Thursday, March 2, 2006

AM to the GM. Topic: Inequalities. Level: Olympiad.

Problem: (2002 MOP Homework - #5) Let $n \ge 2$ be an integer. Show that

$\displaystyle \sum_{i=1}^n kx_k \le nC2+\sum_{i=1}^n x_k^k$

for all nonnegative reals $x_1, x_2, \ldots, x_n$.

Solution: The strange $nC2$ term bothers us. Consider rewriting it as $nC2 = 1+2+\cdots+(n-1)$. Then the RHS becomes

$\displaystyle nC2+\sum_{i=1}^n x_k^k = \sum_{i=1}^n (x_k^k+k-1)$.

We have

$\displaystyle x_k^k+k-1 = x^k+1+1+\cdots+1$ ($k-1$ times)

so by AM-GM,

$x^k+1+1+\cdots+1 \ge kx_k$.

Thus

$\displaystyle nC2+\sum_{i=1}^n x_k^k = \sum_{i=1}^n (x_k^k+k-1) \ge \sum_{i=1}^n kx_k$

as desired. QED.

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Comment: Really a pretty weak inequality because as $n$ or any of the $x_i$'s get big, the RHS is clearly larger than the left. But getting that constant term to go away was tricky. Oftentimes when you have big powers on one side you can AM-GM them with multiple $1$'s to reduce it to a smaller power.

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Practice Problem: Given positive reals $a,b,c$, prove that

$a+b+c \ge 3\sqrt[3]{a\left(\frac{b+c}{2}\right)^2}$.