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

If the figure shows the graph of $y = ax^2 + bx + c$,then . . . . . .

If the minimum value of $f(x)=2x^2+\alpha x+8$ is the same as the maximum value of $g(x)=-3x^2-4x+\alpha^2$, then $\alpha^2=$

Both equations $x^2 + b^2 = 1 - 2bx$ and $x^2 + a^2 = 1 - 2ax$ have exactly one root each,and they share the same root. Then:

Let $\alpha, \beta$ be the roots of the equation $x^{2}-4 \lambda x+5=0$ and $\alpha, \gamma$ be the roots of the equation $x^{2}-(3 \sqrt{2}+2 \sqrt{3}) x+7+3 \lambda \sqrt{3}=0$. If $\beta+\gamma=3 \sqrt{2}$,then $(\alpha+2 \beta+\gamma)^{2}$ is equal to

If $x = \frac{1}{2} \left( \sqrt{3} + \frac{1}{\sqrt{3}} \right)$, then the value of $\frac{\sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}}$ is equal to