$\square$ 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 Canadian Mathematical Olympiad)
$$\begin{aligned}
& \text{Solution: Since } a_{1}+a_{2}+a_{3}+a_{4}+\cdots+a_{100} \leqslant 200, \text{ then} \\
& a_{1}^{2}+a_{2}^{2}+\cdots+a_{100}^{2} \leqslant\left(100-a_{2}\right)^{2}+a_{2}^{2}+a_{3}^{2}+\cdots+a_{100}^{2} \\
= & 100^{2}-200 a_{2}+2 a_{2}^{2}+a_{3}^{2}+\cdots+a_{100}^{2} \\
\leqslant & 100^{2}-\left(a_{1}+a_{2}+a_{3}+a_{4}+\cdots+a_{100}\right) a_{2}+2 a_{2}^{2}+a_{3}^{2}+\cdots+a_{100}^{2} \\
= & 100^{2}+\left(a_{2}^{2}-a_{1} a_{2}\right)+\left(a_{3}^{2}-a_{3} a_{2}\right)+\left(a_{4}^{2}-a_{4} a_{2}\right)+\cdots+\left(a_{100}^{2}-a_{100} a_{2}\right) \\
= & 100^{2}+\left(a_{2}-a_{1}\right) a_{2}+\left(a_{3}-a_{2}\right) a_{3}+\left(a_{4}-a_{2}\right) a_{4}+\cdots+\left(a_{100}-a_{2}\right) a_{100} .
\end{aligned}$$

Since $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{100} \geqslant 0$, it follows that $\left(a_{2}-a_{1}\right) a_{2},\left(a_{3}-a_{2}\right) a_{3},\left(a_{4}-\right.$ $\left.a_{2}\right) a_{4}, \cdots,\left(a_{100}-a_{2}\right) a_{100}$ 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 $\left(a_{2}-a_{1}\right) a_{2},\left(a_{3}-\right.$ $\left.a_{2}\right) a_{3},\left(a_{4}-a_{2}\right) a_{4}, \cdots,\left(a_{100}-a_{2}\right) a_{100}$ are all 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 $i=4$, so 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 this maximum value are $100,0,0, \cdots, 0$ or $50,50,50,50,0, \cdots, 0$.