41. The Dozing Schoolboy. A schoolboy, waking up at the end of an algebra lesson, heard only a fragment of the teacher's phrase: "… I will only say that all the roots are real and positive." Glancing at the board, he saw there a 20th-degree equation assigned as homework, and tried to quickly write it down. He managed to write down only the first two terms $x^{20}-20 x^{19}$, before the teacher erased the board; however, he remembered that the constant term was +1. Could you help our unfortunate hero solve this equation?
41. The roots are positive; their arithmetic mean is $-\frac{(-20)}{20}$, and their geometric mean is $(+1)^{\frac{1}{20}}$. Since both these values coincide, it follows that all roots are equal to 1.

[D. S. Greenstein, A. M. M., 63, 493 (September 1956).]