2022 AIME II Problems/Problem 14

Problem

For positive integers $a$ , $b$ , and $c$ with $a < b < c$ , consider collections of postage stamps in denominations $a$ , $b$ , and $c$ cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to $1000$ cents, let $f(a, b, c)$ be the minimum number of stamps in such a collection. Find the sum of the three least values of $c$ such that $f(a, b, c) = 97$ for some choice of $a$ and $b$ .

Solution 1

Notice that $a$ must equal to $1$ , or else the value $1$ cent isn't able to be represented. At least $b-1$ numbers of $1$ cent stamps will be needed. Using at most $c-1$ stamps of value $1$ and $b$ , it is able to have all the values from $1$ to $c-1$ cents.

To be continued......

~isabelchen

2022 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2022 AIME II Problems/Problem 14

Problem

Solution 1

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