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Discussion of Problem 1874. Football Goal

What shape is the football gate?
Posted by Nodir NAZAROV Komiljonovich 30 Jan 2014 20:17
I tried to figure it out, how 1 tree and two sticks with the length of 2 can form the area more than 4? The largest area would be rectangle. What is the trick on this problem, can anybody explain with Geometry language.

Thanks.
Re: What shape is the football gate?
Posted by Andrew Sboev [USU] 30 Jan 2014 21:48
The key point of the problem statement is
"She knew that the standard football goal was rectangular, but, being creative, she assumed that her goal could have the form of an arbitrary quadrangle."
So, it may be not a rectangle.
Re: What shape is the football gate?
Posted by Nodir NAZAROV Komiljonovich 6 Feb 2014 22:44
I tried trapezoid like:

/|
2/_|
|  |
2|__|
x

and tried to maximize the function - f(x) for total area which answer is sqrt(3) and total area is ~4.33
How to get ~4.8 for the given test?
Re: What shape is the football gate?
Posted by Aleksei Chernenkov 12 Jan 2023 15:20
Try placing sticks a and b as follows:

#
#`.  a
#  `.
#    `.
x #      \
#       \  b
#        \
#         \
=====================
y

Hint: It can be proved that the area is maximal when x = y and angle between a and b is 135 degrees.