Difference between revisions of "1987 AIME Problems/Problem 12"
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== Problem == | == Problem == | ||
− | Let <math> | + | Let <math>m</math> be the smallest [[integer]] whose [[cube root]] is of the form <math>n+r</math>, where <math>n</math> is a [[positive integer]] and <math>r</math> is a [[positive]] [[real number]] less than <math>1/1000</math>. Find <math>n</math>. |
== Solution == | == Solution == | ||
In order to keep <math>m</math> as small as possible, we need to make <math>n</math> as small as possible. | In order to keep <math>m</math> as small as possible, we need to make <math>n</math> as small as possible. | ||
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[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] | ||
+ | {{MAA Notice}} |
Revision as of 18:09, 4 July 2013
Problem
Let be the smallest integer whose cube root is of the form , where is a positive integer and is a positive real number less than . Find .
Solution
In order to keep as small as possible, we need to make as small as possible.
. Since and is an integer, we must have that . This means that the smallest possible should be quite a bit smaller than 1000, so in particular should be less than 1, so and . , so we must have . Since we want to minimize , we take . Then for any positive value of , , so it is possible for to be less than . However, we still have to make sure a sufficiently small exists.
In light of the equation , we need to choose as small as possible to insure a small enough . The smallest possible value for is 1, when . Then for this value of , , and we're set. The answer is .
See also
1987 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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.