In how many ways can a lottery ticket be filled out so that the sum of the 5 marked numbers is at least 228?
Consider the mirror image of a number $a$ to be the number $b$ such that

$$
a+b=91
$$

The mirror image of a possible lottery ticket is the set of five numbers consisting of the mirror images of the marked numbers. Since the mirror image of any lottery number (a positive integer not greater than 90) is also a lottery number, and the mirror images of different numbers are different, the mirror image of any lottery ticket is also a valid lottery ticket.

A ticket and its mirror image are certainly different, because the sum of the numbers marked on them is $5 \cdot 91=455=2 \cdot 227+1$, which is an odd number. From the above decomposition of 455, it is also clear that in one of a ticket and its mirror image, the sum of the five marked numbers will be greater than 227.

The pairing according to the mirror image thus divides the possible $\binom{90}{5}$ lottery tickets into two classes of equal size, such that in every member of one class, the sum of the five marked numbers is at least 228, while in every member of the other class, this sum is at most 227. Thus, there are $\binom{90}{5}: 2=21974634$ ways to fill out a lottery ticket in the prescribed manner.