Here is another elegant Quantum math magazine Brainteaser problem.
“A raft and a motorboat set out downstream from a point A on the riverbank. At the same moment a second motorboat of the same type sets out from point B to meet them. When the first motorboat arrives at B, will the raft (floating with the current) be closer to point A or to the second motorboat? (G. Galperin)”
See the River Traffic Problem
Here is another problem from the “Brainteasers” section of the Quantum magazine.
“Side AE of pentagon ABCDE equals its diagonal BD. All the other sides of this pentagon are equal to 1. What is the radius of the circle passing through points A, C, and E?”
See the Circumscribed House Problem
Here is another Brainteaser from the Quantum magazine.
“Mr. R. A. Scall, president of the Pyramid Bank, lives in a suburb rather far from his office. Every weekday a car from the bank comes to his house, always at the same time, so that he arrives at work precisely when the bank opens. One morning his driver called very early to tell him he would probably be late because of mechanical problems. So Mr. Scall left home one hour early and started walking to his office. The driver managed to fix the car quickly, however, and left the garage on time. He met the banker on the road and brought him to the bank. They arrived 20 minutes earlier than usual. How much time did Mr. Scall walk? (The car’s speed is constant, and the time needed to turn around is zero.) (I. Sharygin)”
I struggled with some of the ambiguities in the problem and made my own assumptions. But it turned out there was a reason they were ambiguous.
See the Walking Banker Problem
This is a fairly straight-forward Brainteaser from the Quantum magazine.
“A man is filling two tanks with water using two hoses. The first hose delivers water at the rate of 2.9 liters per minute, the second at a rate of 8.7 liters per minute. When the smaller tank is half full, he switches hoses. He keeps filling the tanks, and they both fill up completely at the same moment. What is the volume of the larger tank if the volume of the smaller tank is 12.6 liters?”
See The Hose Knows
This is another Brainteaser from the Quantum math magazine .
“How can a polygonal line BDEFG be drawn in a triangle ABC so that the five triangles obtained have the same area?”
I found this problem rather challenging, especially when I first tried to solve it analytically (using hyperbolas). Eventually I arrived at a procedure that would accomplish the result. (revised)
See the Equitable Slice Problem (revised)
(Update 9/22/2021) I goofed. I erroneously and foolishly thought Quantum had not solved the problem. Upon a closer reading I see what they were getting at and revised the posting.
Here is a problem from the Quantum magazine, only this time from the “Challenges” section (these are expected to be a bit more difficult than the Brainteasers).
“Three circles with the same radius r all pass through a point H. Prove that the circle passing through the points where the pairs of circles intersect (that is, points A, B, and C) also has the same radius r.”
Indeed, I found this quite challenging. It took me several weeks to work out my approach and details.
See Three Equal Circles
This is a nifty little problem from the Quantum math magazine.
“Two ants stand at opposite corners of a 1-meter square. A barrier was placed between them in the form of half a 1-meter square attached along the diagonal of the first square, as shown in the picture. One ant wants to walk to the other. How long is the shortest path?”
See the Barrier Minimal Path Problem
This is another delightful Brainteaser from the Quantum math magazine.
“All the vertices of a polygonal line ABCDE lie on a circumference (see the figure), and the angles at the vertices B, C, and D are each 45°.
Prove that the area of the blue part of the circle is equal to the area of the yellow part. (V. Proizvolov)”
I especially liked this problem since I was able to find a solution different from the one given by Quantum. Who knows how many other variations there might be.
See the Circle-Halving Zigzag Problem
This is a nice Brainteaser from the Quantum math magazine.
“Line segment MN is the projection of a circle inscribed in a right triangle ABC onto its hypotenuse AB. Prove that angle MCN is 45°.”
See the Circle Projection Problem.
Here is another problem from the Quantum magazine, only this time from the “Challenges” section (these are expected to be a bit more difficult than the Brainteasers).
“A quadrangle is inscribed in a parallelogram whose area is twice that of the quadrangle. Prove that at least one of the quadrangle’s diagonals is parallel to one of the parallelogram’s sides. (E. Sallinen)”
See Quadrangle in Parallelogram