Difference between revisions of "2024 AIME I Problems/Problem 7"
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+ | Find the maximum real part of <math>(75+117i)z+\frac{96+144i}{z}</math>, where <math>z</math> is a complex number with <math>|z|=4</math>. Here <math>i=\sqrt{-1}</math>. | ||
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==Solution 1== | ==Solution 1== | ||
Revision as of 13:44, 2 February 2024
Problem
Find the maximum real part of , where is a complex number with . Here .
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
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.