Difference between revisions of "1990 AIME Problems/Problem 1"
(→Problem) |
|||
| Line 5: | Line 5: | ||
== Solution == | == Solution == | ||
{{solution}} | {{solution}} | ||
| + | Because there aren't that many perfect squares or cubes, let's look for the smallest perfect square greater than <math>500</math>. This happens to be <math>23^2=529</math>. Notice that there are <math>23</math> squares and <math>8</math> cubes less than or equal to <math>529</math>, but <math>1</math> and <math>2^6</math> are both squares and cubes. Thus, there are <math>529-23-8+2=500</math> numbers in our sequence less than <math>529</math>. Magically, we want the <math>500th</math> term, so our answer is the smallest non-square and non-cube less than <math>529</math>, which is <math>528</math>. (courtesy of 4everwise) | ||
== See also == | == See also == | ||
* [[1990 AIME Problems]] | * [[1990 AIME Problems]] | ||
Revision as of 21:16, 25 February 2007
This is an empty template page which needs to be filled. You can help us out by finding the needed content and editing it in. Thanks.
Problem
The increasing sequence 
consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence.
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
Because there aren't that many perfect squares or cubes, let's look for the smallest perfect square greater than 
. This happens to be 
. Notice that there are 
squares and 
cubes less than or equal to 
, but 
and 
are both squares and cubes. Thus, there are 
numbers in our sequence less than . Magically, we want the 
term, so our answer is the smallest non-square and non-cube less than , which is 
. (courtesy of 4everwise)
