Is ABC Greater Than or Equal to A+B+C?

[MathematicalPhysicist]

and a+b+c>0 and abc>0
proove that abc>=a+b+c

can someone please help?

 

[bogdan]

I'm afraid it's not correct...
if a=0.1,b=0.1,c=0.1 then abc=0.001 and a+b+c=0.3, 0.001<0.3...
[?]

 

[MathematicalPhysicist]

****, damn.
btw, bogdan, did these numbers were the first to pop up your head?

 

[STAii]

It is not really hard to get those numbers.
All you need to know is think for a moment ..
If you have a positive number (call it X), and you multiply it but another positive number (say Y), you can get one of the following results :
XY > X
XY < X
XY = X
Now, if XY = X then Y = 1, obviously this is the changing point between XY > X and XY < X
Try numbers bigger than 1 for Y, say, Y = 2, then XY = 2X, therefore XY > X (remember that X is positive).
Now, try numbers smaller than 1 (and more than 0, remember Y is positive), say, Y = 0.1, then you get XY = X/10, and therefore XY < X
bogdan was trying to disproove the original inequality, logically if he gets a sinlge case that gives a wrong answer in the inquality, then the inquality is wrong.
The inequality says that abc will be bigger or equal to a+b+c,
So, to disproove this try to get smaller value of abc than expected (since this MIGHT turn the inequality wrong, by making left side smaller).
Since values smaller than 1 for a,b,c will make abc smaller and smaller, bogdan have chosen 0.1 for all of them.
I see infinite number of cases that proove the inequality wrong.
Try those :
1- a=0.01, b=0.1, c=0.1
2- a=1, b=1, c=1
3- a=1.1, b=0.0001, c=0.0002

 

[Brad_Ad23]

I wonder if maybe this was supposed to be in the domain of all the positive integers?

 

[MathematicalPhysicist]

Originally posted by Brad_Ad23
I wonder if maybe this was supposed to be in the domain of all the positive integers?

let's say it is, does it change the answer?

 

[bogdan]

Yes...for positive integers it is true...
let's say 2<=a<=b<=c...it doesn't "particularize"...
we have abc>=2*(bc)>=4*c=c+c+c+c>a+b+c...

 

[MathematicalPhysicist]

do you have proof or what you have shown was your proof?

 

[bogdan]

That's the proof...read it carefuly...
(that's not true for integers equal to 1...in specific cases...)

 

[MathematicalPhysicist]

Originally posted by bogdan
That's the proof...read it carefuly...
(that's not true for integers equal to 1...in specific cases...)

thanks.