(a^2-bc)/sqrt(...)<=0

dduclam

1. Let $a,b,c$ be sidelengths of a triangle. Prove that
$\frac{a^2-bc}{\sqrt{4a^2+bc}}+\frac{b^2-ca}{\sqrt{4b^2+ca}}+\frac{c^2-ab}{\sqrt{4c^2+ab}}\le0$
When equality occurs?

**Posted:** Tue Oct 06, 2009 7:26 pm

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Duong Duc Lam

ylg · New Member Offline Joined: 28 Aug 2009 Posts: 6

Clearly when a=b=c the inequality holds
and by SOS we can prove the inequality,i think

**Posted:** Tue Oct 06, 2009 8:42 pm

arqady

**Posted:** Tue Oct 06, 2009 9:19 pm

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Michael Rozenberg

dduclam

Nice, Arqady. Some same form problems are

2. Let $a,b,c$ be sidelengths of a triangle. Prove that
$\frac{a^{2}-bc}{\sqrt{15a^{2}+(b+c)^2}}+\frac{b^{2}-ca}{\sqrt{15b^{2}+(c+a)^2}}+\frac{c^{2}-ab}{\sqrt{15c^{2}+(a+b)^2}}\le0$

3. Let $a,b,c$ be sidelengths of a triangle. Prove that
$\frac{a^{2}-bc}{\sqrt{7a^{2}+b^2+c^2}}+\frac{b^{2}-ca}{\sqrt{7b^{2}+c^2+a^2}}+\frac{c^{2}-ab}{\sqrt{7c^{2}+a^2+b^2}}\le0$

**Posted:** Sun Nov 01, 2009 10:04 am

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Duong Duc Lam

dduclam

**Posted:** Wed Nov 04, 2009 5:40 pm

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Duong Duc Lam