Question

Let $w_1,$ $w_2,$ $\ldots,$ $w_n$ be complex numbers. A line $L$ in the complex plane is called a mean line for the points $w_1,$ $w_2,$ $\ldots,$ $w_n$ if $L$ contains points (complex numbers) $z_1,$ $z_2,$ $\ldots,$ $z_n$ 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.

Answer

Answer: 163

Concepts:complex number centroid slope

Difficulty rating: 1970

Small Hint:

Average the equation $\sum(z_k-w_k)=0$

Big Hint:

A mean line is exactly a line through the centroid of the given points

Solution:

The condition says that the average of the $z_k$ equals the average of the $w_k.$ Since all $z_k$ lie on $L,$ their average lies on $L;$ conversely, any line through the average works by taking all $z_k$ equal to that point. The centroid satisfies $\begin{aligned}\bar x&=\frac{32-7-9+1-14}{5}=\frac35,\\\bar y&=\frac{170+64+200+27+43}{5}\\&=\frac{504}{5}.\end{aligned}$ The line through this point and $(0,3)$ has slope $\frac{\bigl(\frac{504}{5}-3\bigr)}{\frac35}=163.$

1988 AIME Problem 11

Answer: 163

Difficulty rating: 1970

Small Hint:

Big Hint:

Solution:

Problem 11 in Other Years