Problem:
Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of -cent, -cent, and -cent stamps, with exactly of each type. What is the greatest number of stamps Nicolas can use to make exactly in postage? (Note: The amount corresponds to dollars and cents. One dollar is worth cents.)
Answer Choices:
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Solution:
The greatest number of stamps can be found by maximizing the number of small-denomination stamps and minimizing the number of large-denomination stamps. If Nicolas uses all of the stamps and all of the stamps, their value would sum to + = . To make in postage, Nicolas must include at least of the stamps, valued at , and make up the difference with the smaller denominations. The difference of - = is not divisible by cents, so Nicolas must choose an odd number of the stamps. Selecting of the stamps will leave a remainder of - = , which can be made with of the stamps. Therefore the greatest number of stamps can be attained by using of the stamps, of the stamps, and of the stamps, for a total of stamps.
Nicolas’s collection has stamps in total, worth . To maximize the number of stamps that can make , he can first minimize the number of stamps that can form the remainder of - = , then subtract the minimum from the total number of stamps. The amount can be made with stamps: of the cent stamps, of the , and of the stamps. Subtracting this minimum of stamps from the total leaves stamps, which is the greatest number that Nicolas can use to make .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions