Here is another UKMT Senior Challenge problem for 2017.
“The diagram shows a square PQRS with edges of length 1, and four arcs, each of which is a quarter of a circle. Arc TRU has centre P; arc VPW has centre R; arc UV has centre S; and arc WT has centre Q.
What is the length of the perimeter of the shaded region?
A_6___B_(2√2 – 1)π___C_(√2 – 1/2)π ___D_2___E_(3√2 – 2)π”
See Elliptic Circles for a solution.
This is a most surprising and amazing identity from the 1965 Polish Mathematical Olympiads.
“31. Prove that if n is a natural number, then we have
(√2 – 1)n = √m – √(m – 1),
where m is a natural number.”
Here, natural numbers are 1, 2, 3, …
I found it to be quite challenging, as all the Polish Math Olympiad problems seem to be.
See the Amazing Identity
Here is another challenging problem from the Polish Mathematical Olympiads. Its generality will cause more thought than for a simpler, specific problem.
“A cyclist sets off from point O and rides with constant velocity v along a rectilinear highway. A messenger, who is at a distance a from point O and at a distance b from the highway, wants to deliver a letter to the cyclist. What is the minimum velocity with which the messenger should run in order to attain his objective?”
See the Tired Messenger Problem
(Update 1/29/2025) Dan Steinitz Solution
Dan Steinitz from Israel has sent an elegant solution that only involves vectors and geometry without calculus. I have edited slightly his email and added excerpts from his whiteboard solution, though without the Hebrew annotations, which unfortunately I cannot read. But that is the glory of the universal language of mathematics: it can be read and understood in any language.
Here is another typical sum puzzle from Presh Talwalkar.
“Solve the following sums:
_____1/(1×3) + 1/(3×5) + 1/(5×7) + 1/(7×9) + 1/(9×11) =
_____1/(4×7) + 1/(7×10) + 1/(10×13) + 1/(13×16) =
_____1/(2×7) + 1/(7×12) + 1/(12×17) + … =”
The only reason I am including this puzzle is that Talwalkar gets very excited about deriving a formula that can solve sums of this type. This gives me an opportunity to discuss the “formula vs. procedure” way of doing math.
See the Incredible Trick Puzzle for solutions.
This is another physics-based problem from Colin Hughes’s Maths Challenge website (mathschallenge.net) that may take a bit more thought.
“A firework rocket is fired vertically upwards with a constant acceleration of 4 m/s2 until the chemical fuel expires. Its ascent is then slowed by gravity until it reaches a maximum height of 138 metres.
Assuming no air resistance and taking g = 9.8 m/s2, how long does it take to reach its maximum height?”
I can never remember the formulas relating acceleration, velocity, and distance, so I always derive them via integration.
See the Fireworks Rocket for solutions.
This is another stimulating little problem from the 2022 Math Calendar.
“a1 = 1, a2 = 2, …, an+1 = an + 6an-1
x = lim an+1/an as n → ∞
Solve for x.”
As before, recall that all the answers are integer days of the month.
See Stimulating Sequence for a solution.
James Tanton provides another imaginative problem on Twitter.
“I am at point A and want to walk to point B via some point, any point, P on the circle. What point P should I choose so that my journey A → P → B is as short as possible?”
Hint: I got ideas for a solution from two of my posts, “Square Root Minimum” and “Maximum Product”.
This is another infinite series from Presh Talwalkar, but with a twist.
“This problem is adapted from one given in an annual national math competition exam in France. Evaluate the infinite series:
1/2! + 2/3! + 3/4! + …”
The twist is that Talwalkar provides three solutions, illustrating three different techniques that I in fact have used before in series and sequence problems. But this time I actually found a simpler solution that avoids all these. You also need to remember what a factorial is: n! =n(n – 1)(n – 2)…3·2·1.
See a Nice Factorial Sum for solutions.
I came across an interesting problem in the MathsJam Shout for February 2022.
(“MathsJam is a monthly opportunity for like-minded self-confessed maths enthusiasts to get together in a pub and share stuff they like. Puzzles, games, problems, or just anything they think is cool or interesting. Monthly MathsJam nights happen in over 70 locations around the world, on the second-to-last Tuesday of each month. To find your nearest MathsJam, visit the website at www.mathsjam.com.”)
“Given two lines Ax + By + C = 0 and ax + by + c = 0, is there a simple link between the vectors (A, B, C), (a, b, c), and the point where the lines cross?”
The answer, of course, is yes, but the question is somewhat open-ended and I was not able to track down any answer given.
See the Point of Intersection Problem
This is a belated Christmas puzzle from December 2019 MathsMonday.
“A Christmas tree is made by stacking successively smaller cones. The largest cone has a base of radius 1 unit and a height of 2 units. Each smaller cone has a radius 3/4 of the previous cone and a height 3/4 of the previous cone. Its base overlaps the previous cone, sitting at a height 3/4 of the way up the previous cone.
What are the dimensions of the smallest cone, by volume, that will contain the whole tree for any number of cones?”
Recall that the volume of a cone is π r2 h/3.
See Another Christmas Tree Puzzle for a solution.
