Problem 6. On the board, there are five "equations" of the form $x^{2}+\ldots x+\ldots=0$. Two players take turns filling in the dots with natural numbers from 1 to 10, with each number being used only once. The game ends when all numbers are filled in. The player who makes the first move wants to have as many equations as possible with two distinct roots at the end of the game, while his opponent wants to have as few as possible. What is the best result the first player can achieve regardless of the second player's moves?
Answer. 3. Solution. To obtain three equations, each having two distinct roots, the first player needs to use the three initial moves to write the largest of the numbers not yet written in the place of the coefficient of $x$ in the "equation" where no number has yet been written. This is possible because if no more than two pairs of moves have been made, the coefficients could have appeared in a maximum of four "equations."

Let $x^{2}+p x+q=0$ be one of the equations obtained at the end of the game, where the coefficient $p$ was written by the first player on one of the first three moves. Then, obviously, $q \leq p-1$, from which $p^{2}-4 q \geq p^{2}-4(p-1)=(p-2)^{2}>0$, because even on the third move of the first player, $p \geq 6$, as only four numbers were used in the first two pairs of moves. Thus, the first player can ensure three equations with two roots each.

To prevent the first player from obtaining more than three equations with two roots, the second player needs to use the first two moves to write the smallest of the numbers not yet written in the place of the coefficient of $x$ in the "equation" where no number has yet been written. Let $x^{2}+p x+q=0$ be one of the equations obtained at the end of the game, where the coefficient $p$ was written by the second player on one of the first two moves. Then, obviously, $p \leq 4$ and $q \geq p+1$, from which $p^{2}-4 q \geq p^{2}-4(p+1)=p(p-4)-4<0$, meaning the equation has no roots.

- Answer without justification - 0 points. If there is a strategy for only one of the players, the solution is evaluated out of 4 points: 2 points for describing the strategy and 2 points for its justification.