This week's maths problem

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Problem of the week

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On Friday of each week we will post a new maths challenge.
You may submit your solution by the following Thursday.
Each problem contains two different levels of difficulty. You will be awarded from 2 to 12 points for a full answer.
Each series contains 12 problems before the stage winner is determined.
Your score will be published here.

In each series we will give away 3 books as prices. The prices will be drawn by lot among the best ten participants of the series. The books are kindly sponsored by Buchdienst Rattei of Chemnitz.

Suggestions for problems are welcome.

Deadline is the 25th. June 2026

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Series 72

problem 864

Dürer letter K

“Hello Grandad, if I’m not mistaken, this letter is the second-to-last one we need to construct and calculate,” said Maria. “That’s absolutely right!” admitted Grandad.
The smallest circle has a radius of a/30. The three large circles have a radius of a/5 and the other six circles have a radius of a/10. But first things first:
We start with a square ABCD with a side length of a (here a = 10 cm). Points E and F are the midpoints of the corresponding sides of the square. The left-hand part of the letter is a/10 wide and is touched by four medium-sized circles. This leads to the blue problem:

What are the circumference and area of the left-hand part of the letter, shown in red? 6 blue points
Now we continue with the construction.
Point S is easy to identify. Point P divides line segment AB in a ratio of 3 : 1. The circle, whose centre lies on line segment SP, ensures that the width of the diagonal bar at the bottom right is exactly a/10. This line parallel to SP leads to points Q and R. (First, the circle around F and the line parallel to AB must be drawn, on which point R is then to lie.) Now the large circle at the top right is drawn. This touches the sides of the square. A tangent to this circle is constructed from point R. The very small circle at the top right is used to construct the thickness of the slanted bar leading upwards. This is a/30 thick.
Quickly add the curve at the bottom right. First, draw a circle around point B. Place a point M on this circle and draw the circle around M. The position of M should be chosen so that the circle touches the line segment PQ as closely as possible. (Try it out if you don’t want to do the maths.)
Now the letter is complete:

What is the perimeter of the quadrilateral PQRS? 6 red points

Deadline for solution is the 25th. June 2026.

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