This is another problem from the indefatigable Presh Talwalkar.

_ _____Hard Geometry Problem

“In triangle ABC above, angle A is bisected into two 60° angles. If AD = 100, and AB = 2(AC), what is the length of BC?”

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This is another problem from the indefatigable Presh Talwalkar.

_ _____Hard Geometry Problem

“In triangle ABC above, angle A is bisected into two 60° angles. If AD = 100, and AB = 2(AC), what is the length of BC?”

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

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

Presh Talwalkar had another interesting problem.

“A triangle is drawn inside a square with sides 4, 3, and 5, as shown. What is the length of the square’s side?”

The problem looks simple at first, but it takes some care to avoid some hideous quartic equations.

This is another intriguing problem from Presh Talwalkar.

“A car travels 75 miles per hour (mph) downhill, 60 mph on flat roads, and 50 mph uphill. It takes 3 hours to go from town A to B, and it takes 3 hours and 30 minutes for return journey by the same route. What is the distance in miles between towns A and B?”

See the Impossible Car Riddle

Another challenging problem from Presh Talwalkar. I certainly could not have solved it on a timed test at the age of 16.

“**One Of The Hardest GCSE Test Questions – How To Solve The Cosine Problem**

Construct a hexagon from two congruent parallelograms as shown. Given BP = BQ = 10, solve for the cosine of PBQ in terms of x.

This comes from the 2017 GCSE exam, and it confused many people. I received many requests to solve this problem, and I thank Tom, Ben, and James for suggesting it to me.”

See the Parallelogram Cosine Problem

Here is a problem from the famous (infamous?) Putnam exam, presented by Presh Talwalkar. Needless to say, I did not solve it in 30 minutes—but at least I solved it (after making a blizzard of arithmetic and trigonometric errors).

“Today’s problem is from the 1978 test, problem B1 (the easiest of the second set of problems). A convex octagon inscribed in a circle has four consecutive sides of length 3 and four consecutive sides of length 2. Find the area of the octagon.”

My solution is horribly pedestrian and fraught with numerous chances for arithmetic mistakes to derail it, which happened in spades. As I suspected, there was an elegant, “easy” solution (as demonstrated by Talwalkar)—*once you thought of it!* Again, this is like a Coffin Problem. See the Putnam Octagon Problem.

I was astonished that this problem was suitable for 8th graders. First of all the formula for the volume of a cone is one of the least-remembered of formulas, and I certainly never remember it. So my only viable approach was calculus, which is probably not a suitable solution for an 8th grader.

Presh Talwalkar: “This was sent to me as a competition problem for 8th graders, so it would be a challenge problem for students aged 12 to 13. When a conical bottle rests on its flat base, the water in the bottle is 8 cm from its vertex. When the same conical bottle is turned upside down, the water level is 2 cm from its base. What is the height of the bottle? (Note “conical” refers to a right circular cone as is common usage.) I at first thought this problem was impossible. But it actually can be solved. Give it a try and then watch the video for a solution.”

See the Conical Bottle Problem.

Setting aside my chagrin that the following problem was given to pre-university students, I initially found the problem to be among the daunting ones that offer little information for a solution. It also was a bit “inelegant” to my way of thinking, since it involved considering some separate cases. Still, the end result turned out to be unique and satisfying (Talwalkar’s Note 2 was essential for a unique solution, since the problem as stated was ambiguous).

“Kshitij from India sent me this problem from the 1994 India Regional Mathematics Olympiad.

‘**A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is 15000. What are the page numbers on the torn leaf?**’**Note 1**: a ‘leaf’ means a single sheet of paper.**Note 2**: the quoted problem is actual wording from the competition. But let me add an important detail: the book is numbered in the usual sequential way starting with the first page as page 1.” See the Missing Pages Puzzle.

This problem posted by Presh Talwalkar offers a variety of solutions, but I didn’t quite see my favorite approach for such problems. So I thought I would add it to the mix.

“Thanks to Nikhil Patro from India for suggesting this! What is the sum of the corner angles in a regular 5-sided star? What is a + b + c + d + e = ? Here’s a bonus problem: if the star is not regular, what is a + b + c + d + e = ?”