Problems

Can three points with integer coordinates be the vertices of an equilateral triangle?

Prove that there are infinitely many prime numbers \(\{2,3,5,7,11,13...\}\).

In the triangle \(ABC\) the angle \(\angle ABC = 120^{\circ}\). The segments \(AF,\, BE\), and \(CD\) are the bisectors of the corresponding angles of the triangle \(ABC\). Prove that the angle \(\angle DEF = 90^{\circ}\).

In the triangle \(ABC\) the lines \(AE\) and \(CD\) are the bisectors of the angles \(\angle BAC\) and \(\angle BCA\), intersecting at the point \(I\). In the triangle \(BDE\) the lines \(DG\) and \(EF\) are the bisectors of the angles \(\angle BDE\) and \(\angle BED\), intersecting at the point \(H\). Prove that the points \(B,\,H,\, I\) are situated on one straight line.

Jess and Tess are playing a game colouring points on a blank plane. Jess is moving first, she picks a non-colored point on a plane and colours it red. Then Tess makes a move, she picks \(2022\) colourless points on the plane and colours them all green. Jess then moves again, and they take turns. Jess wins if she manages to create a red equilateral triangle on the plane, Tess is trying to prevent that from happening. Will Jess always eventually win?

Can you cover a \(13 \times 13\) square using \(2 \times 2\) and \(3 \times 3\) squares?

A \((2n - 1) \times (2n - 1)\) board is tiled with pieces of the following possible types:

Prove that at least \(4n-1\) of the first type have been used.

Two circles with centres \(A\) and \(C\) are tangent to each other at the point \(B\). Both circles are tangent to the sides of an angle with vertex \(D\). It is known that the angle \(\angle EDF = 60^{\circ}\) and the radius of the smaller circle \(AF=5\). Find the radius of the large circle.

Two circles with centres \(A\) and \(C\) are tangent to each other at the point \(B\). Two points \(D\) and \(E\) are chosen on the circles in such a way that a segment \(DE\) passes through the point \(B\). Prove that the tangent line to one circle at the point \(D\) is parallel to the tangent line to the other circle at the point \(E\).

The number \(n\) is natural. Show that: \[\frac1{\sqrt{1}} +\frac1{\sqrt{2}}+ \frac1{\sqrt{3}} + \dots +\frac1{\sqrt{n}} < 3 \sqrt{n+1} -3.\]