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 \(3\)?
Three kinds of cookies are sold at a store: dark chocolate \((D)\), raspberry with white chocolate \((R)\) and honeycomb \((H)\). Here is a table summarizing the number of people buying cookies this morning.
| \(D\) | \(R\) | \(H\) | \(D, R\) | \(D,H\) | \(R,H\) | \(D,R,H\) | |
|---|---|---|---|---|---|---|---|
| Number of people | 16 | 16 | 10 | 7 | 5 | 3 | 1 |
The column with label \(D,H\), for example, means the number of people who bought both dark chocolate and honeycomb cookies.
How many people bought cookies this morning?
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?
We write all \(26\) different letters of the English alphabet in a line, using each letter exactly once.
How many such arrangements do not contain any of the strings
fish, rat, or bird?
In the trapezium \(ABCD\) the point \(E\) is the point of intersection of the sides \(AB\) and \(CD\), the point \(G\) is the point of intersection of diagonals, the points \(F,H\) are the midpoints of the sides \(BC\) and \(AD\) respectively.
Prove that the points \(E,F,G,H\) lie on one line.
The triangle \(EFG\) is equilateral and \(EF=EG\). 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 he triangle \(EFG\).
Prove that under the homothety transformation a polygon is transferred into a similar polygon. I.e. homothety preserves angles and ratios.
Remember that a still-life in the Game of Life was a pattern that did not change after any number of steps. Show that the following pattern can not be turned into a still life by only changing dead cells into alive cells.
