Difference between revisions of "2022 AIME II Problems/Problem 14"
Bobjoebilly (talk | contribs) m (→Solution) |
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<math>97 > \frac{999}{c}</math>, <math>c>10.3</math> | <math>97 > \frac{999}{c}</math>, <math>c>10.3</math> | ||
− | <math>\text{Case } 1:</math> | + | <math>\text{Case } 1:</math> <math>10.3 < c < 11.7 \rightarrow c = 11 \rightarrow \lfloor \frac{999}{11} \rfloor + \lfloor \frac{10}{b} \rfloor + b-1 = 97</math> |
<math>\lfloor \frac{10}{b} \rfloor + b = 8</math>, <math>b=7</math> | <math>\lfloor \frac{10}{b} \rfloor + b = 8</math>, <math>b=7</math> | ||
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<math>c = 86 \rightarrow</math> <math>\lfloor \frac{999}{86} \rfloor + \lfloor \frac{85}{b} \rfloor + b-1 = 97</math> | <math>c = 86 \rightarrow</math> <math>\lfloor \frac{999}{86} \rfloor + \lfloor \frac{85}{b} \rfloor + b-1 = 97</math> | ||
− | <math>\lfloor \frac{85}{b} \rfloor + b = 87</math>, <math>b=87 > c</math> | + | <math>\lfloor \frac{85}{b} \rfloor + b = 87</math>, <math>b=87 > c \rightarrow \text{no solution}</math> |
<math>c = 87 \rightarrow</math> <math>\lfloor \frac{999}{87} \rfloor + \lfloor \frac{86}{b} \rfloor + b-1 = 97</math> | <math>c = 87 \rightarrow</math> <math>\lfloor \frac{999}{87} \rfloor + \lfloor \frac{86}{b} \rfloor + b-1 = 97</math> | ||
− | <math>\lfloor \frac{86}{b} \rfloor + b = 87</math>, <math>b=86</math> or <math>1</math>. We cannot have | + | <math>\lfloor \frac{86}{b} \rfloor + b = 87</math>, <math>b=86</math> or <math>1</math>. <math>\text{We cannot have }b=1 \text{ since it doesn't satisfy } a<b \text{, and note that if } b=86 \text{ we can have 10 coins of value } c, 1 \text{ of } b, \text{ and } 85</math> <math>\text{of } a \text{ for a total of 96 coins and still be able to make every value from 1 to 1000. Thus } c=87 \text{ yields no solution.}</math> |
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<math>c = 88 \rightarrow</math> <math>\lfloor \frac{999}{88} \rfloor + \lfloor \frac{87}{b} \rfloor + b-1 = 97</math> | <math>c = 88 \rightarrow</math> <math>\lfloor \frac{999}{88} \rfloor + \lfloor \frac{87}{b} \rfloor + b-1 = 97</math> |
Revision as of 16:29, 3 July 2023
Contents
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
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.
So, .
. We can get the answer by solving this equation.
, or
,
,
,
, or .
,
,
The least values of are , , .
~isabelchen ~edited by bobjoebilly
Video Solution
~MathProblemSolvingSkills.com
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.