The number of ordered pairs $(x, y)$ of positive integers satisfying $2^x+3^y=5^{x y}$ is

The polynomial equation $x^3-3 a x^2+\left(27 a^2+9\right) x+2016=0$ has

If $x$ is real, the function $\frac{{(x - a)(x - b)}}{{(x - c)}}$ will assume all real values, provided

Number of integers satisfying inequality, $\sqrt {{{\log }_3}(x) - 1}  + \frac{{\frac{1}{2}{{\log }_3}\,{x^3}}}{{{{\log }_3}\,\frac{1}{3}}} + 2 > 0$ is

Let $a, b, c$ be non-zero real numbers such that $a+b+c=0$, let $q=a^2+b^2+c^2$ and $r=a^4+b^4+c^4$. Then,

A real root of the equation ${\log _4}\{ {\log _2}(\sqrt {x + 8} - \sqrt x )\} = 0$ is