2024 AIME I Problems/Problem 7
Contents
[hide]Problem
Find the maximum real part of , where is a complex number with . Here .
Video Solution in 3 minutes & Cauchy's Inequality by MegaMath
https://www.youtube.com/watch?v=ejmrAJ9TpvM&ab_channel=MegaMathChannel
Solution 1
Let such that . The expression becomes:
Call this complex number . We simplify this expression.
We want to maximize . We can use elementary calculus for this, but to do so, we must put the expression in terms of one variable. Recall that ; thus, . Notice that we have a in the expression; to maximize the expression, we want to be negative so that is positive and thus contributes more to the expression. We thus let . Let . We now know that , and can proceed with normal calculus.
We want to be to find the maximum.
We also find that .
Thus, the expression we wanted to maximize becomes .
~Technodoggo
Solution 2 (Without Calculus)
Same steps as solution one until we get . We also know or . We want to find the line tangent to circle . Using we can substitute and get
~BH2019MV0
Solution 3
Follow Solution 1 to get . We can let and as , and thus we have . Furthermore, we can ignore the negative sign in front of the second term as we are dealing with sine and cosine, so we finally wish to maximize for obviously positive and .
Using the previous fact, we can use the Cauchy-Schwarz Inequality to calculate the maximum. By the inequality, we have:
~eevee9406
Solution 4 (Simple Quadratic Discriminant)
Similar to the solutions above, we find that , where . To maximize this expression, we must maximize . Let this value be . Solving for yields . From the given information we also know that . Substituting in terms of and gives us . Combining fractions, multiplying, and rearranging, gives . This is useful because we want the maximum value of such that this quadratic has real roots which is easy to find using the discriminant. For the roots to be real, . Now all that is left to do is to solve this inequality. Simplifying this expression, we get which means and . Therefore the maximum value of is and
~vsinghminhas
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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