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evarist
Riemann Hypothesis
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Posts: 368
Location: Nam Định Việt Nam
Viet Nam
Post Posted: Mar 16, 2008, 4:57 am • # 1 


Let triangle ABC and an arbitrary point M in plane. Prove or disprove that \frac {MB + MC}{a} + \frac {MC + MA}{b} + \frac {MA + MB}{c}\ge 2\sqrt {3}
I haven't solve it yet but I think the following inequality is true :\frac {MB + MC}{a} + \frac {MC + MA}{b} + \frac {MA + MB}{c}\ge 4\min\lbrace sinA,sinB,sinC\rbrace
Can anyone help me ? :)
 
 
evarist
Riemann Hypothesis
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Posts: 368
Location: Nam Định Việt Nam
Viet Nam
Post Posted: Mar 17, 2008, 3:29 am • # 2 


What about the following :\frac {MA}{b + c} + \frac {MB}{c + a} + \frac {MC}{a + b}\ge \frac {\sqrt {3}}{2}}
\frac {MA}{b + c} + \frac {MB}{c + a} + \frac {MC}{a + b}\ge min\lbrace sinA,sinB,sinC \rbrace :)
 
 
chichi
Hodge Conjecture

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China    Ecuador
Post Posted: Mar 20, 2008, 12:57 pm • # 3 


by triangular inequality we have MB + MC > BC=a etc then
\frac {MB + MC}{a} + \frac {MC + MA}{b} + \frac {MA + MB}{c}>3>\frac{\sqrt{3}}{2}

_________________
What happend to the Canmoo in Hanoi 2007?
 
 
fedja
Birch & Swinnerton Dyer

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Russian Federation    United States
Post Posted: Mar 21, 2008, 5:59 am • # 4 


Are you reading the posts diagonally, chichi?
 
 
evarist
Riemann Hypothesis
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Posts: 368
Location: Nam Định Việt Nam
Viet Nam
Post Posted: Mar 31, 2008, 12:26 am • # 5 


I haven't solve any problem yet but I proved the following inequalities :) :
\frac {HA + HB}{a + b} +\frac {HB + HC}{c + a} +\frac {HC + HA}{c + a}\ge \sqrt {3}
\frac {3}{2}\min\lbrace cotA,cotB,cotC\rbrace\le\frac {HA}{b + c}+\frac{HB}{c+a} + \frac {HC}{a + b}\le\frac {3}{2}\max\lbrac...
 
 
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