The proof use intermediate value theorem? Let $x_1, \dots, x_n$ distinct points in $[\alpha, \beta]$, and $y_1, \dots, y_n$ reals with same sign. Assume that $f: [\alpha, \beta] \rightarrow \mathbb{R}$ is continuous. Then, prove that exists $\lambda \in (\alpha, \beta)$ ( or conform @arovai said $[\alpha,\beta]$ ) such that $\sum\limits_{k=1}^n f(x_k)y_k = f(\lambda) \sum\limits_{k=1}^ny_k$.
I had tried proof using the mean value theorem, but... any hint?
Thanks.

I'd like to provide a higher view, where this is a classic theorem pitched for more general measures.
This is the First Mean Value Theorem of Integration against a weighted counting measure $\mu$, where $\mu(x_j) = y_j$ and $\mu(x) = 0$ otherwise. So if you compare the proof provided at wikipedia to Simon S's answer, you'll notice they are the exact same proof - and this is because they are the same theorem, just with different measures.
