Problems

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 $y$.

It is known that $a>1$. Is it always true that $\lfloor \sqrt{\lfloor \sqrt{a}\rfloor }\rfloor =\lfloor \sqrt{4}a\rfloor$?

Does there exist a function $f(x)$ defined for all real numbers such that $f(\sin x)+f(\cos x)=\sin x$?

When 200 sweets are randomly distributed to a class of schoolchildren, there will always be at least two children who receive the same number of sweets or even no sweets at all. What is the minimum number of children in this class?

A mix of boys and girls are standing in a circle. There are 20 children in total. It is known that each boys’ neighbour in the clockwise direction is a child wearing a blue T-shirt, and that each girls’ neighbour in the anticlockwise direction is a child wearing a red T-shirt. Is it possible to uniquely determine how many boys there are in the circle?

In a $10\times 10$ square, all of the cells of the upper left $5\times 5$ square are painted black and the rest of the cells are painted white. What is the largest number of polygons that can be cut from this square (on the boundaries of the cells) so that in every polygon there would be three times as many white cells than black cells? (Polygons do not have to be equal in shape or size.)

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.

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.