Example 1 Let real numbers $a_{1}, a_{2}, \cdots, a_{100}$ satisfy $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{100} \geqslant 0$, $a_{1}+a_{2} \leqslant 100, a_{3}+a_{4}+\cdots+a_{100} \leqslant 100$, determine the maximum value of $a_{1}^{2}+a_{2}^{2}+\cdots+$ $a_{100}^{2}$, and find the sequence $a_{1}, a_{2}, \cdots, a_{100}$ when the maximum value is achieved. (2000 Year
Solve: Since $a_{1}+a_{2}+a_{3}+a_{4}+\cdots+a_{100} \leqslant 200$, then
$$\begin{aligned}
a_{1}^{2}+ & a_{2}^{2}+\cdots+a_{100}^{2} \leqslant\left(100-a_{2}\right)^{2}+a_{2}^{2}+a_{3}^{2}+\cdots+a_{100}^{2}= \\
& 100^{2}-200 a_{2}+2 a_{2}^{2}+a_{3}^{2}+\cdots+a_{100}^{2} \leqslant \\
& 100^{2}-\left(a_{1}+a_{2}+a_{3}+a_{4}+\cdots+a_{100}\right) a_{2}+ \\
& 2 a_{2}^{2}+a_{3}^{2}+\cdots+a_{100}^{2}= \\
& 100^{2}+\left(a_{2}^{2}-a_{1} a_{2}\right)+\left(a_{3}^{2}-a_{3} a_{2}\right)+ \\
& \left(a_{4}^{2}-a_{4} a_{2}\right)+\cdots+\left(a_{100}^{2}-a_{100} a_{2}\right)= \\
& 100^{2}+\left(a_{2}-a_{1}\right) a_{2}+\left(a_{3}-a_{2}\right) a_{3}+ \\
& \left(a_{4}-a_{2}\right) a_{4}+\cdots+\left(a_{100}-a_{2}\right) a_{100}
\end{aligned}$$

Since $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{100} \geqslant 0$, then $\left(a_{2}-a_{1}\right) a_{2},\left(a_{3}-\right.$ $\left.a_{2}\right) a_{3},\left(a_{4}-a_{2}\right) a_{4}, \cdots,\left(a_{100}-a_{2}\right) a_{100}$ are all non-positive.

Therefore, $a_{1}^{2}+a_{2}^{2}+\cdots+a_{100}^{2} \leqslant 10000$, and equality holds if and only if $a_{1}=100-a_{2}$, $a_{1}+a_{2}+a_{3}+a_{4}+\cdots+a_{100}=200$, and $\left(a_{2}-a_{1}\right) a_{2},\left(a_{3}-a_{2}\right) a_{3},\left(a_{4}-a_{2}\right) a_{4}, \cdots$, $\left(a_{100}-a_{2}\right) a_{100}$ are all 0.

Since $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{100} \geqslant 0$, the last condition holds if and only if for $i \geqslant 1$, $a_{1}=a_{2}=\cdots=a_{i}$, and $a_{i+1}=\cdots=a_{100}=0$.

If $i=1$, then the sequence achieving the maximum value is $100,0,0, \cdots, 0$;
If $i \geqslant 2$, then by $a_{1}+a_{2}=100$, we have $i=4$, thus the corresponding sequence is $50,50,50,50$, $0, \cdots, 0$.

Therefore, the maximum value of $a_{1}^{2}+a_{2}^{2}+\cdots+a_{100}^{2}$ is 10000, and the sequences achieving the maximum value are $100,0$, $0, \cdots, 0$ or $50,50,50,50,0, \cdots, 0$.