Let $a, b, c, d \in \mathbb{R}$ such that $a+3 b+5 c+7 d=14$. Find the minimum possible value of $a^{2}+b^{2}+c^{2}+d^{2}$.
According to the Cauchy-Schwarz inequality, we have:

$$
(a+3 b+5 c+7 d)^{2} \leq\left(a^{2}+b^{2}+c^{2}+d^{2}\right) \times\left(1^{2}+3^{2}+5^{2}+7^{2}\right)
$$

from which: $\left(a^{2}+b^{2}+c^{2}+d^{2}\right) \geq \frac{14^{2}}{1^{2}+3^{2}+5^{2}+7^{2}}=\frac{7}{3}$

We have equality if $b=3 a, c=5 a$ and $d=7 a$ i.e., if $a=\frac{1}{6}, b=\frac{1}{2}, c=\frac{5}{6}$ and $d=\frac{7}{6}$.

Thus, the minimum value of the expression $\left(a^{2}+b^{2}+c^{2}+d^{2}\right)$ is $\frac{7}{3}$.