A person has 10 friends and within a few days invites some of them to visit so that his guests never repeat (on some of the days he may not invite anyone). How many days can he do this for?
How many necklaces can be made from five identical red beads and two identical blue beads?
How many ways can you build a closed line whose vertices are the vertices of a regular hexagon (the line can be self-intersecting)?
How many different four-digit numbers, divisible by 4, can be made up of the digits 1, 2, 3 and 4,
a) if each number can occur only once?
b) if each number can occur several times?
How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?
Prove that there is no graph with five vertices whose degrees are equal to 4, 4, 4, 4, 2.
Eugenie, arriving from Big-island, said that there are several lakes connected by rivers. Three rivers flow from each lake, and four rivers flow into each lake. Prove that she is wrong.
Prove that \(\frac {1}{2} (x^2 + y^2) \geq xy\) for any \(x\) and \(y\).
Prove that for \(a, b, c > 0\), the following inequality is valid: \(\left(\frac{a+b+c}{3}\right)^2 \ge \frac{ab+bc+ca}{3}\).
Prove that for \(x \geq 0\) the inequality is valid: \(2x + \frac {3}{8} \ge \sqrt[4]{x}\).