Problem A. 789. (December 2020)

A. 789. Let \(\displaystyle p(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+1\) be a polynomial with integer coefficients and real roots such that the absolute value of all of its roots are less than \(\displaystyle 1/3\), and all the coefficients of \(\displaystyle p(x)\) are lying in the interval \(\displaystyle [-2019a,2019a]\) for some positive integer \(\displaystyle a\). Prove that if this polynomial is reducible in \(\displaystyle \mathbb{Z}[x]\), then the coefficients of one its factors are less than \(\displaystyle a\).

Submitted by Navid Safaei, Tehran, Iran

(7 pont)

Deadline expired on January 11, 2021.

Statistics:

3 students sent a solution.
7 points:	Bán-Szabó Áron, Fleiner Zsigmond, Füredi Erik Benjámin.

Problems in Mathematics of KöMaL, December 2020