2014 AIME I Problems/Problem 7
Let and be complex numbers such that and . Let . The maximum possible value of can be written as , where and are relatively prime positive integers. Find . (Note that , for , denotes the measure of the angle that the ray from to makes with the positive real axis in the complex plane.
Let and . Then, .
Multiplying both the numerator and denominator of this fraction by gives us:
We know that is equal to the imaginary part of the above expression divided by the real part. Let . Then, we have that:
We need to find a maximum of this expression, so we take the derivative:
Thus, we see that the maximum occurs when . Therefore, , and . Thus, the maximum value of is , or , and our answer is .
Solution 2 (No calculus)
Without the loss of generality one can let lie on the positive x axis and since is a measure of the angle if then and we can see that the question is equivelent to having a triangle with sides and and trying to maximize the angle
using the law of cosines we get: rearranging: solving for we get:
if we want to maximize we need to minimize , using AM-GM inequality we get that the minimum value for hence using the identity we get and our answer is .
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