MathDB

On a blackboard, several polynomials of degree

37

are written, each of them has the leading coefficient equal to

1

. Initially all coefficients of each polynomial are non-negative. By one move it is allowed to erase any pair of polynomials

f, g

and replace it by another pair of polynomials

f_1, g_1

of degree

37

with the leading coefficients equal to

1

such that either

f_1+g_1 = f+g

f_1g_1 = fg

. Prove that it is impossible that after some move each polynomial on the blackboard has

37

distinct positive roots. (8 points)
Alexandr Kuznetsov

Problem Statement