Question 35
Determine the smallest possible value of real number $k$ such that the inequality
$$
\left(\frac{2a}{a-b}\right)^2 + \left(\frac{2b}{b-c}\right)^2 + \left(\frac{2c}{c-a}\right)^2 + k \ge 4 \left(\frac{2a}{a-b} + \frac{2b}{b-c} + \frac{2c}{c-a}\right)
$$
holds for all real numbers $a, b, c$.
$$
\left(\frac{2a}{a-b}\right)^2 + \left(\frac{2b}{b-c}\right)^2 + \left(\frac{2c}{c-a}\right)^2 + k \ge 4 \left(\frac{2a}{a-b} + \frac{2b}{b-c} + \frac{2c}{c-a}\right)
$$
holds for all real numbers $a, b, c$.