## BdMO National Higher Secondary 2020 P3

Mursalin
Posts:68
Joined:Thu Aug 22, 2013 9:11 pm
BdMO National Higher Secondary 2020 P3
$$R$$ হলো এমন সব আয়তের সেট যাদের কেন্দ্র মূলবিন্দুতে এবং পরিসীমা $$1$$ (একটা আয়তের কেন্দ্র হলো তার কর্ণদুটোর ছেদবিন্দু)। $$S$$ হলো এমন একটা ক্ষেত্র যার ভিতরে $$R$$-এর সবগুলো আয়তই আছে। $$S$$-এর সর্বনিম্ন সম্ভাব্য ক্ষেত্রফলকে $$\pi a$$ আকারে লেখা যায় যেখানে $$a$$ একটা বাস্তব সংখ্যা। $$\frac{1}{a}$$-এর মান বের করো।

Let $$R$$ be the set of all rectangles centered at the origin and with perimeter $$1$$ (the center of a rectangle is the intersection point of its two diagonals). Let $$S$$ be a region that contains all of the rectangles in $$R$$ (a region $$A$$ contains another region $$B$$ if $$B$$ is completely inside of $$A$$). The minimum possible area of $$S$$ has the form $$\pi a$$, where $$a$$ is a real number. Find $$\frac{1}{a}$$.
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Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm

### Re: BdMO National Higher Secondary 2020 P3

The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Naeem588
Posts:9
Joined:Sat Apr 03, 2021 1:41 am

### Re: BdMO National Higher Secondary 2020 P3

But here S doesn't contain all rectangles.
Like it doesn't contain the rectangles of base > 0.25
It only contains all possible squares.

Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm
Ok we can assume that we have a rectangle with length $\frac{1}{2}$ and breadth $0$. The circle that comprises all those rectangles has an area of $\frac{1}{16} \pi$. So, $\frac{1}{a} = 16$. No other rectangle with the given criteria can be outisde this circle.