If $x, y, z, t > 0$, find all possible values of $H(x,y,z,t) = \frac{x}{t+x+y}+\frac{y}{x+y+z}+\frac{z}{y+z+t}+\frac{t}{z+t+x}$.
How I think this can be solved:
First off, note that $H$ is an homogeneous function of grade $0$, which implies that for any point $x_1\in\mathbb{R^+}^4$, there exists another point $x_2$ in the hypersphere $x^2+y^2+z^2+t^2=1$ which has the same image (you just have to normalize $x_1$ to find $x_2$). So by adding the condition $x^2+y^2+z^2+t^2=1$ you are not excluding any value from the image of the function. This may be useful to apply some kind of inequality.
I think this problem can be solved by finding a maximum and a minimum value for this expression and then using the Intermediate Value Theorem to justify all values in between the maximum and minimum are in the image of the function, but I haven't succeeded finding that maximum or minimum.
This exercise is right next to another one that is solved using Cauchy-Schwarz inequality, so I suspect it might have something to do with a famous inequality.