Дубанов И.С.

On the board, there are $n$ expressions of the form ${ }^{*} \chi^{2}+*_{X}+*=0$ (where $n$ is an odd number). Two players play a game. They take turns. On a turn, a player is allowed to replace one of the asterisks with a number not equal to zero. After $3 n$ moves, $n$ quadratic equations will be obtained. The first player aims to have as many of these equations as possible have no roots, while the second player tries to prevent this. What is the maximum number of equations without roots that the first player can achieve regardless of the second player's moves?

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Let's outline the strategy of the first player, allowing him to obtain at least ${ }^{n+1 / 2}$ equations that have no roots.

We will call the printing of an expression the first replacement of a star in it with a number. On his first move, and in response to any printing move by the second player, the first player prints one of the remaining expressions, writing 1 before $x$. If the second player writes a number $a$ before $x^{2}$ or instead of the free term in the expression printed by the first, then in response, the first writes the number $\frac{1}{a}$ in the remaining place. The discriminant of the resulting equation ($D=1-4=-3$) is negative, so it has no roots. If the second player writes a number instead of one of the two stars in a previously printed expression, then the first arbitrarily fills in the remaining place in that expression. It is clear that the described strategy allows the first player to print ${ }^{n+1} / 2$ expressions, which he converts into equations that have no roots during the game. It remains to show that the second player, by interfering with the first, can obtain ${ }^{n-1 / 2}$ equations that have roots. Indeed, the second player can ${ }^{n-1 / 2}$ times print expressions, writing the number 1 before $x^{2}$. No matter how the first player plays, the second will be able to place another number in each of the printed expressions. If the place of the free term is not occupied, then by writing -1 there, the second player ensures the presence of roots. If the first player has written a number $c$ instead of the free term, then the second can write a number $b>2 \sqrt{|c|}$ before $x$, and the discriminant of the resulting equation will be positive.

## Answer

${ }^{n+1 / 2} 2$.

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