Difference between revisions of "2024 AIME I Problems/Problem 7"
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+ | ==Solution 3== | ||
+ | Follow Solution 1 to get <math>81a-108b</math>. We can let <math>a=4\cos\theta</math> and <math>b=4\sin\theta</math> as <math>|z|=4</math>, and thus we have <math>324\cos\theta-432\sin\theta</math>. 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 <math>324\cos\theta+432\sin\theta</math> for obviously positive <math>\cos\theta</math> and <math>\sin\theta</math>. | ||
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+ | Using the previous fact, we can use the [[Cauchy-Schwarz Inequality]] to calculate the maximum. By the inequality, we have: | ||
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+ | <math>(324^2+432^2)(\cos^2\theta+\sin^2\theta)\ge(324\cos\theta+432\sin\theta)^2</math> | ||
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+ | <math>540^2\cdot1\ge(324\cos\theta+432\sin\theta)^2</math> | ||
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+ | <math>\boxed{540}\ge324\cos\theta+432\sin\theta</math> | ||
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+ | ~eevee9406 | ||
==See also== | ==See also== |
Revision as of 13:42, 2 February 2024
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 |
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