This is the problem:
Suppose you are given a finite set of coins in the plane, all with different diameters. Show that one of the coins is tangent to at most $5$ of the others.
I can easily imagine a coin with more than $5$ coins tangent to it, so I'm obviously missing something.
On
All the circles have distinct diameters (and henceforth, radii) hence there must be a smallest circle. This is known as the Extreme Principle. Now, consider the smallest circle. At most, how many circles can you draw tangent to it, like you actually can, when all of those tangent circles have bigger radius than the smallest one? Try drawing 2-3 and you will then arrive at the conclusion 5. The problem doesn't says for every one but says for only one circle. Here is a illustration of what you might draw. Image (Imgur Link)
Now, consider only one of the 5 circles. Draw tangents from the radius to the smallest circle. Notice that you can create another tangent circle from this new tangent. Creating a circle anywhere else will make the circle equal to or smaller to the smallest circle which makes a contradiction. The explanation for this is that for every circle, call the length of the arc as the impact of the circle the length measured in unit of length. Then, the impact has to be bigger than 90 degrees, for if it is equal to that, then the circle has the same radius! Try it yourself. Now, the actual impact for all circles looks like 720 degrees which is the degrees for 2 circles. But have a look closely, all the angles overlap! So, consider only the part of the impact which isn't overlapped and that is 360 degrees. You can also do that in cm's or unit lengths.
It didn't say ALL of the coins are tangent to at most five others. One of them may be tangent to a thousand others. But you're asked to show that somewhere among all those coins, there is one that is not tangent to more than five others.
How many coins are tangent to the smallest one?