BALTIC-WAY 2010 Problem 18

Let $p$ be a prime number. For each $k, 1 \leq k \leq p-1$, there exists a unique integer denoted by $k^{-1}$ such that $1 \leq k^{-1} \leq p-1$ and $k^{-1} \cdot k \equiv 1(\bmod p)$. Prove that the sequence

$$1^{-1}, 1^{-1}+2^{-1}, 1^{-1}+2^{-1}+3^{-1}, \ldots, 1^{-1}+2^{-1}+\cdots+(p-1)^{-1}$$

(addition modulo $p$ ) contains at most $(p+1) / 2$ distinct elements.