A firm recorded its expenses in pounds for 100 items, creating a list of 100 numbers (with each number having no more than two decimal places). Each accountant took a copy of the list and found an approximate amount of expenses, acting as follows. At first, he arbitrarily chose two numbers from the list, added them, discarded the sum after the decimal point (if there was anything) and recorded the result instead of the selected two numbers. With the resulting list of 99 numbers, he did the same, and so on, until there was one whole number left in the list. It turned out that in the end all the accountants ended up with different results. What is the largest number of accountants that could work in the company?
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.
A cubic polynomial \(f (x)\) is given. Let’s find a group of three different numbers \((a, b, c)\) such that \(f (a)= b\), \(f (b) = c\) and \(f (c) = a\). It is known that there were eight such groups \([a_i, b_i, c_i]\), \(i = 1, 2, \dots , 8\), which contains 24 different numbers. Prove that among eight numbers of the form \(a_i + b_i + c_i\) at least three are different.
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 the power plant, rectangles that are 2 m long and 1 m wide are produced. The length of the objects is measured by the worker Howard, and the width, irrespective of Howard, is measured by the worker Rachel. The average error is zero for both, but Howard allows a standard measurement error (standard deviation of length) of 3 mm, and Rachel allows a standard error of 2 mm.
a) Find the mathematical expectation of the area of the resulting rectangle.
b) Find the standard deviation of the area of the resulting rectangle in centimetres squared.
In a convex polygon, which has an odd number of vertices equal to \(2n + 1\), two independently of each other random diagonals are chosen. Find the probability that these diagonals intersect inside the polygon.
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.
Investigating one case, the investigator John Smith discovered that the key witness is the one from the Richardson family who, on that fateful day, came home before the others. The investigation revealed the following facts.
1. The neighbour Maria Ramsden, wanting to borrow some salt from the Richardson’s, rang their doorbell, but no one opened the door. At what time though? Who knows? It was already dark...
2. Jill Richardson came home in the evening and found both children in the kitchen, and her husband was on the sofa – he had a headache.
3. The husband, Anthony Richardson, declared that, when he came home, immediately sat down on the sofa and had a nap. He did not see anyone, nor did he hear anything, and the neighbour definitely did not come – the doorbell would have woken him up.
4. The daughter, Sophie, said that when she returned home, she immediately went to her room, and she does not know anything about her father, however, in the hallway, as always, she stumbled on Dan’s shoes.
5. Dan does not remember when he came home. He also did not see his father, but he did hear how Sophie got angry about his shoes.
“Aha,” thought John Smith. “What is the likelihood that Dan returned home before his father?”.