Assume that $ 1a_1+2a_2+\cdots+na_n=1$, where the $a_j$ are real numbers.

geometry inequality

hint: $1 = \left(1a_1+\sqrt{2}(\sqrt{2}a_2)+\cdots + \sqrt{n}(\sqrt{n}a_n)\right)^2\leq (1+2+\cdots + n)(1a_1^2+2a_2^2+\cdots + na_n^2)$

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