TUYMAADA 2020

Each edge of a complete graph with $101$ vertices is marked with $1$ or $-1$ . It is known that absolute value of the sum of numbers on all the edges is less than $150$ . Prove that the graph contains a path visiting each vertex exactly once such that the sum of numbers on all edges of this path is zero.

(Y. Caro, A. Hansberg, J. Lauri, C. Zarb)

Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial $y = x^2 + ax + b$ are drawn in the plane. The numbers $a$ and $b$ are not known. How to draw a unit segment using only ruler and compass?