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nishant kumar agarwal Grade: Upto college level
`        if a,b,c are distinct positive numbers , then the expression (b+c-a)(c+a-b)(a+b-c)-abc is positive or negative........?`
7 years ago

147 Points
```										Dear nishant
let x ,y,z are three positive number
so use AM ≥GM
x+y ≥2(xy)1/2
y+z ≥2(yz)1/2
z+x ≥2(zx)1/2

so (x+y)(y+z)(z+x)≥2(xy)1/22(yz)1/22(zx)1/2
or (x+y)(y+z)(z+x)≥8(xyz)
now put x=b+c-a
y= c+a-b
z= a+b-c
or x+y =2c  , y+z =2a , z+x =2b
(2c)(2a)(2b)    ≥8(b+c-a )( c+a-b)( a+b-c)
abc      ≥(b+c-a )( c+a-b)( a+b-c)
(b+c-a )( c+a-b)( a+b-c) -abc ≤ 0

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```
7 years ago
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