TUYMAADA 2019

In a sequence $a_1, a_2, ..$ of real numbers the product $a_1a_2$ is negative, and to define $a_n$ for $n > 2$ one pair $(i, j)$ is chosen among all the pairs $(i, j), 1 \le i < j < n$ , not chosen before, so that $a_i +a_j$ has minimum absolute value, and then $a_n$ is set equal to $a_i + a_j$ . Prove that $|a_i| < 1$ for some $i$ .

Is it possible to draw in the plane the graph presented in the figure so that all the vertices are different points and all the edges are unit segments? (The segments can intersect at points different from vertices.)

Attachments: