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In a row there are 2023 numbers. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of two neighboring ones.

Find the last number.

We consider a sequence of words consisting of the letters “A” and “B”. The first word in the sequence is “A”, the k-th word is obtained from the $(k-1)$-th by the following operation: each “A” is replaced by “AAB” and each “B” by “A”. It is easy to see that each word is the beginning of the next, thus obtaining an infinite sequence of letters: AABAABAAABAABAAAB …

$\\$ a) Where in this sequence will the 1000th letter “A” be?

$\\$ b) Prove that this sequence is non-periodic.

For which natural $n$ does the number $\frac{n^2}{1,001^n}$ reach its maximum value?

$a_1$, $a_2$, $a_3$, … is an increasing sequence of natural numbers. It is known that $a_{a_k} = 3k$ for any $k.$ Find a$)$ $a_{100}$; b$)$ $a_{2022}$.

A sequence of natural numbers $a_1 < a_2 < a_3 < … < a_n < …$ is such that each natural number is either a term in the sequence, can be expressed as the sum of two terms in the sequence, or perhaps the same term twice. Prove that $a_n \leq n^2 $ for any $n=1, 2, 3,…$

An incredible legend says that one day Stirling was considering the numbers of Stirling of the second kind. During his thoughtfulness, he threw 10 regular dice on the table. After the next throw, he suddenly noticed that in the dropped combination of points there were all of the numbers from 1 to 6. Immediately Stirling reflected: what is the probability of such an event? What is the probability that when throwing 10 dice each number of points from 1 to 6 will drop out on at least one die?

A sequence consists of 19 ones and 49 zeros, arranged in a random order. We call the maximal subsequence of the same symbols a “group”. For example, in the sequence 110001001111 there are five groups: two ones, then three zeros, then one one, then two zeros and finally four ones. Find the mathematical expectation of the length of the first group.

At the ball, there were n married couples. In each pair, the husband and wife are of the same height, but there are no two pairs of the same height. The waltz begins, and all those who came to the ball randomly divide into pairs: each gentleman dances with a randomly chosen lady.

Find the mathematical expectation of the random variable X, “the number of gentlemen who are shorter than their partners”.

At the Antarctic station, there are n polar explorers, all of different ages. With the probability p between each two polar explorers, friendly relations are established, regardless of other sympathies or antipathies. When the winter season ends and it’s time to go home, in each pair of friends the senior gives the younger friend some advice. Find the mathematical expectation of the number of those who did not receive any advice.

A high rectangle of width 2 is open from above, and the L-shaped domino falls inside it in a random way $($see the figure$)$.

a) $k$ $L$-shaped dominoes have fallen. Find the mathematical expectation of the height of the resulting polygon.

b) $7$ $G$-shaped dominoes fell inside the rectangle. Find the probability that the resulting figure will have a height of 12.

Hercules meets the three-headed snake, the Lernaean Hydra and the battle begins. Every minute, Hercules cuts one of the snake’s heads off. With probability ¼ in the place of the chopped off head grows two new ones, with a probability of 1/3, only one new head will grow and with a probability of 5/12, not a single head will appear. The serpent is considered defeated if he does not have a single head left. Find the probability that sooner or later Hercules will beat the snake.

A fair dice is thrown many times. It is known that at some point the total amount of points became equal to exactly 2010.

Find the mathematical expectation of the number of throws made to this point.

A regular dice is thrown many times. Find the mathematical expectation of the number of rolls made before the moment when the sum of all rolled points reaches 2010 $($that is, it became no less than 2010$)$.

On a calculator keypad, there are the numbers from 0 to 9 and signs of two actions $($see the figure$)$. First, the display shows the number 0. You can press any keys. The calculator performs the actions in the sequence of clicks. If the action sign is pressed several times, the calculator will only remember the last click.

a) The button with the multiplier sign breaks and does not work. The Scattered Scientist pressed several buttons in a random sequence. Which result of the resulting sequence of actions is more likely: an even number or an odd number?

b) Solve the previous problem if the multiplication symbol button is repaired.

In the magical land of Anchuria there is a drafts championship made up of several rounds. The days and cities in which the rounds are carried out are determined by a draw. According to the rules of the championship, no two rounds can take place in one city, and no two rounds can take place on one day. Among the fans, a lottery is arranged: the main prize is given to those who correctly guess, before the start of the championship, in which cities and on which days all of the round will take place. If no one guesses, then the main prize will go to the organising committee of the championship. In total, there are eight cities in Anchuria, and the championship is only allotted eight days. How many rounds should there be in the championship, so that the organising committee is most likely to receive the main prize?

A coin is thrown 10 times. Find the probability that it never lands on two heads in a row.

A and B shoot in a shooting gallery, but they only have one six-shot revolver with one cartridge. Therefore, they agreed in turn to randomly rotate the drum and shoot. A goes first. Find the probability that the shot will occur when A has the revolver.

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$)$.

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?