Problem 76 (difficulty: 10/10)

Prove that for any sequence \(\displaystyle a_1,a_2,\ldots,a_n\) of positive real numbers,

\(\displaystyle \frac1{\frac1{a_1}} + \frac2{\frac1{a_1}+\frac1{a_2}} + \frac3{\frac1{a_1}+\frac1{a_2}+\frac1{a_3}} + \ldots + \frac{n}{\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_n}} < 2 (a_1+a_2+\ldots+a_n). \)

(KöMaL N. 189., November 1998)

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