2023 AMC 8
Complete problem set with solutions and individual problem pages
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.)
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Solution 1
Let's use the most stamps to make We have of each stamp, -cent (nickels), -cent (dimes), and -cent (quarters).
If we want the highest number of stamps, we must have the highest number of the smaller value stamps (like the coins above). We can use nickels and dimes to bring our total cost to . However, when we try to use quarters, the cents don’t fit evenly, so we have to give back cents to make the quarter amount . The most efficient way to do this is to give back a -cent (dime) stamp and a -cent (nickel) stamp to have stamps used so far. Now, we just use quarters to get a grand total of .
Solution 2
The value of his entire stamp collection is dollars. To make with stamps, he should remove cents worth of stamps with as few stamps as possible. To do this, he should start by removing as many cent stamps as possible as they have the greatest denomination. He can remove at most of these stamps. He still has to remove cents worth of stamps. This can be done with one and cent stamp. In total, he has stamps in his entire collection. As a result, the maximum number of stamps he can use is .
