Solve 1/a + 1/b + 1/c When a+b+c+2=abc

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The problem involves finding the smallest value of 1/a + 1/b + 1/c given the constraint a + b + c + 2 = abc, where a, b, and c are positive real numbers. A proposed solution is to set a, b, and c all equal to 2, resulting in a minimum value of 1.5. However, there is skepticism about whether this is the absolute minimum, as alternative methods like Lagrange multipliers and AM-GM inequality could yield different results. An example solution is also provided with a combination of values (1, 3, 3), suggesting that the symmetric solution may not always be optimal. Further exploration is encouraged to determine if a lower value than 1.5 exists.
Hobold
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A friend of mine gave me this problem to solve because he couldn't and I've been stuck in it for some time and also can't solve.

If a, b, c are positive real numbers and a + b + c + 2 = abc, find the smallest value of 1/a + 1/b + 1/c.

PS: I'm guessing this is the right section, 'cause this is not a homework.
 
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The only thing I can see that fits for the first part is a, b, and c are all 2.
2+2+2+2=8 and 2x2x2=8. So the last part would be 1.5
smallest value of 1/2 + 1/2 + 1/2=1.5
Unless I am reading something wrong.
 
physi-man said:
The only thing I can see that fits for the first part is a, b, and c are all 2.
2+2+2+2=8 and 2x2x2=8. So the last part would be 1.5
smallest value of 1/2 + 1/2 + 1/2=1.5
Unless I am reading something wrong.

Yep, I got this result as well, but there's nothing that proves that there isn't a combination of numbers which will make 1/a + 1/b + 1/c < 1.5
 
There are two ways that you can do this:
1) Use Lagrange multipliers. You're trying to minimize a function given a constraint so it would work, but might be ugly.

2) Square something crazy. This looks like it's probably a high school math competition problem, which means that 1 is an inappropriate solution. Instead you probably need to do something like square something, which you know is greater than or equal to zero, or maybe use AM/GM. I don't see exactly what it is you would be using but I'll think about it

As for other solutions:

a(1-bc)+b+c+2=0

(-2-b-c)/(1-bc)=a. Now pick values of b and c making this positive. For example if b=c=3, we get (-2-6)/(1-9)=-8/-8=1. So (1,3,3) is also a solution

Of course this isn't a better solution, but you never know if there is one. The symmetric solution is usually optimal when the problem is symmetric.
 
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