1995 AIME Problems/Problem 7
Problem
Given that and
where and are positive integers with and relatively prime, find
Solution
From the givens, , and adding to both sides gives . Completing the square on the left in the variable gives . Since , we have . Subtracting twice this from our original equation gives , so the answer is .
Solution 2
Let . Multiplying with the given equation, , and . Simplifying and rearranging the given equation, . Notice that , and substituting, . Rearranging and squaring, , so , and , but clearly, . Therefore, , and the answer is .
Solution 3
We want . However, note that we only need to find .
Let
From this we have and
Substituting, we have
.
See also
1995 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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All AIME Problems and Solutions |
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