2022 AIME II Problems/Problem 14
For positive integers , , and with , consider collections of postage stamps in denominations , , and 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 cents, let be the minimum number of stamps in such a collection. Find the sum of the three least values of such that for some choice of and .
Notice that we must have , otherwise cent stamp cannot be represented. At least numbers of cent stamps are needed to represent the values less than . Using at most stamps of value and , it can have all the values from to cents. Plus stamps of value , every value up to can be represented. Therefore using stamps of value , stamps of value , and stamps of value , all values up to can be represented in sub-collections, while minimizing the number of stamps.
. We can get the answer by solving this equation.
For , , ,
, , , no solution
, , or , neither values satisfy , no solution
The least values of are , , .
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