Let $a, b, c, d, e \in \mathbb{R}^{+}$ such that $a+b+c+d+e=8$ and $a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16$. What is the maximum value of $e$? (American Olympiads, 1992)
We apply Cauchy-Schwarz to the sequences $(a, b, c, d)$ and $(1,1,1,1)$:

$$
(8-e)^{2} \leqslant 4 \times\left(16-e^{2}\right)
$$

from which we derive $5 e^{2} \leqslant 16 e$ and finally $0 \leqslant e \leqslant \frac{16}{5}$. And, this value of $e$ is valid (we then have $\mathrm{a}=\mathrm{b}=\mathrm{c}=\mathrm{d}=\frac{6}{5}$, which is of course the equality case of Cauchy-Schwarz).