Polish Mathematical Olympiad, 2021

Problem 1

Positive integers $a$, $b$, $n$ satisfy the equality

$$ \frac{a}{b} = \frac{a^2+n^2}{b^2+n^2} $$

Prove that the number $\sqrt{ab}$ is an integer.

Problem 2

In the right triangle $ABC$, the point $M$ is the midpoint of the hypotenuse $AB$. The points $P$ and $Q$ lie on the segments $AM$ and $MB$, respectively, such that $PQ=CQ$. Prove that $AP \leq 2MQ$.

Problem 3

$16$ players participated in the badminton tournament. Each player has played at most one match with any other player, no match ended in a draw. After the tournament, it turned out that each player won a different number of matches. Prove that each player lost a different number of matches.

Problem 4

On the side $AB$ of the scalene triangle $ABC$, points $M$ and $N$ are chosen, such that $AN=AC$ and $BM=BC$. The line parallel to $BC$ passing through the point $M$ and the line parallel to $AC$ through point $N$ intersect at the point $S$. Prove that $\angle CSM = \angle CSN$.

Problem 5

Given two natural numbers $a$ and $b$, which have the same digits in their decimal representation, i.e. each of the digits from $0$ to $9$ occurs the same number of times in the representation of $a$ and in the representation of $b$. Show that if $a + b = 10^{1000}$, then the numbers $a$ and $b$ are divisible by $10$.

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