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Problem of the Day #53: Maximum Root Product May 11, 2011

Posted by Saketh in : potd , trackback

Dasith is investigating sequences of $n$ numbers $x_1$, $x_2$, $x_3$, … $x_n$ that satisfy the following relationship for some predetermined constant $k$.

\[k\displaystyle\prod\limits_{i=1}^n \frac{1}{x_i} + \displaystyle\sum\limits_{i=1}^n \frac{1}{x_i} = 1 \]

Right now, he is considering the case where $n=3$. He knows that the sum of his three numbers is $s$. Help him determine, in terms of $k$ and $s$, the maximum possible value of the product of his numbers.

Bonus: Can you extend your result for higher values of $n$?

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