This is a straight-forward problem by Geoffrey Mott-Smith from 1954.
“Three tangent circles of equal radius r are drawn, all centers being on the line OE. From O, the outer intersection of this axis with the left-hand circle, line OD is drawn tangent to the right-hand circle. What is the length, in terms of r, of AB, the segment of this tangent which forms a chord in the middle circle?”
See An Intercept Problem
This is a challenging problem from the 1986 American Invitational Mathematics Exam (AIME).
“Let triangle ABC be a right triangle in the xy-plane with a right angle at C. Given that the length of the hypotenuse AB is 60, and that the medians through A and B lie along the lines y = x + 3 and y = 2x + 4 respectively, find the area of triangle ABC.”
I have included a sketch to indicate that the sides of the right triangle are not parallel to the Cartesian coordinate axes.
The AIME (American Invitational Mathematics Examination) is an intermediate examination between the American Mathematics Competitions AMC 10 or AMC 12 and the USAMO (United States of America Mathematical Olympiad). All students who took the AMC 12 (high school 12th grade) and achieved a score of 100 or more out of a possible 150 or were in the top 5% are invited to take the AIME. All students who took the AMC 10 (high school 10th grade and below) and had a score of 120 or more out of a possible 150, or were in the top 2.5% also qualify for the AIME.
See the Challenging Triangle Problem.
This is a thoughtful puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings.
“A ladder is leaning against a wall. The base of the ladder starts sliding away from the wall, with the top of the ladder sliding down the wall. As the ladder slides, you watch the red point in the middle of the ladder. What figure does the red point trace? What about other points on the ladder?”
See the Ladder Locus Puzzle
Here is a slightly different kind of problem from the Polish Mathematical Olympiads.
“106. A beam of length a is suspended horizontally by its ends by means of two parallel ropes of lengths b. We twist the beam through an angle φ about the vertical axis passing through the centre of the beam. How far will the beam rise?”
See the Twisting Beam 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.
Here is another good problem from Five Hundred Mathematical Challenges:
“Problem 100. A hexagon inscribed in a circle has three consecutive sides of length a and three consecutive sides of length b. Determine the radius of the circle.”
This problem made me think of the Putnam Octagon Problem. Again my approach might be considered a bit pedestrian. 500 Math Challenges had a slightly slicker solution.
See the Lop-sided Hexagon Problem
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
Having fallen under the spell of Catriona Shearer’s geometric puzzles again, I thought I would present the latest group assembled by Ben Orlin, which he dubs “Felt Tip Geometry”, along with a bonus of two more recent ones that caught my fancy as being fine examples of Shearer’s laconic style. Orlin added his own names to the four he assembled and I added names to my two, again ordered from easier to harder.
See Geometric Puzzle Munificence.
(Update 4/16/2020) Ben Orlin has another set of Catriona Shearer puzzles 11 Geometry Puzzles That Drive Mathematicians to Madness which I will leave you to see and enjoy. But I wanted to emphasize some observations he included that I think are spot on. 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