CZ-SK-MO 2021

A fraction with $1010$ squares in the numerator and $1011$ squares in the denominator serves as a game board for a two player game. $$\frac{\square + \square +...+ \square}{\square + \square +...+ \square+ \square}$$ Players take turns in moves. In each turn, the player chooses one of the numbers $1, 2,. . . , 2021$ and inserts it in any empty field. Each number can only be used once. The starting player wins if the value of the fraction after all the fields is filled differs from number $1$ by less than $10^{-6}$ . Otherwise, the other player wins. Decide which of the players has a winning strategy.

(Pavel Šalom)

Let $I$ denote the center of the circle inscribed in the right triangle $ABC$ with right angle at the vertex $A$ . Next, denote by $M$ and $N$ the midpoints of the lines $AB$ and $BI$ . Prove that the line $CI$ is tangent to the circumscribed circle of triangle $BMN$ .

(Patrik Bak, Josef Tkadlec)

The different non-zero real numbers a, b, c satisfy the set equality $\{a + b, b + c, c + a\} = \{ab, bc, ca\}$ . Prove that the set equality $\{a, b, c\} = \{a^2 -2, b^2 - 2, c^2 - 2\}$ also holds. . (Josef Tkadlec)

Find all natural numbers $n$ for which equality holds $n + d (n) + d (d (n)) +... = 2021$ , where $d (0) = d (1) = 0$ and for $k> 1$ , $d (k)$ is the superdivisor of the number $k$ (i.e. its largest divisor of $d$ with property $d <k$ ).

(Tomáš Bárta)

We call a string of characters neat when it has an even length and its first half is identical to the other half (eg. abab). We call a string nice if it can be split on several neat strings (e.g. abcabcdedefto abcabc, dede, and ff). By string reduction we call an operation in which we wipe two identical adjacent characters from the string (e.g. the string abbac can be reduced to aac and further to c). Prove any string containing each of its characters in even numbers can be obtained by a series of reductions from a suitable nice string.

(Martin Melicher)

An acute triangle $ABC$ is given. Let us denote $X$ for each of its inner points $X_a, X_b, X_c$ its images in axial symmetries sequentially along the lines $BC, CA, AB$ . Prove that all $X_aX_bX_c$ triangles have a common interior point.

(Josef Tkadlec)