Difference between revisions of "1999 AIME Problems/Problem 3"
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We conclude that the answer is <math>1 + 9 + 10 + 18 = \boxed{38}.</math> | We conclude that the answer is <math>1 + 9 + 10 + 18 = \boxed{38}.</math> | ||
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+ | ==Solution 4: Graphing== | ||
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+ | If we graphed <cmath> y = \sqrt{x^2-19x+99} </cmath> we would see that only four values of x return integer values of y: 10, 9, 1, 18. Thus, the answer if <math>\boxed{38}</math>. | ||
== See also == | == See also == |
Revision as of 18:36, 22 April 2020
Problem
Find the sum of all positive integers for which is a perfect square.
Solution 1
If for some positive integer , then rearranging we get . Now from the quadratic formula,
Because is an integer, this means for some nonnegative integer . Rearranging gives . Thus or , giving or . This gives or , and the sum is .
Solution 2
Suppose there is some such that . Completing the square, we have that , that is, . Multiplying both sides by 4 and rearranging, we see that . Thus, . We then proceed as we did in the previous solution.
Solution 3
When , we have
So if and is a perfect square, then
or .
For , it is easy to check that is a perfect square when and ( using the identity
We conclude that the answer is
Solution 4: Graphing
If we graphed we would see that only four values of x return integer values of y: 10, 9, 1, 18. Thus, the answer if .
See also
1999 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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