2007 National, Junior 11
- Fahim Shahriar
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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
Notre Dame College
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Re: 2007 National, Junior 11
It is also in mathematical quickies and গনিতের মজা মজার গণিত ।
the solution is like this.
\[a^{2}+b^{2}+c^{2}=ab+bc+ca\]
\[2a^{2}+2b^{2}+2c^{2}=2ab+2bc+2ca\]
\[(a-b)^{2}+(b-c)^{2}+(c-a)^{2}=0\]
\[a=b=c\]
the solution is like this.
\[a^{2}+b^{2}+c^{2}=ab+bc+ca\]
\[2a^{2}+2b^{2}+2c^{2}=2ab+2bc+2ca\]
\[(a-b)^{2}+(b-c)^{2}+(c-a)^{2}=0\]
\[a=b=c\]
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
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Re: 2007 National, Junior 11
This problem is also a problem of BdMO National 2008 Junior!Fahim Shahriar wrote: ↑Sun Jan 27, 2013 1:06 amIf $a,b,c$ are the sides of a triangle such that $a^2+b^2+c^2=ab+bc+ca$. Prove that the triangle is equilateral.
Question repeat!