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
similar as above, but to get the intuition why we chose to consider 23^2 = 529 , consider this
we need n - T = 500, where n = #integers in the list 1,2,..,n and T is the set of numbers which are either k^2 or k^3 and <=n firstly, we clearly need n > 500 so we think of taking the smallest square greater than 500 and let that be equal to n( u could try letting n = 512 = 8^3 to with similiar logic, but quickly realise that it fails). This is done so that set T is easy to calculate so n = 529, set T = 23+8-2 by PIE hence n-T = 500 so our answer is 529-1 = 528
See also
| 1990 AIME (Problems • Answer Key • Resources) | ||
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Followed by Problem 2 | |
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