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#### Pranav

##### Well-known member

- Nov 4, 2013

- 428

**Problem:**

Given that $a,b$ and $c$ are the sides of $\Delta ABC$ such that $$z=\log_{2^a+2^{-a}} \left(4(ab+bc+ca)-(a+b+c)^2\right)$$ then $z$ has a real value if and only if

A)a=b=2c

B)3a=2b=c

C)a-b=3c

D)None of these

**Attempt:**

I am not sure where to start with this kind of problem. I wrote the expression inside the logarithm as follows:

$$-(a^2+b^2+c^2-2(ab+bc+ca))$$

Since, anything inside the logarithm must be greater than zero, I have

$$a^2+b^2+c^2-2(ab+bc+ca)<0 \Rightarrow a^2+b^2+c^2<2(ab+bc+ca)$$

But $a^2+b^2+c^2 >ab+bc+ca$, I get $ab+bc+ca>0$. I doubt this is going to help.

Any help is appreciated. Thanks!