An abstract artist took a wooden \(5\times 5\times 5\) cube and divided each face into unit squares. He painted each square in one of three colours – black, white, and red – so that there were no horizontally or vertically adjacent squares of the same colour. What is the smallest possible number of squares the artist could have painted black following this rule? Unit squares which share a side are considered adjacent both when the squares lie on the same face and when they lie on adjacent faces.
Author: A. Glazyrin
In the coordinate space, all planes with the equations \(x \pm y \pm z = n\) (for all integers \(n\)) were carried out. They divided the space into tetrahedra and octahedra. Suppose that the point \((x_0, y_0, z_0)\) with rational coordinates does not lie in any plane. Prove that there is a positive integer \(k\) such that the point \((kx_0, ky_0, kz_0)\) lies strictly inside some octahedron from the partition.
For the anniversary of the London Mathematical Olympiad, the mint coined three commemorative coins. One coin turned out correctly, the second coin on both sides had two heads, and the third had tails on both sides. The director of the mint, without looking, chose one of these three coins and tossed it at random. She got heads. What is the probability that the second side of this coin also has heads?
In a convex hexagon, independently of each other, two random diagonals are chosen. Find the probability that these diagonals intersect inside the hexagon (inside – that is, not at the vertex).
The shooter shoots at 3 targets until he shoots everything. The probability of a hit with one shot is \(p\).
a) Find the probability that he needs exactly 5 shots.
b) Find the mathematical expectation of the number of shots.
Ten tennis players came to the competitions, 4 of them were from Russia. According to the rules for the first round, the tennis players are broken into pairs randomly. Find the probability that in the first round, all Russian tennis players will play only with other Russian tennis players.
At a conference there were 18 scientists, of which exactly 10 know the eye-popping news. During the break (coffee break), all scientists are broken up into random pairs, and in each pair, anyone who knows the news, tells this news to another if he did not already know it.
a) Find the probability that after the coffee break, the number of scientists who know the news will be 13.
b) Find the probability that after the coffee break the number of scientists who know the news will be 14.
c) Denote by the letter \(X\) the number of scientists who know the eye-popping news after the coffee break. Find the mathematical expectation of \(X\).
A table of size \(3 \times 3\) (as for playing tic-tac-toe) is given. Four chips are put (one each) on four randomly selected cells. Find the probability that among these four chips there are three that stand in a row vertically, horizontally or diagonally.
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.
A ticket for a train costs 50 pence, and the penalty for a ticketless trip is 450 pence. If the free rider is discovered by the controller, he pays both the penalty and the ticket price. It is known that the controller finds the free rider on average once out of every 10 trips. The free rider got acquainted with the basics of probability theory and decided to adhere to a strategy that gives the mathematical expectation of spending the smallest possible. How should he act: buy a ticket every time, never buy one, or throw a coin to determine whether he should buy a ticket or not?