15. Let real numbers $a_{1}, a_{2}, \cdots, a_{100}$ satisfy: (i) $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{100} \geqslant 0$; (ii) $a_{1}+a_{2} \leqslant 100$; (iii) $a_{3}+a_{4} +\cdots+a_{100} \leqslant 100$. Find the maximum value of $a_{1}^{2}+a_{2}^{2}+\cdots+a_{100}^{2}$ and the values of $a_{1}, a_{2}, \cdots, a_{100}$ when the maximum value is achieved.
15. For $i \geqslant 3, 0 \leqslant a_{i} \leqslant a_{2}, a_{i}\left(a_{i}-a_{2}\right) \leqslant 0$, and equality holds if and only if $a_{i} \in\left\{0, a_{2}\right\}$.

When $i=3,4, \cdots, 100$, we have $\sum_{i=3}^{100} a_{i}^{2} \leqslant a_{2} \sum_{i=3}^{100} a_{i} \leqslant 100 a_{2}$ by (iii), and equality holds if and only if $\sum_{i=3}^{100} a_{i}=100$ or $a_{2}=0$.

From (i) and (ii), $0 \leqslant a_{2} \leqslant 100-a_{1} \leqslant 100-a_{2}$, so $0 \leqslant a_{2} \leqslant 50$. Therefore, $2 a_{2}\left(a_{2}-50\right) \leqslant 0$, and equality holds if and only if $a_{2}=0$ or $a_{2}=50$.
$$
\sum_{i=1}^{100} a_{i}^{2}=a_{1}^{2}+a_{2}^{2}+\sum_{i=3}^{100} a_{i}^{2} \leqslant\left(100-a_{2}\right)^{2}+a_{2}^{2}+100 a_{2}=10000+2 a_{2}\left(a_{2}-50\right) \leqslant 10000 .
$$

Equality holds if:
(a) $\left\{a_{3}, a_{4}, \cdots, a_{100}\right\} \subseteq \left\{0, a_{2}\right\}$.
(b) $\sum_{i=3}^{100} a_{i}=100$ or $a_{2}=0$
(c) $a_{1}=100-a_{2}$
(d) $a_{2} \in\{0, 50\}$. Thus, $a_{1}, a_{2}, \cdots, a_{100}$ are either $100,0,0, \cdots, 0$ or $50,50,50,50,0,0, \cdots, 0$.