This is a fairly straight-forward problem from Presh Talwalkar.
“A triangle is divided by 8 parallel lines that are equally spaced, as shown below. Starting from the top small triangle, color each alternate stripe in blue and color the remaining stripes in red. If the blue stripes have a total area of 145, what is the total area of the red stripes?”
See the Triangle Stripes Problem
This is a Maths Item of the Month (MIOM) problem that seems opaque at first. (“The Maths Item of the Month is a monthly problem aimed at teachers and students of GCSE and A level Mathematics.”)
“Two fixed circles, C1 and C2, intersect at A and B. P is on C1. PA and PB produced meet C2 at A’ and B’ respectively. How does the length of the chord A’B’ change as P moves?”
Just start noticing relationships and the answer falls out nicely.
(MIOM problems often appear on MathsMonday and are also produced by Mathematics Education Innovation (MEI).)
See the Triangle Projection Problem
This is a nice problem from the UKMT Senior Mathematics Challenge for 2022:
“Five line segments of length 2, 2, 2, 1 and 3 connect two corners of a square as shown in the diagram. What is the shaded area?
A 8____B 9____C 10____D 11____E 12”
The pleasure of solving this problem may be lessened if one is under a time crunch, as is the case with all these timed tests.
See the Broken Diagonal Problem
This is an interesting problem from the Scottish Mathematics Council (SMC) 2014 Senior Math Challenge .
“Two straight sections of a road, each running from east to west, and located as shown, are to be joined smoothly by a new roadway consisting of arcs of two circles of equal radius. The existing roads are to be tangents at the joins and the arcs themselves are to have a common tangent where they meet. Find the length of the radius of these arcs.”
See the Road Construction Problem
This turned out to be a challenging puzzle from the 1980 Canadian Math Society’s magazine, Crux Mathematicorum.
“Proposed by Leon Bankoff, Los Angeles, California.
Professor Euclide Paracelso Bombasto Umbugio has once again retired to his tour d’ivoire where he is now delving into the supersophisticated intricacies of the works of Grassmann, as elucidated by Forder’s Calculus of Extension. His goal is to prove Neuberg’s Theorem:
If D, E, F are the centers of squares described externally on the sides of a triangle ABC, then the midpoints of these sides are the centers of squares described internally on the sides of triangle DEF. [The accompanying diagram shows only one internally described square.]
Help the dedicated professor emerge from his self-imposed confinement and enjoy the thrill of hyperventilation by showing how to solve his problem using only highschool, synthetic, Euclidean, ‘plain’ geometry.”
Alas, my plane geometry capability was inadequate to solve the puzzle that way, so I had to resort to the sledge hammer of analytic geometry, trigonometry, and complex variables.
See Neuberg’s Theorem
Here is another elegant Quantum math magazine Brainteaser problem.
“Two squares are inscribed in a semicircle as shown in the figure at left. Prove that the area of the big square is four times that of the small one.”
See More Squares in Semicircle
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”.
See Minimum Path Via Circle
This seemingly impossible problem from Presh Talwalkar turned out to be quite solvable upon reflection.
“A similar question was given to students in Thailand. For real numbers x, y, what is the minimum value of
√((x – 4)2 + (y – 10)2) + √((x – 44)2 + (y – 19)2)”
See the Square Root Minimum
Via Alex Bellos I found another Russian math magazine with fun problems. It is called Kvantik and Tanya Khovanova has a description (2015):
“Kvant [Quantum] was a very popular science magazine in Soviet Russia. It was targeted to high-school children and I was a subscriber. Recently I discovered that a new magazine appeared in Russia. It is called Kvantik, which means Little Kvant. It is a science magazine for middle-school children. The previous years’ archives are available online in Russian. I looked at 2012, the first publication year, and loved it.”
Unfortunately, the magazine is in Russian and the later issues are only partially given online. To get the full magazine you need to subscribe. I used Google Translate and the mathematical context to render the English. Here is an interesting geometric problem that I would have thought to be quite challenging for middle schoolers.
“The vertices of the two squares are joined by two segments, as in the figure. It is given that these segments are equal. Find the angle between them.
See the Two Squares Problem
(Update 8/22/2022, 9/1/2022) Simpler Solution, Simplest Solution!
Here is a problem from the 2022 Math Calendar.
“Two small circles of radius 4 are inscribed in a large semicircle as shown. Find the radius of the large semicircle.”
As before, recall that all the answers are integer days of the month.
As seemed to be implied by the original Math Calendar diagram, I made explicit that the upper circle was tangent to the midpoint of the chord. Otherwise, the problem is insufficiently constrained.
See Two and a Half Circles