Problems

Elections are approaching in Problemland! There are three candidates for president: \(A\), \(B\), and \(C\).

An opinion poll reports that \(65\%\) of voters would be satisfied with \(A\), \(57\%\) with \(B\), and \(58\%\) with \(C\). It also says that \(28\%\) would accept \(A\) or \(B\), \(30\%\) \(A\) or \(C\), \(27\%\) \(B\) or \(C\), and that \(12\%\) would be content with all three candidates.

Show that there must have been a mistake in the poll.

You are creating passwords of length \(8\) using only the letters \(A\), \(B\), and \(C\). Each password must use all three letters at least once.

How many such passwords are there?

How many numbers from \(1\) to \(1000\) are divisible by \(2\) or \(3\)?

At the space carnival, visitors can try two special attractions: the Zero-Gravity Room or the Laser Maze. By the end of the day:

• \(100\) visitors have tried at least one of the two attractions,

• \(50\) visitors tried the Laser Maze,

• \(20\) visitors tried both attractions.

How many visitors tried only the Zero-Gravity Room?

The triangle \(EFG\) is isosceles with \(EF=EG\). A circle with center \(A\) is tangent to the sides \(EF\) and \(EG\) at the points \(C\) and \(B\) respectively. It is also tangent to the circle circumscribed around the triangle \(EFG\) at the point \(H\). Prove that the midpoint of the segment \(BC\) is the center of the circle inscribed into the triangle \(EFG\).

We are going to play the following game: look at the grid below, where each cell is painted black or white

choose any two cells that are next to each other and switch both of their colours. By switch, we mean: if a cell is black, change it to white, and if it is white, change it to black. Is it possible to turn the entire grid black?

Under a homothety transformation, a line \(l\) is sent to a line \(l'\) which is parallel to \(l\).

Show that a homothety is uniquely determined by where it sends any two distinct points.

Consider a homothety with center \(O\) and coefficient \(k\). Which lines are sent to themselves by this homothety? (Hint: the answer will depend on \(k\))

On the first day Robinson Crusoe tied the goat with a single piece of rope by putting one peg into the ground. What shape did the goat graze?