s(n)\) the sum of the divisors of \(n\). For example, the divisors of \(n=6\) are \(1\), \(2\), \(3\) and \(6\), so \(s(6)=1+2+3+6=12\). Prove that, for all \(n\ge1\), \[\sum_{k=1}^ns(k)=s(1)+s(2)+...+s(n)\le\frac{\pi^2}{12}n^2+\frac{n\log n}{2}+\frac{n}{2}.\]

It is impossible to completely tile the plane using only regular pentagons. However, can you identify at least three different types of pentagons (each with at least two different corresponding sides AND angles) that can be used to tile the plane in three distinct ways? Here essentially different means the tilings with different pattern.

Draw how to tile the whole plane with figures, made from squares \(1\times 1\), \(2\times 2\), \(3\times 3\), where squares are used the same amount of times in the design of the figure.

Consider a segment on a line of length \(3m\). Jack chose \(4\) random points on the segment and measured all the distances between points. Prove that at least one of the distances is less or equal than \(1m\).

Twelve lines are drawn on the plane, passing through a point \(A\). Prove that there are two of them with angle less than \(17^{\circ}\) between them.

Inside a square of area \(6\), there are three polygons, each of area \(3\). Show that some two of them overlap and the area of the overlap is at least \(1\).

A Wimbledon doubles court is 78 ft \(\times\) 36 ft. After a long practice match, there were 79 tennis balls in the court area. Show that some two of them were no further than \(6\sqrt{2}\) ft away.