Difference between revisions of "2008 AIME I Problems/Problem 8"
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=== Solution 3: Complex Numbers === | === Solution 3: Complex Numbers === | ||
Adding a series of angles is the same as multiplying the complex numbers whose arguments they are. In general, <math>\arctan\frac{1}{n}</math>, is the argument of <math>n+i</math>. The sum of these angles is then just the argument of the product | Adding a series of angles is the same as multiplying the complex numbers whose arguments they are. In general, <math>\arctan\frac{1}{n}</math>, is the argument of <math>n+i</math>. The sum of these angles is then just the argument of the product | ||
− | |||
<cmath>(3+i)(4+i)(5+i)(n+i)</cmath> | <cmath>(3+i)(4+i)(5+i)(n+i)</cmath> | ||
− | + | and expansion give us <math>(48n-46)+(48+46n)i</math>. Since the argument of this complex number is <math>\frac{\pi}{4}</math>, its real and imaginary parts must be equal; then, we can we set them equal to get | |
− | and expansion give us <math>(48n-46)+(48+46n)i</math>. Since the argument of this complex number is <math>\frac{\pi}{4}</math>, its real and imaginary parts must be equal | + | <cmath>48n - 46 = 48 + 46n.</cmath> |
− | + | Therefore, <math>n=boxed{47}</math>. | |
− | < | ||
− | Therefore, | ||
==Solution 4 Sketch == | ==Solution 4 Sketch == |
Revision as of 19:04, 16 August 2023
Problem
Find the positive integer such that
Contents
[hide]Solution
Solution 1
Since we are dealing with acute angles, .
Note that , by tangent addition. Thus, .
Applying this to the first two terms, we get .
Now, .
We now have . Thus, ; and simplifying, .
Solution 2 (generalization)
From the expansion of , we can see that and If we divide both of these by , then we have which makes for more direct, less error-prone computations. Substitution gives the desired answer.
Solution 3: Complex Numbers
Adding a series of angles is the same as multiplying the complex numbers whose arguments they are. In general, , is the argument of . The sum of these angles is then just the argument of the product and expansion give us . Since the argument of this complex number is , its real and imaginary parts must be equal; then, we can we set them equal to get Therefore, .
Solution 4 Sketch
You could always just bash out (where a,b,c,n are the angles of the triangles respectively) using the sum identities again and again until you get a pretty ugly radical and use a triangle to get and from there you use a sum identity again to get and using what we found earlier you can find by division that gets us
~YBSuburbanTea
See also
2008 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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