Let \(n\geq 2\) be an integer. Fix \(2n\) points in space, so that no four points lie on a common plane. Suppose there are \(n^2+1\) segments between these points. Show that these segments must form at least \(n\) triangles.
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?
A book club with 37 members is reading the following books: “Brave New World", “Dracula" and “Flatland". Each member chose one of the books, though some people chose more than one book. We know that:
23 people chose “Brave New World";
18 people chose “Dracula";
26 people chose “Flatland";
7 people chose all three books.
How many people chose at least two books?
Jan wants to paint a map with \(95\) countries on it, where only one colour can be used for each country. He has \(33\) different colours to paint with and he must use each of yellow, blue, green, purple and red at least once. How many ways of painting the map are there?
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?