Week of 1/13

1. Consider a 5 x 5 square, and four beetles placed at each of the corners. Each beetle starts moving toward the beetle on its right at unit speed.

As time progresses, what shape to the beetles trace out? How long does it take for the beetles to intersect, and where do they intersect? What happens if we change the set-up, so three beetles, are placed at the corners of an equilateral triangle of length 5. How long does it take for them to meet up, and where do they meet?

(Elem, S)

2. Prove that it is impossible to write a group G as a union of two proper subgroups.

(UpDiv, S)

3. Consider a regular n-gon inscribed in the unit circle. Show that the product of the distances from one vertex to all other vertices is n.

4. Find (with proof) all ordered pairs (x, y) \in \mathbb{R}^2 such that \frac{\log x}{\log y} = \frac{x}{y}

(LowDiv, T?)

Thanks to yesmanapple, bryanic, Anon and Zach for the problems of this week.

Note: Do not post solutions below.

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