Here is a fairly straight-forward problem from *500 Mathematical Challenges*.

**“Problem 256.** Let *n* be a positive integer. Show that (*x* – 1)^{2} is a factor of *x ^{n} – n*(

*x*– 1) – 1.”

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Here is a fairly straight-forward problem from *500 Mathematical Challenges*.

**“Problem 256.** Let *n* be a positive integer. Show that (*x* – 1)^{2} is a factor of *x ^{n} – n*(

Here is a problem from *Five Hundred Mathematical Challenges* that I indeed found quite challenging.

**“Problem 235. **Two fixed points *A* and *B* and a moving point *M *are taken on the circumference of a circle. On the extension of the line segment *AM *a point *N *is taken, outside the circle, so that lengths MN = MB*. *Find the locus of *N.*”

Since one of the first hurdles I faced with this problem was trying to figure out what type of shape was being generated, I thought I would omit my usual drawings illustrating the problem statement. There turned out to be a lot of cases to consider, but the result was most satisfying. I also included the case when *N* is inside the circle. Again Visio was my main tool to handle all the examples with the concomitant requirement to prove whatever Visio suggested.

See the Curve Making Puzzle

Here is another good problem from *Five Hundred Mathematical Challenges*:

**“Problem 100.** A hexagon inscribed in a circle has three consecutive sides of length *a* and three consecutive sides of length *b*. Determine the radius of the circle.”

This problem made me think of the Putnam Octagon Problem. Again my approach might be considered a bit pedestrian. 500 Math Challenges had a slightly slicker solution.

See the Lop-sided Hexagon Problem

The following problem from *Five Hundred Mathematical Challenges* was a challenge indeed, even though it appeared to be a standard travel puzzle.

“**Problem 118**. Andy leaves at noon and drives at constant speed back and forth from town A to town B. Bob also leaves at noon, driving at 40 km per hour back and forth from town B to town A on the same highway as Andy. Andy arrives at town B twenty minutes after first passing Bob, whereas Bob arrives at town A forty-five minutes after first passing Andy. At what time do Any and Bob pass each other for the nth time?”

See the Perpetual Meetings Problem

Here is another simply amazing problem from *Five Hundred Mathematical Challenges*:

“**Problem 154**. Show that three solutions, (x1,.y1), (x2,.y2), (x3, y3), of the four solutions of the simultaneous equations

____________(x – h)² + (y – k)² = 4(h² + k²)

______________________xy = hk

are vertices of an equilateral triangle. Give a geometrical interpretation.”

Again, I don’t see how anyone could have discovered this property involving a circle, a hyperbola, and an equilateral triangle. It seems plausible when h.=.k, but it is not at all obvious for h.≠.k. For some reason, I had difficulty getting a start on a solution, until the obvious approach dawned on me. I don’t know why it took me so long.

See the Amazing Triangle Problem.

This is truly an amazing result from *Five Hundred Mathematical Challenges*.

“**Problem 119**. Two unequal regular hexagons ABCDEF and CGHJKL touch each other at C and are so situated that F, C, and J are collinear.

Show that

(i) the circumcircle of BCG bisects FJ (at O say);

(ii) ΔBOG is equilateral.”

I wonder how anyone ever discovered this.

See the Magic Hexagons

If you will pardon the pun, this is a diabolical problem from the collection *Five Hundred Mathematical Challenges*.

“**Problem 5**. Calculate the sum

__________

It has a non-calculus solution, but that involves a bunch of manipulations that were not that evident to me, or at least I doubt if I could have come up with them. I was able to reframe the problem using one of my favorite approaches, power series (or polynomials). The calculations are a bit hairy in any case, but I was impressed that my method worked at all.

See the Number of the Beast