USAMO Problem 2

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

Note.\mathbf{Note.} The root-mean-square of nn numbers a1,a2,,ana_1, a_2, \cdots, a_n is defined to be [a12+a22++an2n]1/2\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}

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