Solution: 2014 Fall Midterm - 6

Author: Michiel Smid

Question

Consider a square with sides of length 3. Assume this square contains 10 points. Which of the following is true?

(a)

At least two of these $n$ points are at distance at most $\left. 1 \middle/ \sqrt{2} \right.$.

(b)

None of the above.

(c)

At least two of these $n$ points are at distance at most $1$.

(d)

At least two of these $n$ points are at distance at most $\sqrt{2}$.

Solution

This is too hard to draw out, so I’ll explain

Split the square into 9 pieces. 3 rows and 3 columns. You now have 9 squares that are each 1 unit long

Put a point in each of them as far away from the others.

For instance, the point in the upper right square will try to move to the upper right corner to get even further from the other points in other squares.

You’ve just put down 9 points. Put 1 more point in one of the boxes. So let’s use the upper right square as example.

Assuming you put the 10th point in that box, then you’ll put it in the bottom left part of the upper right box.

The length of the right of the box is ${sqrt{1}}$. Distance from the points is hypotenuse.

$\text{Hypotenuse}^2 = {sqrt{1}}^2 + {sqrt{1}}^2$

$\text{Hypotenuse}^2 = 2 {sqrt{1}}^2$

$\text{Hypotenuse}^2 = 2$

$\text{Hypotenuse} = sqrt{2}$