Problems

What has a greater value: $300!$ or ${100}^{300}$?

A numerical sequence is defined by the following conditions: $$a_1=1,a_{n+1}=a_n+\lfloor \sqrt{a_n}\rfloor .$$

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.

Carry out the following experiment 10 times: first, toss a coin 10 times in a row and record the number of heads, then toss the coin 9 times in a row and again, record the number of heads. We call the experiment successful, if, in the first case, the number of heads is greater than in the second case. After conducting a series of 10 such experiments, record the number of successful and unsuccessful experiments. Collect the statistics in the form of a table.

a) Anton throws a coin 3 times, and Tina throws it two times. What is the probability that Anton gets more heads than Tina?

b) Anton throws a coin $n+1$ times, and Tanya throws it $n$ times. What is the probability that Anton gets more heads than Tina?

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?

Anna, Boris and Fred decided to go to a children’s Christmas party. They agreed to meet at the bus stop, but they do not know who will come to what time. Each of them can come at a random time from 15:00 to 16:00. Fred is the most patient of them all: if he comes and finds that neither Anna nor Boris are at the bus stop, then Fred will wait for one of them for 15 minutes, and if he waits for more than 15 minutes and no one arrives he will go to the Christmas party by himself. Boris is less patient: he will only wait for 10 minutes. Anna is very impatient: she will not wait at all. However, if Boris and Fred meet, they will wait for Anna until 16:00. What is the probability that all of them will go to the Christmas party?

On a roulette, any number from 0 to 2007 can be determined with the same probability. The roulette is spun time after time. Let $P_k$ denote the probability that at some point the sum of the numbers that are determined by a ball being thrown on the roulette is $k$. Which number is larger: $P_{2007}$ or $P_{2008}$?

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).