# Circles in Circles

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

“Inside a circle there are two intersecting circles. One of them touches the big circle in point A, the other in point B. Prove that if segment AB meets the smaller circles at one of their common points, then the sum of their radii equals the radius of the big circle. Is the converse true?  (A. Vesyolov)”

# Right Angles in a Square

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)”

# Rolling Wheels Puzzle

Here is another Quantum math magazine Brainteaser.

“Two wheels roll toward each other with identical angular velocity. At the moment of collision they contact each other at the same points that touched the ground before they began rolling. Could the radii of the wheels differ?”

See the Rolling Wheels Puzzle for solution.

# Wisdom of Old

Here is another Brainteaser from the Quantum magazine.

“King Arthur ordered a pattern for his quarter-circle shield. He wanted it to be painted in three colors: yellow, the color of kindness; red, the color of courage: and blue the color of wisdom. When the artist brought in his work, the king’s armor-bearer said there was more courage than wisdom on the shield. But the artist managed to prove that the proportions of both virtues were equal. Can you tell how? (A. Savin)”

This is another relatively simple problem, though it may look a bit daunting at first.

# More Squares in Semicircle

Here is another elegant Quantum math magazine Brainteaser problem.

“Two squares are inscribed in a semicircle as shown in the figure at left. Prove that the area of the big square is four times that of the small one.”

# Meeting on the Bridge

Here is another Brainteaser from the Quantum math magazine.

“Nick left Nicktown at 10:18 A.M. and arrived at Georgetown at 1:30 P.M., walking at a constant speed. On the same day, George left Georgetown at 9:00 A.M. and arrived at Nicktown at 11:40 A.M., walking at a constant speed along the same road. The road crosses a wide river. Nick and George arrived at the bridge simultaneously, each from his side of the river. Nick left the bridge 1 minute later than George. When did they arrive at the bridge?”

See Meeting on the Bridge for solutions.

# Playing with Triangles

Here is another elegant Quantum math magazine Brainteaser from the imaginative V. Proizvolov.

“Two isosceles right triangles are placed one on the other so that the vertices of each of their right angles lie on the hypotenuse of the other triangle (see the figure at left). Their other four vertices form a quadrilateral. Prove that its area is divided in half by the segment joining the right angles. (V. Proizvolov)”

# River Traffic Problem

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 for solutions.

# Circumscribed House 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 for solutions.

# Walking Banker 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 for solutions.