2007 iTest Problems/Problem 57
The following problem is from the Ultimate Question of the 2007 iTest, where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substituted.
Problem
How many positive integers are within of exactly perfect squares? (Note: is considered a perfect square.)
Solution
This problem is essentially asking for how many are there perfect squares from to .
To find the bounds, note that the difference between consecutive perfect squares are odd numbers. As it increases, the distance between perfect squares increase. Let be the difference between the minimum perfect square and the next perfect square. Since there are perfect squares in the range, the last difference is , and the sum of the differences is . This equals , so writing an equation and solving for yields . The sum of the odd numbers from to is , so we found a place to start.
Using the boundries as reference, we can make a table to find values of and find the number of perfect squares.
Value of | Value of | Value of | Smallest PS in bound | Largest PS in bound | Number of PS |
If gets below , then there will be more perfect squares because while , so more perfect squares would be gained than lost. Similarly, if gets above , then there will be less perfect squares because while , so more perfect squares would be lost than gained.
Based on the table, there are integers that satisfy the criteria.
See Also
2007 iTest (Problems) | ||
Preceded by: Problem 56 |
Followed by: Problem 58 | |
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