Question 60
Suppose that the real numbers $a_{1}, a_{2}, \ldots, a_{100}$ satisfy
$$
\begin{gathered}
a_{1} \geq a_{2} \geq \cdots \geq a_{100} \geq 0 \\
a_{1}+a_{2} \leq 100 \\
a_{3}+a_{4}+\cdots+a_{100} \leq 100
\end{gathered}
$$
Determine the maximum possible value of $a_{1}^{2}+a_{2}^{2}+\cdots+a_{100}^{2}$, and find all possible sequences $a_{1}, a_{2}, \ldots, a_{100}$ which achieve this maximum.
$$
\begin{gathered}
a_{1} \geq a_{2} \geq \cdots \geq a_{100} \geq 0 \\
a_{1}+a_{2} \leq 100 \\
a_{3}+a_{4}+\cdots+a_{100} \leq 100
\end{gathered}
$$
Determine the maximum possible value of $a_{1}^{2}+a_{2}^{2}+\cdots+a_{100}^{2}$, and find all possible sequences $a_{1}, a_{2}, \ldots, a_{100}$ which achieve this maximum.