2019 AIME I Problems/Problem 12
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[hide]Problem
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 1
Notice that we must have However, , so Then, the real part of is . Since , let . Then, It follows that , and the requested sum is .
(Solution by TheUltimate123)
Solution 2
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
Solution 3
I would like to use a famous method, namely the coni method.
Statement .If we consider there complex number in argand plane then .
According to the question given, we can assume , respectively.
WLOG,. According to the question .
So,.
Now, .
. WLOG, .where .
So,. Solving, .get ,
±. So, possible value of .
. ~ftheftics.
Proof of this method
Note that if we translate a triangle, the measures of all of its angles stay the same. So we can translate on the complex plane so that . Let the images of be respectively. Then, we can use the formula:
. (This is known as Euler's Theorem.)
Using Euler's theorem (represent each complex number in polar form, then use exponent identities), we can show that . So this method is valid. ~Math4Life2020
Solution 4
It is well known that is perpendicular to iff is a pure imaginary number. Here, we have that , , and . This means that this is equivalent to being a pure imaginary number. Plugging in , we have that being pure imaginary. Factoring and simplifying, we find that this is simply equivalent to being pure imaginary. We let , so this is equivalent to being pure imaginary. Expanding the product, this is equivalent to being pure imaginary. Taking the real part of this, and setting this equal to , we have that . Since , we have that . By the quadratic formula, , and taking the positive root gives that , so the answer is
~smartninja2000
Solution 5 (Official MAA)
The arguments of the two complex numbers differ by if the ratio of the numbers is a pure imaginary number. Thus three distinct complex numbers and form a right triangle at if and only if has real part equal to Hence must have real part equal to If the real part of is which is when The requested sum is
See also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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