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. Adding in $\lfloor \frac{999}{c} \rfloor$ stamps of value $c$ , every value up to $1000$ is able to be represented. Therefore using $\lfloor \frac{999}{c} \rfloor$ stamps of value $c$ , $\lfloor \frac{c-1}{b} \rfloor$ stamps of value $b$ , and $b-1$ stamps of value $1$ all values up to $1000$ are able to be represented in sub-collections, while minimizing the number of stamps. $\lfloor \frac{999}{c} \rfloor + \lfloor \frac{c-1}{b} \rfloor + b-1 = 97$

To be continued......

~isabelchen

2022 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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2022 AIME II Problems/Problem 14

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Solution 1

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