Extreme principle/solutions

These are the solutions to the problems on the extreme principle.

1

First, we note that it is impossible for a coin $\omega$ to be tangent to six coins $\omega_1, \omega_2, \hdots, \omega_6$ if each of the $\omega_i$ is bigger than $\omega$.

Now consider the smallest coin. By the above observation, it is tangent to at most five of the others.

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