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 \(y\).
The functions \(f\) and \(g\) are defined on the entire number line and are reciprocal. It is known that \(f\) is represented as a sum of a linear and a periodic function: \(f (x) = kx + h (x)\), where \(k\) is a number, and \(h\) is a periodic function. Prove that \(g\) is also represented in this form.
Is there a positive integer \(n\) such that \[\sqrt{n}{17\sqrt{5} + 38} + \sqrt{n}{17\sqrt{5} - 38} = 2\sqrt{5}\,?\]
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?
a) There is an unlimited set of cards with the words “abc”, “bca”, “cab” written. Of these, the word written is determined according to this rule. For the initial word, any card can be selected, and then on each turn to the existing word, you can either add on a card to the left or to the right, or cut the word anywhere (between the letters) and put a card there. Is it possible to make a palindrome with this method?
b) There is an unlimited set of red cards with the words “abc”, “bca”, “cab” and blue cards with the words “cba”, “acb”, “bac”. Using them, according to the same rules, a palindrome was made. Is it true that the same number of red and blue cards were used?
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.
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.