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

If ${\log }_{10}x+{\log }_{10}y\ge 2$, then the smallest possible value of $x+y$ is

Let $A\mathit{,}\mathit{ }G$ and $H$ be the arithmetic mean, geometric mean and harmonic mean, respectively of two distinct positive real numbers. If $\alpha$ is the smallest of the two roots of the equation $A\left(G-H\right)x^2+G\left(H-A\right)x+H\left(A-G\right)=0,$ then 

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

The function $f\left(x\right)=\sin x+\frac{1}{\sin x}$ in the interval $\left(0, \pi \right)$ has a

If $\left(a,b\right)$ is a variable point lying on the circle $x^2+y^2=c^2$, then the least possible value of $\frac{1}{a^2}+\frac{1}{b^2}$ is -

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 $y=\frac{x}{a{x}^2+b}(a,b>0)$.

If $x, y\in (0,\infty )$, then the least possible value of $x^{100}+\frac{100}{x}+y^{50}+\frac{50}{y}$ is

The value of $25 {\sin }^2\theta +16{cosec}^2\theta$ is always greater than or equal to _____

The product of $n$ positive numbers is $1$. Then the minimum value of their sum is______.