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Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on Function of Complex Numbers and Integers.

## Function of Complex Numbers – AIME I, 1999

Let f(z) =(a+bi)z where a,b are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin given that |a+bi|=8 and that $b^{2}$=$\frac{m}{n}$ where m and n are relatively prime positive integers, find m+n.

• is 107
• is 259
• is 840
• cannot be determined from the given information

### Key Concepts

Functions

Integers

Complex Numbers

But try the problem first…

Source

AIME I, 1999, Question 9

Complex Numbers from A to Z by Titu Andreescue

## Try with Hints

First hint

Let z=1+i f(1+i)=(a+bi)(1+i)=(a-b)+(a+b)i The image point must be equidistant from (1,1) and(0,0) then the image point lie on the line with slope -1 and which passes through $(\frac{1}{2},\frac{1}{2})$ that is x+y=1

Second Hint

putting x=(a-b) and y=(a+b) gives 2a=1 and $a=\frac{1}{2}$

Final Step

and $(\frac{1}{2})^{2} +b^{2}=8^{2}$ then $b^{2}=\frac{255}{4}$ then 255+4=259.