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)”

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

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.

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

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

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.

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

Here is yet another (belated) collection of beautiful geometric problems from Catriona Agg (née Shearer).

This is a straight-forward problem from the Scottish Mathematical Council (SMC) Senior Mathematics Challenge.

“A circle has radius 1 cm and *AB* is a diameter. Two circular arcs of equal radius are drawn with centres *A* and *B*. These arcs meet on the circle as shown. Calculate the shaded area.”

There are several possible approaches and the SMC offers two examples.

See the Moon Quarters Problem

Here is a familiar puzzle from the *Mathigon* Puzzle Calendars for 2021.

“Given a line and two points A and B, which point P on the line forms the largest angle APB?”

See the Max Angle Puzzle

An excellent application of the solution to this puzzle can be found at Numberphile, where Ben Sparks explains an optimal rugby goal-kicking strategy.

**(Update 3/23/2023) Solution Construction**