# Problems of the Week

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# 2017-06-19 Basic

$\large \frac{2^n+2^{n-1}}{2^{n+1}-2^n} = \, ?$

Suppose that a human can have at most 1,000,000 hairs on his or her head.

In a city with at least 1,000,002 people, are there always at least two people with the exact same number of hairs on each of their heads?

A circle is inscribed in a quadrilateral, as shown above. The two blue sides have lengths 17 and 13, respectively. What is the sum of the lengths of the other two red sides?

When jumping off a ledge feet-first, why is it advisable to bend your knees on landing on the ground?

There are 2 identical regular tetrahedrons. Each is painted with 1 red face, 1 blue face, 1 green face, and 1 yellow face.

Will they always be painted in the same way?


Clarification: They are painted in the same way if you cannot tell them apart from each other.

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