Let $x$ and $y$ be two positive real numbers such that $x+y=1$. Then,the minimum value of $\frac{1}{x}+\frac{1}{y}$ is

The minimum value of $\frac{9 \cdot 3^{2x} + 6 \cdot 3^x + 4}{9 \cdot 3^{2x} - 6 \cdot 3^x + 4}$ is

Let $a, b, c$ be the lengths of three sides of a triangle satisfying the condition $(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0$. If the set of all possible values of $x$ is the interval $(\alpha, \beta)$,then $12(\alpha^2+\beta^2)$ is equal to.

The probability of selecting integers $a \in [-5, 30]$ such that $x^{2}+2(a+4)x-5a+64 > 0$ for all $x \in \mathbb{R}$ is:

Let $a, b, c$ and $d$ be any four real numbers. Then $a^{n} + b^{n} = c^{n} + d^{n}$ holds for any natural number $n$ if:

If $x = \frac{1}{2} \left( \sqrt{7} + \frac{1}{\sqrt{7}} \right)$,then the value of $\frac{\sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}}$ is equal to