What has a greater value: \(300!\) or \(100^{300}\)?
A numerical sequence is defined by the following conditions: \[a_1 = 1, \quad a_{n+1} = a_n + \lfloor \sqrt{a_n}\rfloor .\]
Prove that among the terms of this sequence there are an infinite number of complete squares.
Prove that in a three-digit number, that is divisible by 37, you can always rearrange the numbers so that the new number will also be divisible by 37.
The frog jumps over the vertices of the hexagon \(ABCDEF\), each time moving to one of the neighbouring vertices.
a) How many ways can it get from \(A\) to \(C\) in \(n\) jumps?
b) The same question, but on condition that it cannot jump to \(D\)?
c) Let the frog’s path begin at the vertex \(A\), and at the vertex \(D\) there is a mine. Every second it makes another jump. What is the probability that it will still be alive in \(n\) seconds?
d)* What is the average life expectancy of such frogs?
Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.
Is it possible to:
a) load two coins so that the probability of “heads” and “tails” were different, and the probability of getting any of the combinations “tails, tails,” “heads, tails”, “heads, heads” be the same?
b) load two dice so that the probability of getting any amount from 2 to 12 would be the same?
The figure shows the scheme of a go-karting route. The start and finish are at point \(A\), and the driver can go along the route as many times as he wants by going to point \(A\) and then back onto the circle.
It takes Fred one minute to get from \(A\) to \(B\) or from \(B\) to \(A\). It also takes one minute for Fred to go around the ring and he can travel along the ring in an anti-clockwise direction (the arrows in the image indicate the possible direction of movement). Fred does not turn back halfway along the route nor does not stop. He is allowed to be on the track for 10 minutes. Find the number of possible different routes (i.e. sequences of possible routes).
An exam is made up of three trigonometry problems, two algebra problems and five geometry problems. Martin is able to solves trigonometry problems with probability \(p_1 = 0.2\), geometry problems with probability \(p_2 = 0.4\), and algebra problems with probability \(p_3 = 0.5\). To get a \(B\), Martin needs to solve at least five problems, where the grades are as follows \((A+, A, B, C, D)\).
a) With what probability does Martin solve at least five problems?
Martin decided to work hard on the problems of any one section. Over a week, he can increase the probability of solving the problems of this section by 0.2.
b) What section should Martin complete, so that the probability of solving at least five problems becomes the greatest?
c) Which section should Martin deal with, so that the mathematical expectation of the number of solved problems becomes the greatest?
\(N\) people lined up behind each other. The taller people obstruct the shorter ones, and they cannot be seen.
What is the mathematical expectation of the number of people that can be seen?
In the centre of a rectangular billiard table that is 3 m long and 1 m wide, there is a billiard ball. It is hit by a cue in a random direction. After the impact the ball stops passing exactly 2 m. Find the expected number of reflections from the sides of the table.