An algebra problem by Paola Ramírez

Algebra Level pending

Let $a$ , $b$ , and $c$ be positive real numbers such that $a+b+c=1$ and always satify the following inequality.

$\left(\frac{1}{a}-1\right)+\left(\frac{1}{b}-1\right)+\left(\frac{1}{c}-1\right)\geq n$

Find the maximum value of $n$ .

2 solutions

Utsav Playz
Oct 2, 2019

For a, b, c to be the maximum. WLOG, Lets say a=b=c From this we get,

3a=1

a = 1/3

Hence a=b=c= 1/3 We can easily notice that this is the maximum value of a, b, and c

Putting the values in the equation, we get

(3-1) + (3-1) + (3-1) >= n

2+2+2 >= n

6 >= n

Hence, 6 is the answer.

Kaustubh Mishra
Feb 19, 2017

Just apply AM>=HM on a b and c and add -3 on both side