## 2007 National, Junior 11

For students of class 6-8 (age 12 to 14)
Fahim Shahriar
Posts: 138
Joined: Sun Dec 18, 2011 12:53 pm

### 2007 National, Junior 11

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

nafistiham
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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.

$a^{2}+b^{2}+c^{2}=ab+bc+ca$
$2a^{2}+2b^{2}+2c^{2}=2ab+2bc+2ca$
$(a-b)^{2}+(b-c)^{2}+(c-a)^{2}=0$
$a=b=c$
$\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0$
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.

samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### Re: 2007 National, Junior 11

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!