A Minimization that is not so Clear

Algebra Level 5

For a positive integer $n$ , define $S_n$ to be the minimum value of the sum $\sum_{k = 1}^n \sqrt{(2k-1)^2 + (a_k)^2},$ where $a_1, a_2,\ldots, a_n$ are positive real numbers whose sum is $17$ . There is a unique positive integer $n$ for which $S_n$ is also an integer. Find this $n$ .

The answer is 12.

1 solution

Alan Yan
Oct 22, 2015

By Minkowski's Inequality,

$S_n \ge \sqrt {\left(\sum_{k = 1}^n (2k - 1)\right)^2 + \left(\sum_{k = 1}^n a_k\right)^2} = \sqrt{n^4 + 17^2}.$

Let $n^4 + 17^2 = m^2 \implies (m - n^2)(m+n^2) = 289 \implies \boxed{n = 12}$ .

I have. Tried it using RMS>=AM. At the end i got n^2+17/sqrt2

Aakash Khandelwal - 5 years, 7 months ago

However in that case, equality cannot hold.

Alan Yan - 5 years, 7 months ago

Oh! Just missed that. Thank you.

Aakash Khandelwal - 5 years, 7 months ago

A Minimization that is not so Clear

The answer is 12.

1 solution

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