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Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1988 based on Complex Plane.

## Problem on Complex Plane – AIME I, 1988

Let w_1,w_2,….,w_n be complex numbers. A line L in the complex plane is called a mean line for the points w_1,w_2,….w_n if L contains points (complex numbers) z_1,z_2, …..z_n such that $\sum_{k=1}^{n}(z_{k}-w_{k})=0$ for the numbers $w_1=32+170i, w_2=-7+64i, w_3=-9+200i, w_4=1+27i$ and $w_5=-14+43i$, there is a unique mean line with y-intercept 3. Find the slope of this mean line.

• is 107
• is 163
• is 634
• cannot be determined from the given information

### Key Concepts

Integers

Equations

Algebra

But try the problem first…

Source

AIME I, 1988, Question 11

Elementary Algebra by Hall and Knight

## Try with Hints

First hint

$\sum_{k=1}^{5}w_k=3+504i$

and $\sum_{k-1}^{5}z_k=3+504i$

Second Hint

taking the numbers in the form a+bi

$\sum_{k=1}^{5}a_k=3$ and $\sum_{k=1}^{5}b_k=504$

Final Step

or, y=mx+3 where $b_k=ma_k+3$ adding all 5 equations given for each k

or, 504=3m+15

or, m=163.