Problems
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: \[\frac1{\sqrt{1}} +\frac1{\sqrt{2}}+ \frac1{\sqrt{3}} + \dots +\frac1{\sqrt{n}} < 3 \sqrt{n+1} -3.\]
If \(n\) is a positive integer, we denote by \(s(n)\) the sum of the divisors of \(n\). For example, the divisors of \(n=6\) are \(1,2,3,6\), so \(s(6)=1+2+3+6=12\). Prove that, for all \(n\geq1\), \[s(1)+s(2)+\cdots+s(n)\leq n^2.\] Denote by \(t(n)\) is instead the sum of the squares of the divisors of \(n\) (e.g., \(t(6)=1^2+2^2+3^2+6^2=50\)), can you find a similar inequality for \(t(n)\)?
There are \(16\) cities in the kingdom. We would like to build roads between these cities so that one can get from any city to any other without passing through more than one city on the way. To save cost, we would like to have no more than four roads coming out of each city. Prove that such a system of roads is unfortunately impossible to build.
Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a square, or slightly harder in a shape of a given rectangle.
Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a shape like this
Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a given triangle.
Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of half a circle.
