2014 AIME I Problems/Problem 14
Let be the largest real solution to the equation
There are positive integers , , and such that . Find .
The first step is to notice that the 4 on the right hand side can simplify the terms on the left hand side. If we distribute 1 to , then the fraction becomes of the form . A similar cancellation happens with the other four terms. If we assume is not the highest solution (which is true given the answer format) we can cancel the common factor of from both sides of the equation.
Then, if we make the substitution , we can further simplify.
If we group and combine the terms of the form and , we get this equation:
Then, we can cancel out a from both sides, knowing that is not a possible solution given the answer format. After we do that, we can make the final substitution .
Using the quadratic formula, we get that the largest solution for is . Then, repeatedly substituting backwards, we find that the largest value of is . The answer is thus
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