Problems

The sequence of numbers $a_1,a_2,\dots$ is given by the conditions $a_1=1$, $a_2=143$ and

for all $n\ge 2$.

Prove that all members of the sequence are integers.

In the number $1234096\dots$ each digit, starting with the 5th digit, is equal to the final digit of the sum of the previous 4 digits. Will the digits 8123 ever occur in a row in this number?

The sequence of numbers $a_n$ is given by the conditions $a_1=1$, $a_{n+1}=a_n+1/a_n^2$ ($n\ge 1$).

Is it true that this sequence is limited?

Definition. The sequence of numbers $a_0,a_1,\dots ,a_n,\dots$, which, with the given $p$ and $q$, satisfies the relation $a_{n+2}=p{a}_{n+1}+q{a}_n$ ($n=0,1,2,\dots$) is called a linear recurrent sequence of the second order.

The equation $$x^2-px-q=0$$ is called a characteristic equation of the sequence $\{a_n\}$.

Prove that, if the numbers $a_0$, $a_1$ are fixed, then all of the other terms of the sequence $\{a_n\}$ are uniquely determined.

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).

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

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.

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).

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 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.