Problems

The tower in the castle of King Arthur is crowned with a roof, which is a triangular pyramid, in which all flat angles at the top are straight. Three roof slopes are painted in different colours. The red roof slope is inclined to the horizontal at an angle $\alpha$, and the blue one at an angle $\beta$. Find the probability that a raindrop that fell vertically on the roof in a random place fell on the green area.

If one person spends one minute waiting, we will say that one human-minute is spent aimlessly. In the queue at the bank, there are eight people, of which five plan to carry out simple operations, which take 1 minute, and the others plan to carry out long operations, taking 5 minutes. Find:

a) the smallest and largest possible total number of aimlessly spent human-minutes;

b) the mathematical expectation of the number of aimlessly spent human-minutes, provided that customers queue up in a random order.

There are 9 street lamps along the road. If one of them does not work but the two next to it are still working, then the road service team is not worried about it. But if two lamps in a row do not work then the road service team immediately changes all non-working lamps. Each lamp does not work independently of the others.

a) Find the probability that the next replacement will include changing 4 lights.

b) Find the mathematical expectation of the number of lamps that will have to be changed on the next replacement.

What is the smallest number of cells that can be chosen on a $15\times 15$ board so that a mouse positioned on any cell on the board touches at least two marked cells? (The mouse also touches the cell on which it stands.)

What is the largest number of horses that can be placed on an $8\times 8$ chessboard so that no horse touches more than seven of the others?

Harry thought of two positive numbers $x$ and $y$. He wrote down the numbers $x+y$, $x-y$, $xy$ and $x/y$ on a board and showed them to Sam, but did not say which number corresponded to which operation.

Prove that Sam can uniquely figure out $x$ and $y$.

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).