1986 USAMO 第 3 题

What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?

$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be

$$

\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}

$$

大于 1 的最小整数 $n$ 是多少，前 $n$ 个正整数的均方根是整数？

$\mathbf{注.}$$n$个数$a_1, a_2, \cdots, a_n$的均方根定义为

$$\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}$$