Question 67
Let $p$ be a prime number. For each $k$, $1 \le k \le p-1$, there exists an integer $m$, $1 \le m \le p-1$, such that $mk \equiv 1 \pmod{p}$. We will denote this integer by $\frac{1}{k}$. Prove that the sequence
$$
1, 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, \dots, 1+\frac{1}{2}+\dots+\frac{1}{p-1}
$$
(addition modulo $p$) contains at most $(p+1)/2$ distinct elements.
$$
1, 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, \dots, 1+\frac{1}{2}+\dots+\frac{1}{p-1}
$$
(addition modulo $p$) contains at most $(p+1)/2$ distinct elements.