A.M., G.M., H.M. and their Relations

If $a_1,a_2,.....a_n$ are positive real numbers whose product is a fixed number $e$, the minimum value of $a_1+a_2+a_3+.......+a_{n-1}+2a_n$ is

If $a_1, a_2, .......a_n$ are positive real numbers whose product is a fixed number $e$, the minimum value of $a_1+a_2+ .......+2a_n$ is

Let $\sin \beta$ be the $GM$ of $\sin \alpha$ and $\cos \alpha ; \tan \gamma$ be the $AM$ of $\sin \alpha$ and $\cos \alpha$.

What is the value of $\sec 2\gamma$?

Let $\sin \beta$ be the $GM$ of $\sin \alpha$ and $\cos \alpha ; \tan \gamma$ be the $AM$ of $\sin \alpha$ and $\cos \alpha$.

What is $\cos 2\beta$ equal to?

A quadratic equation is given by

$\left(3+2\sqrt{2}\right)x^2-\left(4+2\sqrt{3}\right)x+\left(8+4\sqrt{3}\right)=0$

What is the GM of the roots of the equation?

A quadratic equation is given by

$\left(3+2\sqrt{2}\right)x^2-\left(4+2\sqrt{3}\right)x+\left(8+4\sqrt{3}\right)=0$

What is the HM of the roots of the equation?

If $G$ is the geometric mean of numbers $1, 2, 2^2, 2^3,\dots , 2^{n-1}$, then what is the value of $1+2{\log }_{2}G$?

Given that $m\left(\theta \right)={\cot }^2\theta +n^2 {\tan }^2\theta +2n$, where $n$ is a fixed positive real number.

Under what condition does $m$ attain the least value?

Given that $m\left(\theta \right)={\cot }^2\theta +n^2 {\tan }^2\theta +2n$, where $n$ is a fixed positive real number.

What is the least value of $m\left(\theta \right)$?

Let $x$ and $y$ be positive real numbers. What is the smallest possible value of $\frac{16}{x}+\frac{108}{y}+xy$?

If $a,b,c$ are positive real numbers, then minimum value of $\frac{\left(a+b+c\right)\left(a^2+b^2+c^2\right)}{abc}$ is

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 number of different possible values for the sum $x+y+z,$ where $x, y, z$ are real number such that $x^4+4y^4+16z^4+64=32xyz$ is

The number of three digit numbers $\overline{abc}$ such that the arithmetic mean of $b \& c$ and the square of their geometric mean are equal is

If $x>m\; ,\; y>n\; ,\; z>\text{r}(x,\; y,\; z\; >\; 0)$ such that $\left|\begin{array}{ccc}\text{x}& \text{n}& \text{r}\\ \text{m}& \text{y}& \text{r}\\ \text{m}& \text{n}& \text{z}\end{array}\right|=0$.

 The greatest value of $\frac{\text{xyz}}{\left(\text{x}-\text{m}\right)\left(\text{y}-\text{n}\right)\left(\text{z}-\text{r}\right)}$  is 

If $x>m\; ,\; y>n\; ,\; z>\text{r}(x,\; y,\; z\; >\; 0)$ such that $\left|\begin{array}{ccc}\text{x}& \text{n}& \text{r}\\ \text{m}& \text{y}& \text{r}\\ \text{m}& \text{n}& \text{z}\end{array}\right|=0$.

The value of $\frac{\text{m}}{\text{x}-\text{m}}+\frac{\text{y}}{\text{y}-\text{n}}+\frac{\text{r}}{\text{z}-\text{r}}$  is

If $x>\mathrm{m},\mathrm{y}>\mathrm{n},\mathrm{z}>r(\mathrm{x},\mathrm{y,}\mathrm{z\; >\; 0)}$ such that $\left|\begin{array}{ccc}x& n& r\\ m& y& r\\ m& n& z\end{array}\right|=0$. The value of $\frac{x}{x-m}+\frac{y}{y-n}+\frac{z}{z-r}$, is

If $a, b, c, d$ are four positive real numbers such that $abcd = 1,$ then the minimum value of the expression $\left(1+a\right)\left(1+b\right) \left(1+c\right) \left(1+d\right)$ is equal to

Let $\underset{r=1}{\overset{16}{\Sigma }}\mathrm{s}\mathrm{i}{\mathrm{n}}^{-1}\left({\mathrm{x}}_{\mathrm{r}}\right)\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}\left({\mathrm{x}}_{\mathrm{r}}\right)={\pi }^2$, $x_r>0$ $\forall$ $r=1,2,...,16$ then the number of divisors of the sum of series $\underset{r=1}{\overset{16}{\Sigma }}{x}_r^2$ equals:

Find the maximum value for the function $y=\frac{x}{a{x}^2+b}(a,b>0)$.