Putnam 1967

Consider polynomial forms $ax^2 - bx + c$ with integer coefficients which have two distinct zeros in the open interval $0 < x < 1$. Exhibit with a proof the least positive integer value of $a$ for which such a polynomial exists.

Find the sum of the first $n$ terms in the binomial expansion of $(2 - 1)^{-n}$ where $n$ is a positive integer.