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Prove that 2/(b+c)+2/(c+a)+2/(a+b)<<1/a+...

Prove that `2/(b+c)+2/(c+a)+2/(a+b)<<1/a+1/b+1/c ,w h e r ea ,b ,c>>0.`

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To prove that \[ \frac{2}{b+c} + \frac{2}{c+a} + \frac{2}{a+b} < \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \] where \( a, b, c > 0 \), we will use the concept of the Arithmetic Mean (AM) and Harmonic Mean (HM). ...
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CENGAGE ENGLISH-INEQUALITIES INVOLVING MEANS -Illustration
  1. Prove that (a b+x y)(a x+b y)>4a b x y(a , b ,x ,y >0)dot

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  2. Prove that b^2c^2+c^2a^2+a^2b^2gtabc(a+b+c), where a,b,c gt 0 .

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  3. Prove that 2/(b+c)+2/(c+a)+2/(a+b)<<1/a+1/b+1/c ,w h e r ea ,b ,c>>0.

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  4. If a ,b , a n dc are distinct positive real numbers such that a+b+c=1,...

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  5. Find the minimum value of 4^sin^(2x)+4^cos^(2x) .

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  6. If (log)2(a+b)+(log)2(c+d)geq4. Then find the minimum value of the exp...

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  7. Find all real solutions to 2^x+x^2=2-(1)/(2^x).

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  8. Find all positive real solutions to 4x+(18)/(y)=14,2y+(9)/(z)=15,9z+(1...

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  9. Let A;G;H be the arithmetic; geometric and harmonic means between thre...

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  10. about to only mathematics

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  11. If a+b+c=1, then prove that 8/(27a b c)>{1/a-1}{1/b-1}{1/c-1}> 8.

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  12. If y z+z x+x y=12 ,w h e r ex ,y ,z are positive values, find the grea...

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  13. If a ,b ,c are positive, then prove that a//(b+c)+b//(c+a)+c//(a+b)geq...

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  14. Prove that 2^n >1+nsqrt(2^(n-1)),AAn >2 where n is a positive integer.

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  15. If S+a1+a2+a3++an ,a1 in R^+ for i=1ton , then prove that S/(S-a1)+S/...

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  16. If a1+a2+a3+......+an=1 AA ai > 0, i=1,2,3,......,n, then find the ma...

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  17. If a , b , c , are positive real numbers, then prove that (2004, 4M) {...

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  18. Prove that (sec^4alpha)/(tan^2beta)+(sec^4beta)/(tan^2alpha)ge8. If ea...

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  19. Prove that [(x^2+y^2+z^2)/(x+y+z)]^(x+y+z)> x^x y^y z^z >[(x+y+z)/3]^(...

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  20. Prove that 1^1xx2^2xx3^3xxxxu nlt=[(2n+1)//3]n(n+1)//2,n in Ndot

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