2007 National, Junior 11

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Fahim Shahriar
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2007 National, Junior 11

Unread post by Fahim Shahriar » Sun Jan 27, 2013 1:06 am

If $a,b,c$ are the sides of a triangle such that $a^2+b^2+c^2=ab+bc+ca$. Prove that the triangle is equilateral.
Name: Fahim Shahriar Shakkhor
Notre Dame College

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nafistiham
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Re: 2007 National, Junior 11

Unread post by nafistiham » Sun Jan 27, 2013 1:19 pm

It is also in mathematical quickies and গনিতের মজা মজার গণিত ।
the solution is like this.

\[a^{2}+b^{2}+c^{2}=ab+bc+ca\]
\[2a^{2}+2b^{2}+2c^{2}=2ab+2bc+2ca\]
\[(a-b)^{2}+(b-c)^{2}+(c-a)^{2}=0\]
\[a=b=c\]
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
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samiul_samin
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Re: 2007 National, Junior 11

Unread post by samiul_samin » Sat Feb 23, 2019 10:16 pm

Fahim Shahriar wrote:
Sun Jan 27, 2013 1:06 am
If $a,b,c$ are the sides of a triangle such that $a^2+b^2+c^2=ab+bc+ca$. Prove that the triangle is equilateral.
This problem is also a problem of BdMO National 2008 Junior!
Question repeat!

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