2017 Putnam B3

Suppose that $f(x) = \sum_{i=0}^\infty c_i x^i$ is a power series for which each coefficient $c_i$ is $0$ or $1$.

Show that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational.

假设 $f(x) = \sum_{i=0}^\infty c_i x^i$ 是一个幂级数，其中每个系数 $c_i$ 为 $0$ 或 $1$。

证明如果 $f(2/3) = 3/2$，则 $f(1/2)$ 一定是无理数。