Here is another problem from Presh Talwalkar which he says is adapted from India’s Civil Services Exam.
“There are three runners X, Y, and Z. Each runs with a different uniform speed in a 1000 meters race. If X gives Y a start of 50 meters, they will finish the race at the same time. If X gives Z a start of 69 meters, they will finish the race at the same time. Suppose Y and Z are in a [1000 meter] race. How much of a start should Y give to Z so they would finish the race at the same time?”
Even though Talwalkar’s original graphic showed all the runners in a 1000 meter race, it was not immediately clear to me from the wording that the race between Y and Z was also 1000 meters. But that was the case, so I made it explicit.
See the Three Runners Puzzle
Here is yet another problem from Presh Talwalkar. This one is rather elegant in its simplicity of statement and answer.
“Solve For The Angle – Viral Puzzle
I thank Barry and also Akshay Dhivare from India for suggesting this problem! This puzzle is popular on social media. What is the measure of the angle denoted by a “?” in the following diagram? You have to solve it using elementary geometry (no trigonometry or other methods). It’s harder than it looks. I admit I did not solve it. Can you figure it out?”
See the Shy Angle Problem.
Yet another interesting problem from Presh Talwalkar.
“Two side-by-side squares are inscribed in a semicircle. If the semicircle has a radius of 10, can you solve for the total area of the two squares? If no, demonstrate why not. If yes, calculate the answer.”
This puzzle shares the characteristics of all good problems where the information provided seems insufficient.
See the Sum of Squares Puzzle.
This is another challenging puzzle from Presh Talwalkar that seems difficult to know where to start.
“Given the figure shown at left, what is the value of x?”
See the Chord Progression Puzzle
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?”
See Hard Geometric Problem
(Update 7/18/2020, 7/20/2020) Alternative Solution Continue reading
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.
See Tipsy 3-4-5 Triangle
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