Problem:
Find the sum of all positive integers such that, given an unlimited supply of stamps of denominations , and cents, cents is the greatest postage that cannot be formed.
Solution:
Chicken McNugget Theorem states how the largest number of "nuggets" that can't be bought with boxes of a and b nuggets where is
Using this here, we know that the largest amount of cents that won't be able to be bought initially is plugging in Chicken McNugget theorem on and to get which bounds to be at least 24 as less than that we'd be able to create 91 cents.
Let's use casework from here.
A key observation is that can't be created, but that can be created, which would make every single cent value after that able to be created (just add multiples of 5 cents).
The key observation here is that will create , but can't have been formed as a result of adding to , which means it would have been formed solely as a result of multiples of and
Since we also had , we experiment with possible pairs that can create , and get that the only possible sets that works is and .
Adding, we get
OLD
By the Chicken McNugget theorem, the least possible value of such that cents cannot be formed satisfies , so must be at least .
For a value of to work, we must not only be unable to form the value , but we must also be able to form the values through , as with these five values, we can form any value greater than by using additional cent stamps.
Notice that we must form the value without forming the value . If we use any cent stamps when forming , we could simply remove one to get . This means that we must obtain the value using only stamps of denominations and .
Recalling that , we can easily figure out the working pairs that can used to obtain , as we can use at most stamps without going over. The potential sets are , and .
The last two obviously do not work, since they are too large to form the values through , and by a little testing, only and can form the necessary values, so . .
The problems on this page are the property of the MAA's American Mathematics Competitions