10.22 Find the largest 5-digit number $A$ with the following properties: its 4th digit is greater than the 5th digit, its 3rd digit is greater than the sum of the 4th and 5th digits, its 2nd digit is greater than the sum of the 3rd, 4th, and 5th digits, and its first digit is greater than the sum of the other digits.
(43rd Moscow Mathematical Olympiad, 1980)
[Solution] Let the desired five-digit number be $\overline{a b c d e}$, where $a \in\{1,2,3, \cdots, 9\}, b, c, d, e \in\{0,1,2, \cdots, 9\}$.
From the problem, we have
$$
\left\{\begin{array}{l}
d>e, \\
c>d+e, \\
b>c+d+e, \\
a>b+c+d+e .
\end{array}\right.
$$

Thus,
$$
a>2(c+d+e)>4(d+e)>8 e .
$$

Therefore, $e$ can only be 0 or 1.
If $e=1$, then $d \geqslant 2, c \geqslant 4, b \geqslant 8, a \geqslant 16$, which contradicts $a \in\{1,2, \cdots, 9\}$.
Thus,
$$
e=0 .
$$

Then $\quad d \geqslant 1, c \geqslant 2, b \geqslant 4, a \geqslant 8$.
To find the largest five-digit number, let $a=9$, since $b<6$, we can take
$$
b=5, \quad c=2, \quad d=1, \quad e=0 \text {. }
$$

Therefore, the largest five-digit number that meets the requirements is
$$
95210 \text {. }
$$