Here is another engaging problem from Presh Talwalkar.
“___________Triangle Area 1984 AIME
Point P is in the interior of triangle ABC, and the lines through P are parallel to the sides of ABC. The three triangles shown in the diagram have areas of 4, 9, and 49. What is the area of triangle ABC?”
See the Pinwheel Area Problem
Here is another challenging problem from the 2004 Pi in the Sky Canadian magazine for high school students.
“Problem 4. Find the real solutions of the system
________________ (x + y)^5 = z,
________________ (y + z)^5 = x,
________________ (z + x)^5 = y.”
See the Quintic Nightmare
This is another fairly simple puzzle from Futility Closet.
“If an equilateral triangle is inscribed in a circle, then the distance from any point on the circle to the triangle’s farthest vertex is equal to the sum of its distances to the two nearer vertices (q = p + r).
(A corollary of Ptolemy’s theorem.)”
See A Tidy Theorem
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 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
Here is another train puzzle, this time from J. A. H. Hunter’s Entertaining Mathematical Teasers:
“Mike had made the [train] trip many times. ‘That’s the morning express from Tulla we’re passing,’ he said. ‘It left Tulla one hour after we pulled out from Brent, but we’re just 25% faster.’ ‘That’s right, and we’re also passing Cove, two-thirds the distance between Brent and Tulla,’ Martin agreed. ‘So we’re both right on schedule.’ Obviously a couple of train buffs! Assuming constant speeds and no stops, how long would it be before they reached Tulla?”
See the Train Buffs
It is hard to believe a whole year has passed since I started this blog. What is even more surprising is that by February I thought I was about done. I had more or less uploaded the math curiosities and problems I had been thinking about over the years and had presented most of the math essays I had written. There are of course only a finite number of math problems in the world, so I thought I was about done. But much to my surprise I kept finding one more thing that interested me, either an essay or math problem. So here I am. We will have to see what the next year brings.
What to say on this anniversary? I think I will give a retrospective about how the website has been received this past year. This is a challenge, since virtually all my visitors have been silent (which means I haven’t faced criticism, but then I generally haven’t received the necessary correctives either). There are ways to glean some information about visitors and I extracted what I could from the simple plugin I use to count visitors to different posts. I hear that Google analytics provides lots of details, such as the country of origin of a visitor (which would be interesting) and the like, but I have avoided Google and the other social networks as much as possible. After all, I am only an old curmudgeon with old-school notions of privacy.
See One Year Anniversary
I have been subverted again by a recent post by Ben Orlin, “Geometry Puzzles for a Winter’s Day,” which is another collection of Catriona Shearer’s geometric puzzles, this time her favorites for the month of November 2019 (which Orlin seems to have named himself). I often visit Orlin’s blog, “Math with Bad Drawings”, so it is hard to kick my addiction to Shearer’s puzzles if he keeps presenting collections. Her production volume is amazing, especially as she is able to maintain the quality that makes her problems so special.
The Stained Glass puzzle generated some discussion about needed constraints to ensure a solution. Essentially, it was agreed to make explicit that the drawing had vertical and horizontal symmetry in the shapes, that is, flipping it horizontally or vertically kept the same shapes, though some of the colors might be swapped.
See Geometric Puzzle Madness
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.
(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
This is a somewhat elegant problem from the 1987 Discover magazine’s Brain Bogglers by Michael Stueben:
“Each dot in the figure at left represents a factory. On which of the city’s 63 intersections should a warehouse be built to make the total distance between it and all the factors as short as possible? (A much simpler solution than counting and totaling the distances is available.)”
Note that the distance is the taxicab distance I discussed in my article South Dakota Travel Problem rather than the distance along straight lines between the warehouse and factories.
See the Factory Location Problem