Difference between revisions of "2022 AIME II Problems/Problem 14"
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For positive integers <math>a</math>, <math>b</math>, and <math>c</math> with <math>a < b < c</math>, consider collections of postage stamps in denominations <math>a</math>, <math>b</math>, and <math>c</math> 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 <math>1000</math> cents, let <math>f(a, b, c)</math> be the minimum number of stamps in such a collection. Find the sum of the three least values of <math>c</math> such that <math>f(a, b, c) = 97</math> for some choice of <math>a</math> and <math>b</math>. | For positive integers <math>a</math>, <math>b</math>, and <math>c</math> with <math>a < b < c</math>, consider collections of postage stamps in denominations <math>a</math>, <math>b</math>, and <math>c</math> 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 <math>1000</math> cents, let <math>f(a, b, c)</math> be the minimum number of stamps in such a collection. Find the sum of the three least values of <math>c</math> such that <math>f(a, b, c) = 97</math> for some choice of <math>a</math> and <math>b</math>. | ||
| − | ==Solution== | + | ==Solution 1== |
| + | |||
| + | Notice that <math>a</math> must equal to <math>1</math>, or else the value <math>1</math> cent isn't able to be represented. At least <math>b-1</math> number or <math>1</math> cent stamps will be needed. | ||
| + | |||
| + | To be continued | ||
| + | |||
| + | ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen] | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2022|n=II|num-b=13|num-a=15}} | {{AIME box|year=2022|n=II|num-b=13|num-a=15}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 11:55, 19 February 2022
Problem
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 .
Solution 1
Notice that must equal to 
, or else the value cent isn't able to be represented. At least 
number or cent stamps will be needed.
To be continued
See Also
| 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 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
