Problems

A piece containing exactly \(4\) black cells is cut out from a regular \(8\) by \(8\) chessboard. You are only allowed to cut along the edges of the cells and the piece must be connected - namely you cannot have cells attached only with a vertex, they have to share a common edge.

Find the largest possible area of such a piece.

In a distant village, there are \(3\) houses and \(3\) wells. Inhabitants of each house want to have access to all \(3\) wells. Is it possible to build non-intersecting straight paths from each house to each well? All houses and well must be level (that is, none of them are higher up, like on a mountain, nor are any of them on lower ground, like in a valley).

Today you saw two infinitely long buses with seats numbered as \(1,2,3,...\) carrying infinitely many guests each arriving at the full hotel. How do you accommodate everyone?

Find all \(n\) such that a closed system of \(n\) gears in a plane can rotate. We call a system closed if the first gear wheel is connected to the second and the \(n\)th, the second is connected to the first and the third, the third is connected to the second and the fourth, the fourth is connected to the third and the fifth, and so on until the \(n\)th is connected to the \(n-1\)th and the first. In the picture, we have a closed system of three gears.

Show that \(\sqrt[3]{3}\) is irrational.

Suppose \(x,y\) are real numbers such that \(x < y + \varepsilon\) for every \(\varepsilon > 0\). Show that \(x \leq y\).

Could you meet a person inhabiting this planet who asks you “Am I a Goop?"

On this planet you meet a couple called Tom and Betty. You hear Tom ask someone: “Are Betty and I both Goops?"
What kind is Betty?

You learn that one of the aliens living on this planet is a wizard. You learnt that by overhearing a certain question being asked on the planet. What question could that have been?

Suppose you meet a person inhabiting this planet and they ask you “Am I a Crick?" What would you conclude?