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.
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?
Kate and Gina agreed to meet at the underground in the first hour of the afternoon. Kate comes to the meeting place between noon and one o’clock in the afternoon, waits for 10 minutes and then leaves. Gina does the same.
a) What is the probability that they will meet?
b) How will the probability of a meeting change if Gina decides to come earlier than half past twelve, and Kate still decides to come between noon and one o’clock?
c) How will the probability of a meeting change if Gina decides to come at an arbitrary time between 12:00 and 12:50, and Kate still comes between 12:00 and 13:00?
James furiously cuts a rectangular sheet of paper with scissors. Every second he cuts a random piece by an unsystematic rectilinear cut into two parts.
a) Find the mathematical expectation of the number of sides of a polygon (made from a piece of paper) that James randomly picks up after an hour of such work.
b) Solve the same problem if at first the piece of paper had the form of an arbitrary polygon.
The upper side of a piece of square paper is white, and the lower one is red. In the square, a point F is randomly chosen. Then the square is bent so that one randomly selected vertex overlaps the point F. Find the mathematical expectation of the number of sides of the red polygon that appears.
Ben is going to bend a square sheet of paper \(ABCD\). Ben calls the fold beautiful, if the side \(AB\) crosses the side \(CD\) and the four resulting rectangular triangles are equal. Before that, Jack selects a random point on the sheet \(F\). Find the probability that Ben will be able to make a beautiful fold through the point \(F\).
In the triangle \(ABC\), the angle \(A\) is equal to \(40^{\circ}\). The triangle is randomly thrown onto a table. Find the probability that the vertex \(A\) lies east of the other two vertices.
One day in autumn the Scattered Scientist glanced at his ancient wall clock and saw that three flies fell asleep on the dial. The first one slept exactly at the 12 o’clock mark on the clock, and the other two just as neatly settled on the marks of 2 hours and 5 hours. The scientist made measurements and determined that the hour hand does not threaten the flies, but the minute one will sweep them all in turn. Find the probability that exactly 40 minutes after the Scientist noticed the flies, exactly two flies out of three were swept away by the minute hand.
On one island, one tribe has a custom – during the ritual dance, the leader throws up three thin straight rods of the same length, connected in the likeness of the letter capital \(\pi\), \(\Pi\). The adjacent rods are connected by a short thread and therefore freely rotate relative to each other. The bars fall on the sand, forming a random figure. If it turns out that there is self-intersection (the first and third bars cross), then the tribe in the coming year are waiting for crop failures and all sorts of trouble. If there is no self-intersection, then the year will be successful – satisfactory and happy. Find the probability that in 2019, the rods will predict luck.