# Distance to Flag Problem

The following puzzle is from the Irishman Owen O’Shea.

“The figure shows the location of three flags [at A, B, and C] in one of the fields on a neighbor’s farm.  The angle ABC is a right angle.  Flag A is 40 yards from Flag B.  Flag B is 120 yards from flag C.  Thus, if one was to walk from A to B and then on to C, one would walk a total of 160 yards.

Now there is a point, marked by flag D, [directly] to the left of flag A.  Curiously, if one were to walk from flag A to flag D and then diagonally across to flag C, one would walk a total distance of 160 yards.

The question for our puzzlers is this: how far is it from flag D to flag A?”

This problem has a simple solution.  But it also suggests a more advanced alternative approach.

See the Distance to Flag Problem for a solution.

# Elliptic Circles

Here is another UKMT Senior Challenge problem for 2017.

“The diagram shows a square PQRS with edges of length 1, and four arcs, each of which is a quarter of a circle. Arc TRU has centre P; arc VPW has centre R; arc UV has centre S; and arc WT has centre Q.

What is the length of the perimeter of the shaded region?

A_6___B_(2√2 – 1)π___C_(√2 – 1/2 ___D_2___E_(3√2 – 2)π”

See Elliptic Circles for a solution.

# Pythagorean Parabola Puzzle

Since the changes in Twitter (now X), I have not been able to see the posts, not being a subscriber.  But I noticed poking around that some twitter accounts were still viewable.  However, like some demented aging octogenarian they had lost track of time, that is, instead of being sorted with the most recent post first, they showed a random scattering of posts from different times.  So a current post could be right next to one several years ago.  That is what I discovered with the now defunct MathsMonday site.  I found a post from 10 May 2021 that I had not seen before, namely,

“The points A and B are on the curve y = x2 such that AOB is a right angle.  What points A and B will give the smallest possible area for the triangle AOB?”

See the Pythagorean Parabola Puzzle for solution.

(Update 9/1/2023) Elegant Alternative Solution by Oscar Rojas

# 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)”

# Missing Interval Puzzle

Henk Reuling posted a deceptively simple-looking geometric problem on Twitter.

“I found this old one cleaning up my ‘downloads’ [source unknown] I haven’t been able to solve it, so help!

According to the given information in the figure, what is the length of the missing interval on the diagonal of the square?”

See the Missing Interval Puzzle for solutions.

# An Intercept Problem

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 for solutions.

# Challenging Triangle 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 for solutions.

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 for solutions.

# Twisting Beam Problem

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 for solutions.