On the dining room table, there is a choice of six dishes. Every day Valentina takes a certain set of dishes (perhaps, she does not take a single dish), and this set of dishes should be different from all of the sets that she took in the previous days. What is the maximum number of days that Valentina will be able to eat according to such rules and how many meals will she eat on average during the day?
There are 30 students in the class. Prove that the probability that some two students have the same birthday is more than 50%.
Are there any irrational numbers \(a\) and \(b\) such that the degree of \(a^b\) is a rational number?
Three people play table tennis, and the player who lost the game gives way to the player who did not participate in it. As a result, it turned out that the first player played 10 games and the second played 21 games. How many games did the third player play?
One of \(n\) prizes is embedded in each chewing gum pack, where each prize has probability \(1/n\) of being found.
How many packets of gum, on average, should I buy to collect the full collection prizes?
You are given 10 different positive numbers. In which order should they be named \(a_1, a_2, \dots , a_{10}\) such that the sum \(a_1 +2a_2 +3a_3 +\dots +10a_{10}\) is at its maximum?
Prove that, if \(b=a-1\), then \[(a+b)(a^2 +b^2)(a^4 +b^4)\dotsb(a^{32} +b^{32})=a^{64} -b^{64}.\]
Solve the equation \(xy = x + y\) in integers.
Determine all solutions of the equation \((n + 2)! - (n + 1)! - n! = n^2 + n^4\) in natural numbers.
Find all functions \(f (x)\) defined for all real values of \(x\) and satisfying the equation \(2f (x) + f (1 - x) = x^2\).