20. Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8}$ be positive integers such that
$$
a_{1}^{2}+\left(2 a_{2}\right)^{2}+\left(3 a_{3}\right)^{2}+\left(4 a_{4}\right)^{2}+\left(5 a_{5}\right)^{2}+\left(6 a_{6}\right)^{2}+\left(7 a_{7}\right)^{2}+\left(8 a_{8}\right)^{2}=204 \text {. }
$$

Find the value of $a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}+a_{8}$.
20. Ans: 8
Observe that
$$
a_{1}^{2}+\left(2 a_{2}\right)^{2}+\cdots+\left(8 a_{8}\right)^{2} \geq 1^{2}+2^{2}+\cdots+8^{2}=204 .
$$

So the only possiblility is that $a_{i}=1$ for each $i=1,2, \ldots, 8$. Hence $a_{1}+\cdots+a_{8}=8$.