PROBLEM 1. Consider the distinct rational numbers $a_{1}, a_{2}, \ldots, a_{n}$, such that among any four of them, there are two whose product is -1.

a) Give an example of 6 such numbers;

b) Find the maximum value of $n$.
PROBLEM 1. Consider the distinct rational numbers $a_{1}, a_{2}, \ldots, a_{n}$, such that among any four of them, there are two whose product is -1.

a) Give an example of 6 such numbers;

b) Find the maximum value of $n$.

## Grading Rubric.

$(3 p)$

a) For example, the numbers $2, 3, 4, -\frac{1}{2}, -\frac{1}{3}, -\frac{1}{4}$ satisfy the conditions of the problem; in general, we can take the numbers $a, b, c, -\frac{1}{a}, -\frac{1}{b}, -\frac{1}{c}$, where $a, b$, and $c$ are distinct numbers from $\mathbb{Q}^{*}$.

(3p) b) If $n \geq 7$, then at least 4 of the $n$ numbers have the same sign. Therefore, in this case, there are 4 numbers among which we cannot choose two whose product is -1.

(1p) Thus, the maximum value of $n$ is 6.