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
This is a tantalizing problem from the 1977 Crux Mathematicorum.
“278. Proposed by W.A. McWorter, Jr., The Ohio State University.
If each of the medians of a triangle is extended beyond the sides of the triangle to 4/3 its length, show that the three new points formed and the vertices of the triangle all lie on an ellipse.”
See the Elliptical Medians Problem
This is a somewhat challenging problem from the 1997 American Invitational Mathematics Exam (AIME).
“A car travels due east at 2/3 miles per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at √2/2 miles per minute. At time t = 0, the center of the storm is 110 miles due north of the car. At time t = t1 minutes, the car enters the storm circle, and at time t = t2 minutes, the car leaves the storm circle. Find (t1 + t2)/2.”
See the Storm Chaser Problem for solutions.
This is a nifty problem from Presh Talwalkar.
“This is from a Manga called Q.E.D. I thank Sparky from the Philippines for the suggestion!
A string of beads is formed from 25 circles of the same size. The string passes through the center of each circle. The area enclosed by the string inside each circle is shaded in blue, and the remaining areas of the circles are shaded in orange. What is the value of the orange area minus the blue area? Calculate the area in terms of r, the radius of each circle.”
See the String of Beads Puzzle for solutions.
This is a fairly extensive clock problem by Geoffrey Mott-Smith from 1954.
“The clock shown in the illustration has just struck five. A number of things are going to happen in this next hour, and I am curious to know the exact times.
See After Five O’Clock for solutions.
This is a slightly challenging problem from Dan Griller.
“Every pupil at the Euler Academy studies French or Spanish. At the start of the year, one third of the French students also studied Spanish, and 2 fifths of the Spanish students also studied French. After one term, six of the double-linguists dropped French, so that now only a quarter of the French students study Spanish. How many pupils are at the Euler Academy?”
Just to be clear, “French students” means Euler Academy pupils studying French, and similarly for “Spanish students.”
See the Language Students Puzzle for solution.
Since the changes in Twitter (now X), I have not been able to see the posts, not being a subscriber. But I noticed poking around that some twitter accounts were still viewable. However, like some demented aging octogenarian they had lost track of time, that is, instead of being sorted with the most recent post first, they showed a random scattering of posts from different times. So a current post could be right next to one several years ago. That is what I discovered with the now defunct MathsMonday site. I found a post from 10 May 2021 that I had not seen before, namely,
“The points A and B are on the curve y = x2 such that AOB is a right angle. What points A and B will give the smallest possible area for the triangle AOB?”
See the Pythagorean Parabola Puzzle for solution.
(Update 9/1/2023) Elegant Alternative Solution by Oscar Rojas
Continue reading
This essay is slightly tangential to my usual fare, but it is prompted by a most amazing video that convinced me that the impact of AI this time is not hype, but rather a real threat to our society. I found the video at 3 Quarks Daily and it was of Johnny Cash singing a song called “Barbie Girl” to the tune of his trademark Folsom Prison Blues—only it wasn’t the late Johnny Cash (1932–2003), it was AI!
See Voice Stealing
(Update 11/2/2023) I was wondering why the seeming lack of interest in this post, and then I tried the link to the video and found it has been removed from the public. There must be a reason, probably copyright issues somewhere, so even though I got a copy when it was public, I don’t think I should post it. This is really too bad, since it is in incredible example of what may be our dystopian future. I just rooted around and found another link that seems to be working. I have updated the text above.
This is another puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings.
“A mysterious square has materialized in the middle of the MCG, hovering in mid-air. The heights above the ground of three of its corners are 13, 21 and 34 metres. The fourth corner is higher still. How high?”
See the Floating Square Puzzle for solutions.
(Update 8/13/2023) Alternative Solution Continue reading
