Frightful Fraction Minimization

Algebra Level 4

Given non-zero real values of $a,b$ and $c$ such that their sum is $x$ , we denote $f(x)$ as the minimum value of $3 + \frac{a+b}c+ \frac{b+c}a+\frac{c+a}b$ . If $f(x) = 9$ for all integers $-10\le x \le 10$ except at $x=P$ , find the value of $P$ .

The answer is 0.

1 solution

Aruna Uthayakumar
Jun 19, 2015

If x=0, a+b+c=0 and this implies a+b=-c , b+c=-a , c+a=-b

Now

f(x)= 3 + (a+b)/c + (b+c)/a + (c+a)(b) 
= 3  + (-c/c) + (-a/a) + (-b/b) 
= 3 + (-1) + (-1) + (-1)
= 3 - 3
= 0

Hence, f(x) = 0 for x=0

Frightful Fraction Minimization

The answer is 0.

1 solution

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