The Futility Closet website had the following problem:

“In isosceles triangle ABC, CD = AB and BE is perpendicular to AC. Show that CEB is a 3-4-5 right triangle.”

See a Triangle Puzzle

The Futility Closet website had the following problem:

“In isosceles triangle ABC, CD = AB and BE is perpendicular to AC. Show that CEB is a 3-4-5 right triangle.”

See a Triangle Puzzle

Here is another problem from the “Challenges” section of the *Quantum* magazine.

“Point L divides the diagonal AC of a square ABCD in the ratio 3:1, K is the midpoint of side AB. Prove that angle KLD is a right angle. (Y. Bogaturov)”

This is a classic example of a mixture problem from Dan Griller that recalls my agonies of beginning algebra.

“In Cauchy Village, full fat milk has 3.5% fat content, semi-skimmed milk has a 1.5% fat content, and skimmed milk has a 0.2% fat content. How many liters of full fat milk must be added to 100 liters of skimmed milk to produce semi-skimmed milk?”

See the Milk Mixing Puzzle for solution.

This is a nice puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings.

“Two ants are on a cylindrical glass that is 5 centimetres in diameter. The ants are on opposite sides of the glass, 5 centimetres down from the glass’s rim. If both ants are on the outside of the glass, what is the shortest distance required for one ant to crawl to the other? What if one ant is on the outside of the glass and the other is on the inside?”

See the Ant Connection Problem for solution.

This is a nifty problem from Presh Talwakar.

“This is adapted from the 1994 Putnam, A2. Thanks to Nirman for the suggestion!

Let *R* be the region in the first quadrant bounded by the *x*-axis, the line *y* = *x*/2, and the ellipse *x*^{2}/9 + *y*^{2} = 1. Let *R*‘ be the region in the first quadrant bounded by the *y*-axis, the line *y* = *mx* and the ellipse. Find the value of *m* such that *R* and *R*‘ have the same area.”

See the Putnam Ellipse Areas Problem for solution.

One of the physics blogs I enjoy reading is by the mathematical physicist Peter Woit, called *Not Even Wrong*. A recent post provided a tantalizing teaser:

“I want to [link to] an insightful explanation of the history of string theory, discussing the implications of how it was sold to the public. It’s by a wonderful young physicist I had never heard of before, Angela Collier. She has a Youtube channel, and her latest video is string theory lied to us and now science communication is hard.

… It’s as hilarious as it is brilliant, and you have to see for yourself.”

Collier delivered her talk lucidly and thoroughly—*all while playing a frenetic video game*! She claimed she used the length of the game to time her talk. Of course we can walk and talk, and ride bicycles and talk, but I have never seen anyone split their mental concentration between a fast-paced video game and an esoteric physics explanation of the history of string theory and supersymmetry—for over 50 minutes! And there was something about her presentation that was completely captivating. It was definitely a serious scientific talk, but the ludicrousness of the game-playing echoed how ridiculous the continued, misplaced fascination with string theory is. Naturally I had to learn more about this provocative physicist.

See A New Day

This is a Catriona Agg problem presented by itself, since it turned out to be the most challenging one I ever tried. Usually I can solve her problems in a few minutes or maybe hours, or sometimes days if they are especially challenging. But this problem has taken me weeks and I had to rely on a non-geometric argument. The problem is full of fascinating and unexpected relationships, but I couldn’t find a way to use them to prove the answer.

See the Curious Sunbeam Problem

**(Update 5/5/2023) Alternative Solution** Continue reading

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

This is a straight-forward problem from *Five Hundred Mathematical Challenges*.

“A boat has sprung a leak. Water is coming in at a uniform rate and some has already accumulated when the leak is detected. At this point, 12 men of equal skill can pump the boat dry in 3 hours, while 5 men require 10 hours. How many men are needed to pump it dry in 2 hours?”

See the Bailing Water Problem for solution.

This is a fun problem from the 1949 *Eureka* magazine.

“The following problems were set at the Archimedeans’ 1949 Problems Drive. Competitors were allowed five minutes for each question. [This is problem #9.]

A pillar is in the form of a truncated right circular cone. The diameter at the top is 1 ft., at the bottom it is 2 ft. The slant height is 15 ft. A streamer is wound exactly five times round the pillar starting at the top and ending at the bottom. What is the shortest length the streamer can have?”

See the Pillar Wrapping Problem for solution.