1990 AIME Problems/Problem 1
Contents
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 1
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 biggest non-square and non-cube less than , which is .
Solution 2
This solution is similar as Solution 1, but to get the intuition why we chose to consider , consider this:
We need , where is an integer greater than 500 and is the set of numbers which contains all .
Firstly, we clearly need , so we substitute n for the smallest square or cube greater than . However, if we use , the number of terms in will exceed . Therefore, , and the number of terms in is by the Principle of Inclusion-Exclusion, fulfilling our original requirement of . As a result, our answer is .
See also
1990 AIME (Problems • Answer Key • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
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.