Solution: 2017 Fall Midterm - 8
Author: Michiel SmidQuestion
Consider a square with sides of length 17. This square contains $n$ points. What is the minimum
value of $n$ such that we can guarantee that at least two of these points have distance at most
$\left. 17 \middle/ \sqrt{2} \right.$?
(a)
4
(b)
5
(c)
6
(d)
7
Solution
Placing 4 points furthest from each other creates a square.
The distance between the two points is $ 17 $, which is just the length of side
Now, we place 4 points in each corner of the square. Place the fifth point in the middle of the square.
Now, let’s calculate the distance between the middle point and any other point.
$ a^2 + b^2 = c^2 $
$ {( \frac{1}{2}17)}^2 + {( \frac{1}{2}17)}^2 = c^2 $
$ \frac{1}{4}17^2 + \frac{1}{4}17^2 = c^2 $
$ \frac{1}{2}17^2 = c^2 $
$ sqrt{ \frac{1}{2}17^2} = c $
$ sqrt{ \frac{1}{2}}17 = c$