Difference between revisions of "2019 AIME I Problems/Problem 12"
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==Solution== | ==Solution== | ||
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+ | We will use the fact that segments <math>AB</math> and <math>BC</math> are perpendicular in the complex plane if and only if <math>\frac{a-b}{b-c}\in i\mathbb{R}</math>. To prove this, when dividing two complex numbers you subtract the angle of one from the other, and if the two are perpendicular, subtracting these angles will yield an imaginary number with no real part. | ||
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+ | Now to apply this: | ||
+ | <cmath>\frac{f(z)-z}{f(f(z))-f(z)}\in i\mathbb{R}</cmath> | ||
+ | <cmath>\frac{z^2-19z-z}{(z^2-19z)^2-19(z^2-19z)-(z^2-19z)}</cmath> | ||
+ | <cmath>\frac{z^2-20z}{z^4-38z^3+341z^2+380z}</cmath> | ||
+ | <cmath>\frac{z(z-20)}{z(z+1)(z-19)(z-20)}</cmath> | ||
+ | <cmath>\frac{1}{(z+1)(z-19)}\in i\mathbb{R}</cmath> | ||
+ | |||
+ | The factorization of the nasty denominator above is made easier with the intuition that <math>(z-20)</math> must be a divisor for the problem to lead anywhere. Now we know <math>(z+1)(z-19)\in i\mathbb{R}</math> so using the fact that the imaginary part of <math>z</math> is <math>11i</math> and calling the real part r, | ||
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+ | <cmath>(r+1+11i)(r-19+11i)\in i\mathbb{R}</cmath> | ||
+ | <cmath>r^2-18r-140=0</cmath> | ||
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+ | solving the above quadratic yields <math>r=9+\sqrt{221}</math> so our answer is <math>9+221=\boxed{230}</math> | ||
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==See Also== | ==See Also== | ||
{{AIME box|year=2019|n=I|num-b=11|num-a=13}} | {{AIME box|year=2019|n=I|num-b=11|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 22:39, 14 March 2019
The 2019 AIME I takes place on March 13, 2019.
Problem 12
Given , there are complex numbers with the property that , , and are the vertices of a right triangle in the complex plane with a right angle at . There are positive integers and such that one such value of is . Find .
Solution
We will use the fact that segments and are perpendicular in the complex plane if and only if . To prove this, when dividing two complex numbers you subtract the angle of one from the other, and if the two are perpendicular, subtracting these angles will yield an imaginary number with no real part.
Now to apply this:
The factorization of the nasty denominator above is made easier with the intuition that must be a divisor for the problem to lead anywhere. Now we know so using the fact that the imaginary part of is and calling the real part r,
solving the above quadratic yields so our answer is
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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