11.1. On the board, there are $n$ different integers. The product of the two largest is 77. The product of the two smallest is also 77. For what largest $n$ is this possible?

(R. Zhenodarov, jury)
Answer. For $n=17$.

Solution. The numbers $-11, -7, -6, -5, \ldots, 6, 7, 11$ provide an example for $n=17$.

Assume that there are at least 18 such numbers. Then, at least 9 of them will have the same sign (all positive or all negative). Among these 9 numbers, the absolute values of the two largest will be at least 8 and 9, respectively. Then their product cannot be equal to 77.

Comment. Proving that $n \leqslant 17-$ is worth 5 points.

Providing an example for $n=17-2$ points.