I've been struggling with the following problem for hours:
Consider the expression $p^2\frac{x}{y+z}+q^2\frac{y}{x+z}+r^2\frac{z}{x+y}$, where $p,q,r>0$ are parameters. Choose $x,y,z\ge0$ so that the value of the expression is a minimum.
Any ideas to begin with? :/
Converting to a new set of variables, we have
$p^2 u + q^2 v + r^2 w$ to be minimised, where $\dfrac{1}{1+u} + \dfrac{1}{1+v}+ \dfrac{1}{1+w} = 2$
From the constraint, we have $w = \dfrac{1- uv}{2 uv + u + v}$, so we can write the unconstrained problem as:
$$\text{Minimise} \quad M = p^2 u + q^2 v + r^2 \dfrac{1- uv}{2 uv + u + v}$$
This should be possible to address easily by taking derivatives w.r.t. $u, v$ and setting them to zero. Using this approach, I get one solution which could be positive: $$ u = \frac{-p + q + r}{2p} \quad \text{and} \quad v = \frac{p-q + r}{2q} \quad \text{and} \quad w = \frac {p + q - r}{2 r}$$
(Note these need not be always positive, unless $p, q, r$ can form a triangle. Also notable is that the weights are smaller, larger the square.)
With this, the minimum value of $M$ is $\quad \dfrac{(p+q+r)^2}{2} - (p^2 + q^2 + r^2)$ which suggests there should be a nice inequality way of getting here as well.