problem from "Scuola Normale Superiore" admission test 2017 This is a problem from "Scuola Normale Superiore" admission test 2017.
Can anybody help me to solve it?
Let $ \alpha, \beta, \gamma $ and $ \delta $ $ \in \mathbb{R}$. We call $ S $ the set of points $ (x, y, z) $ of the Euclidean space such that $ z \leq min(\alpha x + \beta y, \gamma x + \delta y) $.
Suppose then that for some $ r, s \in \mathbb{R} $ happens that for every $ (x, y, z) \in S $ we have $ z \leq rx + sy $.
Prove that in this case there is a number $ t \in \mathbb{R} $ with $ 0 \leq t \leq 1 $ such that $ r = t \alpha + (1-t) \gamma $ and $ s = t \beta + (1-t) \delta $.
Note: in the solution we can assume that $ (\alpha, \beta) \neq (0,0) $, $ (\gamma, \delta) \neq (0,0) $ and that the lines $ \alpha x + \beta y = 0 $ and $ \gamma x + \delta y = 0 $ are distinct

Hint: you are looking at the intersection of two half-spaces, bounded by two planes $\pi: z=\alpha x + \beta y$ and $\sigma: z=\gamma x + \delta y$ both including the origin. If the two planes are distinct, their intersection must be a line, and you can imagine obtaining $\pi$ by slowly rotating $\sigma$ around that line and viceversa. The planes that you obtain as you go through that rotation (in the form $z=t\cdot(\alpha x + \beta y)+(1-t)\cdot(\gamma x + \delta y)$ for $0\leq t\leq 1$) are all the planes that bound half-spaces containing the intersection of your original half-spaces - why? Begin by answering why the latter planes must all contain the entire line $\pi\cap\sigma$. 
Note that the hypothesis that $\pi\neq\sigma$ is not necessary (why?) it just leaves out an unnecessary complication.
