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 another problem from the 2020 Math Calendar.

As a hint, recall that all the answers are integer days of the month. And the solution employs a technique familiar to these pages.

See Autumn Sum

The following problem comes from a 1961 exam set collected by Ed Barbeau of the University of Toronto. The discontinued exams (by 2003) were for 5^{th} year Ontario high school students seeking entrance and scholarships for the second year at a university.

“If *s _{n}* denotes the sum of the first

.”

Unfortunately, the “Grade XIII” exam problem sets were not provided with answers, so I have no confirmation for my result. There may be a cunning way to manipulate the series to get a solution, but I could not see it off-hand. So I employed my tried and true power series approach to get my answer. It turned out to be power series manipulations on steroids, so there must be a simpler solution that does not use calculus. I assume the exams were timed exams, so I am not sure how a harried student could come up with a quick solution. I would appreciate any insights into this.

See Serious Series

(**Update 1/18/2021**) Another Solution Continue reading

This is another problem from the 2020 Math Calendar.

“Find the difference between the highest and lowest roots of

*f*(*x*) = *x*^{3} – 54*x*^{2} + 969*x* – 5780”

See Root Difference

Here is another problem from the 2020 Math Calendar to stimulate your mind.

Remember that the answers to Math Calendar problems must all be whole numbers representing days of the month.

See New Years Sum

This is a delightful and surprising problem from Presh Talwalkar.

“This puzzle was created by a MindYourDecisions fan in India. What is the value of the infinite product? The numerators are the odd nth roots of [Euler’s constant] e and the denominators are even nth roots of e.”

See Euler Magic

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

This 2005 four-star problem from Colin Hughes at *Maths Challenge* is also a bit challenging.

“**Problem**

For any set of real numbers, R = {x, y, z}, let sum of pairwise products,

________________S = xy + xz + yz.

Given that x + y + z = 1, prove that S ≤ 1/3.”

Again, I took a different approach from Maths Challenge, whose solution began with an unexplained premise.

See the Pairwise Products

This is another stimulating math problem from Colin Hughes’s *Maths Challenge* website (mathschallenge.net).

“**Problem**

Find the exact value of the following infinite series:

____________1/2! + 2/3! + 3/4! + 4/5! + …”

See the Unexpected Sum

This is a stimulating problem from the UKMT Senior Math Challenge for 2017. The additional problem “for investigation” is particularly challenging. (I have edited the problem slightly for clarity.)

“The parabola with equation y = x² is reflected about the line with equation y = x + 2. Which of the following is the equation of the reflected parabola?

A_x = y² + 4y + 2_____B_x = y² + 4y – 2_____C_x = y² – 4y + 2

D_x = y² – 4y – 2_____E_x = y² + 2

**For investigation:** Find the coordinates of the point that is obtained when the point with coordinates (x, y) is reflected about the line with equation y = mx + b.”

See Flipping Parabolas.