Difference between revisions of "2019 AIME I Problems/Problem 12"

Revision as of 18:31, 25 February 2021

Problem

Given $f(z) = z^2-19z$ , there are complex numbers $z$ with the property that $z$ , $f(z)$ , and $f(f(z))$ are the vertices of a right triangle in the complex plane with a right angle at $f(z)$ . There are positive integers $m$ and $n$ such that one such value of $z$ is $m+\sqrt{n}+11i$ . Find $m+n$ .

Solution 1

Notice that we must have $\frac{f(f(z))-f(z)}{f(z)-z}=-\frac{f(f(z))-f(z)}{z-f(z)}\in i\mathbb R .$ However, $f(t)-t=t(t-20)$ , so $\begin{align*} \frac{f(f(z))-f(z)}{f(z)-z}&=\frac{(z^2-19z)(z^2-19z-20)}{z(z-20)}\\ &=\frac{z(z-19)(z-20)(z+1)}{z(z-20)}\\ &=(z-19)(z+1)\\ &=(z-9)^2-100. \end{align*}$ Then, the real part of $(z-9)^2$ is $100$ . Since $\text{Im}(z-9)=\text{Im}(z)=11$ , let $z-9=a+11i$ . Then, $100=\text{Re}((a+11i)^2)=a^2-121\implies a=\pm\sqrt{221}.$ It follows that $z=9+\sqrt{221}+11i$ , and the requested sum is $9+221=\boxed{230}$ .

(Solution by TheUltimate123)

Solution 2

We will use the fact that segments $AB$ and $BC$ are perpendicular in the complex plane if and only if $\frac{a-b}{b-c}\in i\mathbb{R}$ . 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: $\frac{f(z)-z}{f(f(z))-f(z)}\in i\mathbb{R}$ $\frac{z^2-19z-z}{(z^2-19z)^2-19(z^2-19z)-(z^2-19z)}$ $\frac{z^2-20z}{z^4-38z^3+341z^2+380z}$ $\frac{z(z-20)}{z(z+1)(z-19)(z-20)}$ $\frac{1}{(z+1)(z-19)}\in i\mathbb{R}$

The factorization of the nasty denominator above is made easier with the intuition that $(z-20)$ must be a divisor for the problem to lead anywhere. Now we know $(z+1)(z-19)\in i\mathbb{R}$ so using the fact that the imaginary part of $z$ is $11i$ and calling the real part r,

$(r+1+11i)(r-19+11i)\in i\mathbb{R}$ $r^2-18r-140=0$

solving the above quadratic yields $r=9+\sqrt{221}$ so our answer is $9+221=\boxed{230}$

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 , $A= f(f(z)),B=f(z),C= x$ respectively.

WLOG, $Z_1= \frac{f(f(z))-f(z)}{z-f(z)}$ . According to the question $\arg{Z_1}=\frac{\pi}{2}$ .

So, $\Re (Z_1)=0$ .

Now, $Z_1=\frac{z(z-19)(z+1)(z-20)}{-z(z-20)}$ .

$\implies Z_1= -(z^2-18z-19)$ . WLOG, $z=a+11i$ .where $a=m+\sqrt{n}$ .

So, $\Re (Z_1)= -(a^2-18a-140)$ . Solving, $a^2-18a-140=0$ .get ,

$a=9$ ± $\sqrt{221}$ . So, possible value of $a=9+\sqrt{221}$ .

$m+n=\boxed{230}$ . ~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 $ABC$ on the complex plane so that $B=0$ . Let the images of $A, B, C$ be $A', B', C'$ respectively. Then, we can use the formula:

$re^{i\theta}=rcos(\theta)+i \cdot rsin(\theta)$ . (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 $arg(\frac{A'}{C'})=arg(A')-arg(C')=\angle(A'B'C')=\angle(ABC)$ . So this method is valid. ~Math4Life2020

Solution 4

It is well known that $AB$ is perpendicular to $CD$ iff $\frac{d-c}{b-a}$ is a pure imaginary number. Here, we have that $A=z$ , $B,C=f(z)$ , and $D=f(f(z))$ . This means that this is equivalent to $\frac{f(f(z))-f(z)}{f(z)-z}$ being a pure imaginary number. Plugging in $f(z)=z^2-19z$ , we have that $\frac{(z^2-19z)-19(z^2-19z)-(z^2-19z)}{z^2-19z-z}$ being pure imaginary. Factoring and simplifying, we find that this is simply equivalent to $(z-19)(z+1)$ being pure imaginary. We let $z=a+bi$ , so this is equivalent to $(a+bi-19)(a+bi+1)$ being pure imaginary. Expanding the product, this is equivalent to $a^2+abi+a+abi-b^2+bi-19a-19bi-19$ being pure imaginary. Taking the real part of this, and setting this equal to $0$ , we have that $a^2-18a-b^2-19=0$ . Since $b=11$ , we have that $a^2-18a-140=0$ . By the quadratic formula, $a=9 \pm \sqrt{221}$ , and taking the positive root gives that $a=9+ \sqrt{221}$ , so the answer is $9+221=230$

~smartninja2000

Solution 5 (Official MAA)

The arguments of the two complex numbers differ by $90^\circ$ if the ratio of the numbers is a pure imaginary number. Thus three distinct complex numbers $A,\,B,$ and $C$ form a right triangle at $B$ if and only if $\tfrac{C-B}{B-A}$ has real part equal to $0.$ Hence $\begin{align*} \frac{f(f(z))-f(z)}{f(z)-z}&=\frac{(z^2-19z)^2-19(z^2-19z)-(z^2-19z)}{(z^2-19z)-z}\\ &=\frac{(z^2-19z)(z^2-19z-19-1)}{z^2-20z}\\ &=\frac{z(z-19)(z+1)(z-20)}{z(z-20)} \\ &=z^2-18z-19 \end{align*}$ must have real part equal to $0.$ If $z=x+11i,$ the real part of $z^2-18z-19$ is $x^2-11^2-18x-19,$ which is $0$ when $x=9\pm\sqrt{221}.$ The requested sum is $9+221=230.$

@@ Line 72: / Line 72: @@
 ~smartninja2000
+==Solution 5 (Official MAA)==
+The arguments of the two complex numbers differ by <math>90^\circ</math> if the ratio of the numbers is a pure imaginary number. Thus three distinct complex numbers <math>A,\,B,</math> and <math>C</math> form a right triangle at <math>B</math> if and only if <math>\tfrac{C-B}{B-A}</math> has real part equal to <math>0.</math> Hence <cmath>\begin{align*}
+\frac{f(f(z))-f(z)}{f(z)-z}&=\frac{(z^2-19z)^2-19(z^2-19z)-(z^2-19z)}{(z^2-19z)-z}\\
+&=\frac{(z^2-19z)(z^2-19z-19-1)}{z^2-20z}\\
+&=\frac{z(z-19)(z+1)(z-20)}{z(z-20)} \\
+&=z^2-18z-19
+\end{align*}</cmath> must have real part equal to <math>0.</math> If <math>z=x+11i,</math> the real part of <math>z^2-18z-19</math> is <math>x^2-11^2-18x-19,</math> which is <math>0</math> when <math>x=9\pm\sqrt{221}.</math> The requested sum is <math>9+221=230.</math>
 ==See also==
 {{AIME box|year=2019|n=I|num-b=11|num-a=13}}
 {{MAA Notice}}

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