Problems

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.

Alice the fox and Basilio the cat have grown \(20\) counterfeit bills on a money tree and now write seven-digit numbers on them. Each bill has \(7\) empty cells for numbers. Basilio calls out one digit "1" or "2" (he doesn’t know the others), and Alice writes the number into any empty cell of any bill and shows the result to Basilio. When all the cells are filled, Basilio takes as many bills with different numbers as possible (out of several with the same number, he takes only one), and the rest is taken by Alice. What is the largest number of bills Basilio can get, regardless of Alice’s actions?

Detective Nero Wolf investigates a crime. He’s got \(80\) people involved in the case, among whom one is a criminal and another is a witness to the crime (but it is not known who either of them are). Each day the detective may invite one or more of these \(80\) people, and if there is a witness among those invited, but not the perpetrator, the witness will report who the perpetrator is. Can the detective solve a case in \(12\) days?

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. Prove that it is not possible to build a system of roads in such a way that one can get from any city to any other without passing through more than one city on the way, and with at most four roads coming out of each city.

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