The government of Bosnia and Herzegovina has decided to implement a new system of license plates. Each license plate must contain 8 digits, each of which can be $0,1,2,3,4,5,6,7,8$ or 9. Furthermore, two distinct license plates must always have at least two different digits. For example, if it puts the license plate 00000000 into circulation, the government cannot put the license plate 00010000 into circulation.
Find the maximum number of license plates that the government can put into circulation.
We will show that the maximum number of available plates is equal to $10^{7}$.
First, we say that two plates belong to the same family if their first seven digits are different. Each family contains ten plates, so there are $10^{7}$ distinct families. However, the government cannot put more than one plate from each family into circulation. Therefore, it can put at most $10^{7}$ plates into circulation.
Conversely, the government can manage to create $10^{7}$ plates as follows: the last digit of each plate is equal to the negative of the sum, modulo 10, of the first seven digits of the plate. Thus, it creates one plate per family.
Moreover, the sum of the digits of each plate is divisible by 10. But if two plates differ in exactly one digit, the sums of their digits cannot be equal to each other (modulo 10). Therefore, two distinct plates always differ in at least two digits, which concludes the proof.

Graders' Comments This difficult exercise was fully solved by only one student. Several students found interesting constructions or lower bounds, allowing them, for example, to assert that at least $10^{4}$ or $8!=40320$ license plates could be put into place.
However, very few students managed to obtain the appropriate construction to reach $10^{7}$ plates. The most reliable way to achieve such a construction was to look at what would happen with 1 digit, then 2 digits, then 3 digits, and to identify regular patterns that appeared in all relevant copies: an integer was equal to the sum of the others.
Furthermore, and independently, very few students attempted to find an upper bound on the number of possible plates, even though a simple application of the pigeonhole principle would have sufficed here, and many students seemed to have an intuition about it. It is always a pity not to seek upper and lower bounds when trying to estimate a quantity, and even if these bounds are not necessarily easy to find.