KBB3 Problem 2

Given $x_1, \cdots, x_n, y_1, \cdots y_n$ non-negative real numbers such that:

$x_1 + \cdots + x_n = 1$

$x_1y_1 + \cdots + x_ny_n = n^2$

Find the minimum value of

$\displaystyle \frac{x_1(y_1^2 + n^2)}{y_1 + 1} + \cdots + \frac{x_n(y_n^2 + n^2)}{y_n + 1}$

Solution

For each $i$,

$(y_i^2 + n^2)(n^2+1) \geq (y_in + n)^2 \iff (y_i - n^2)^2 \geq 0$

The first inequality is also imminent by Cauchy. Thus

$\displaystyle \frac{x_i(y_i^2 + n^2)}{y_i+1} \geq \frac{n^2}{n^2+1}x_i(y_i+1)$

When summed over all $i$, we have

$\displaystyle LHS \geq \frac{n^2}{n^2+1}(\sum x_iy_i + \sum x_i) = n^2$

The minimum occurs when $y_i = n^2 \forall i$ and it happens for any combination of $x_i$s whose sum equal 1.