BdMO National Higher Secondary 2019/7

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samiul_samin
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BdMO National Higher Secondary 2019/7

Unread post by samiul_samin » Mon Mar 04, 2019 9:40 am

Given three cocentric circles $\omega_1$,$\omega_2$,$\omega_3$ with radius $r_1,r_2,r_3$ such that $r_1+r_3\geq {2r_2}$.Constrat a line that intersects $\omega_1$,$\omega_2$,$\omega_3$ at $A,B,C$ respectively such that $AB=BC$.

samiul_samin
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Re: BdMO National Higher Secondary 2019/7

Unread post by samiul_samin » Thu Mar 14, 2019 11:01 am

Diagram
2019-03-14 08.51.48-1-3.png
This diagram is for \[r_1+r_3=2r_2\]

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math_hunter
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Re: BdMO National Higher Secondary 2019/7

Unread post by math_hunter » Fri Mar 15, 2019 7:47 pm

Have you got the solution of this problem???

soyeb pervez jim
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Re: BdMO National Higher Secondary 2019/7

Unread post by soyeb pervez jim » Sat Mar 16, 2019 12:51 am

May be not for all cases $AB=BC$ can't be drawn even if $r_1+r_3\geq 2r_2$. I think $2r_{2}^{2} \geq r_{1}^{2}+r_{3}^{2}$ also must hold

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Anindya Biswas
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Re: BdMO National Higher Secondary 2019/7

Unread post by Anindya Biswas » Sat Feb 06, 2021 9:07 pm

This problem is just asking for a triangle whose $2$ sides and median on the third side is given.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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Anindya Biswas
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Re: BdMO National Higher Secondary 2019/7

Unread post by Anindya Biswas » Sun Feb 07, 2021 1:06 am

samiul_samin wrote:
Mon Mar 04, 2019 9:40 am
Given three cocentric circles $\omega_1$,$\omega_2$,$\omega_3$ with radius $r_1,r_2,r_3$ such that $r_1+r_3\geq {2r_2}$.Constrat a line that intersects $\omega_1$,$\omega_2$,$\omega_3$ at $A,B,C$ respectively such that $AB=BC$.
Draw a parallelogram with side lengths $r_1$ and $r_3$ and with a diagonal of length $2r_2$. The other diagonal of this parallelogram is the required segment.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

hriditapaul
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Re: BdMO National Higher Secondary 2019/7

Unread post by hriditapaul » Sun Feb 28, 2021 2:21 pm

Can you show the proof?

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Anindya Biswas
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Re: BdMO National Higher Secondary 2019/7

Unread post by Anindya Biswas » Mon Mar 01, 2021 1:13 pm

hriditapaul wrote:
Sun Feb 28, 2021 2:21 pm
Can you show the proof?
Diagonals of parallelogram bisects each other.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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