Try this beautiful problem from Pre-RMO, 2019 based on Geometry of plane figures.
From a square with sides of length 5, triangular pieces from the four corners are removed to form a regular octagon. Find the area removed to the nearest integer.
Area of Plane figures
Algebra
Approximation in number theory
Answer: 4.
Pre-RMO, 2019
Geometry Revisited by Coxeter .
A figure is taken such that side of triangle is x and sides of octagon are $x\sqrt 2$ and $5-2x$
Here $5-2x$=$x\sqrt 2$ then x=$\frac{5}{2+\sqrt 2}$
Area removed =2x^2=$\frac{2(25)}{2(1+2^\frac{1}{2})}$ which is nearly 4.3 then the nearest integer is 4
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