## Task 3 - 070513

The largest five-digit number is sought, for which the following applies:

a) The tens digit represents a number that is half as large as the thousands digit.

b) The units and the hundreds digit can be swapped without changing the five-digit number.
A five-digit natural number with the digits [abcde] can be written as

$$
z=10000 \cdot a+1000 \cdot b+100 \cdot c+10 \cdot d+e
$$

where $a$ can be any digit from 0 to 9, while $b, c, d, e$ can only be from 1 to 9. With the statement a), $d$ can be replaced by $b$: $\frac{b}{2}=d$. $b$ can no longer be 9. The statement b) leads to $e=c$. Thus, we have:

$$
z=10000 a+1000 b+100 c+5 b+c=10000 a+1005 b+101 c
$$

A sum is maximized when each summand is as large as possible. Given the constraints, $a$ and $c$ can be at most 9, and $b$ can be at most 8. The desired number is thus

$$
z=10101 \cdot 9+1005 \cdot 8=90909+8040=98949
$$