equal semicircles in a circle with 4 times larger diameter
Source: IV Soros Olympiad 1997-98 R3 9.12 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics
June 1, 2024
combinatoricsgeometrycombinatorial geometry
Problem Statement
One day, Professor Umzar Azum decided to fry dumplings for dinner. He took out a frying pan, opened a pack of dumplings, but suddenly thought about the question: how many dumplings could he fit in his frying pan? Measuring the sizes of the frying pan and dumplings, the professor came to the conclusion that the dumplings have the shape of a semicircle, the diameter of which is four times smaller than the diameter of the frying pan. Show how on the frying pan it is possible to place (without overlap):
a) pieces of dumplings;
b) pieces of dumplings; .
(The problem boils down to placing, without overlapping, the appropriate number of identical semicircles inside a circle with a diameter four times larger.)Note: We (the authors of the problem) do not know the answer to the question whether it is possible to place 25 semicircles in a circle with a diameter four times smaller, and even more so we do not know what the largest number of such semicircles is. We will welcome any progress in solving the problem and evaluate it accordingly.