OPM 2022

The Proenc has a new $8\times 8$ chess board and requires composing it into rectangles that do not overlap, so that: (i) each rectangle has as many white squares as black ones; (ii) there are no two rectangles with the same number of squares. Determines the maximum value of $n$ for which such a decomposition is possible. For this value of $n$ , determine all possible sets ${A_1,... ,A_n}$ , where $A_i$ is the number of rectangle $i$ in squares, for which a decomposition of the board under the conditions intended actions is possible.

Let $[AD]$ be a median of the triangle $[ABC]$ . Knowing that $\angle ADB = 45^o$ and $\angle A CB = 30^o$ , prove that $\angle BAD = 30^o$ .

In a badminton competition, $16$ players participate, of which $10$ are professionals and $6$ are amateurs. In the first phase, eight games are drawn. Among the eight winners of these games, four games are drawn. The four winners qualify for the semi-finals of the competition. Assuming that, whenever a professional player and an amateur play each other, the professional wins the game, what is the probability that an amateur player will reach the semi-finals of the competition?

Given two natural numbers $a < b$ , Xavier and Ze play the following game. First, Xavier writes $a$ consecutive numbers of his choice; then, repeat some of them, also of his choice, until he has $b$ numbers, with the condition that the sum of the $b$ numbers written is an even number. Ze wins the game if he manages to separate the numbers into two groups with the same amount. Otherwise, Xavier wins. For example, for $a = 4$ and $b = 7$ , if Xavier wrote the numbers $3,4,5,6,3,3,4$ , Ze could win, separating these numbers into groups $3,3 ,4,4$ and $3,5,6$ . For what values of $a$ and $b$ can Xavier guarantee victory?