Find the minimum value of $a^{2}+b^{2}+c^{2}+d^{2}$ given that $a+2 b+3 c+4 d=12$.
## By Cauchy-Schwarz,

$$
a^{2}+b^{2}+c^{2}+d^{2} \geqslant \frac{(a+2 b+3 c+4 d)^{2}}{1+4+9+16}=\frac{144}{30}=\frac{24}{5}
$$

Conversely, this value is achieved for $a=\frac{2}{5}, b=\frac{4}{5}, c=\frac{6}{5}, d=\frac{8}{5}$.