2. Alberto and Barbara write numbers on the blackboard. Alberto starts and writes the real number $x$. Then Barbara writes the number 1. The two then take turns, and on each turn, they write a number. On his turn, Alberto multiplies the last number written by $x^{2}$ and writes the result. On her turn, Barbara adds $x+1$ to the last number written and writes the result.

They stop when 2020 numbers are written on the blackboard. Which of the following statements is always true?

(A) There is at least one negative number written on the blackboard (B) All the numbers written on the blackboard are positive (C) Alberto has written only positive numbers (D) Alberto has written only negative numbers (E) Barbara has written only positive numbers
2. The answer is (E). We can proceed by elimination. (A) is false because if $x>0$ then at each turn Alberto and Barbara multiply and add positive numbers to those already on the board. Since the first two numbers are $x$ and 1, there are no negative numbers on the board. (B) and (C) are false because $x$ can be negative. (D) is false because $x$ can be positive.

To prove that (E) is indeed true, observe that the first numbers written by Barbara are $1, 1+x+x^{2}, 1+x+x^{2}+x^{3}+x^{4}=1+x+x^{2}(1+x+x^{2})$, and that in general Barbara will write expressions of the form $1+x+\ldots+x^{2n}$. A number of this type is necessarily positive. Indeed, if $x=1$ this is obvious, and in the other case the sum can be calculated using the properties of geometric progressions: we have $1+x+\ldots+x^{2n}=\frac{x^{2n+1}-1}{x-1}$, which is positive because the numerator and denominator have the same sign (positive if $x>1$, negative if $x<1$).