E35 - Three Triangles in a Regular Hexagon

Here is another problem from the Catriona Shearer collection. ABCDEF is a regular hexagon. Lines have been drawn to create three triangles, whose interiors have been shaded. What fraction of the hexagon is shaded?

The points labeled M_i are the midpoints of  their sides, and the unlabeled point marks the center of the hexagon.

E34. Tangent circles in a rectangle

This problem was called to my attention to Apratim Roy. It comes from a collection, on Twitter, of many dozens of interesting geometry problems, at the level of high school geometry, by Catriona Shearer, a math teacher in the UK, which she has posted on Twitter, @Cshearer41.

In the diagram below. you are given circles O and P, externally tangent at B. Circle O is tangent to two sides of rectangle DEFG, and circle P is tangent to three sides of the rectangle, as shown. Find the measure of angle ABC.

It is quite surprising that you don’t need to be given any measurements to answer the problem. (But no fair using that fact in your proof; that is, you can’t just prove a special case such as two circles of equal size.) I also like this problem because it is hard, but not too hard, and because there are several different nice ways to solve it.

A 14. Triangles in a regular hexagon

I came across a neat problem in Quora. When I found it, there were no solutions, and with crucial help from my student Apratim Roy, I was able to find a very elegant solution. Unfortunately, when I went back to Quora I was unable to find the original problem, so I cannot give credit to the proposer. If anyone wants, send me an email and I will send you my solution, or provide a hint.

Here is the problem: Let ABCDEF be a regular hexagon, and P a point inside the hexagon. Suppose Area(PAB) = 3, Area(PCD) = 5, and Area(PEF) = 8. Find Area(PBC).

E33. Tiling a checkerboard

This problem was shown to me by my student, Apratim Roy. Though it involves only elementary concepts, I found it rather difficult, and thought the solution was very surprising and elegant.

You are given a standard 8 x 8 checkerboard, with one square removed, and 21 3 x 1 tiles. In other words, there are exactly enough tiles to cover the modified board. Your task is to find a way to do this, without cutting any tile.

(a) Find out what square must be removed for the task to be possible. (4 possible answers).
(b) Describe the tiling. (Many possible answers).

You might guess that the removed square needs to be one of the four corners of the checkerboard, but you would be wrong.

A13. Points of tangency of an ellipse and a circle

Let E be the ellipse with equation x²/4 + y² = 1 and C(r) be the circle with center (1,0) and radius r. For which values of r do the curves E and C(r) have point(s) of tangency?

This is fairly routine, but still a bit challenging to find all solutions.

A12. Fifth powers final digit - generalized.

If numbers are expressed in base b, for which b is it true that n⁵ and n end in the same digit for all positive integers n?
This is an obvious generalization of Problem E32.

E32. Fifth powers final digit

The following rather neat problem occurs in Challenging Problems in Algebra by Alfred S. Posamentier and Charles T. Salkind. I think it is suitable for a bright high school student or, with some hints, even for average high school students.

Prove that n and n⁵ always end in the same digit (in ordinary base-10 representation).