Let $m$ be the minimum possible value of $\log _3\left(3^{y_1}+3^{y_2}+3^{y_3}\right)$, where $y _1, y _2, y _3$ are real numbers for which $y _1+ y _2+ y _3=9$. Let $M$ be the maximum possible value of $\left(\log _3 x _1+\log _3 x _2+\log _3 x _3\right)$, where $x_1, x_2, x_3$ are positive real numbers for which $x_1+x_2+x_3=9$. Then the value of $\log _2\left(m^3\right)+\log _3\left(M^2\right)$ is. . . . . . 

If $a,\;b,\;c$ are in $A.P.$ and $a,\;c - b,\;b - a$ are in $G.P. $ $(a \ne b \ne c)$, then $a:b:c$ is

If $p,q,r$ are in $G.P$ and ${\tan ^{ - 1}}p$, ${\tan ^{ - 1}}q,{\tan ^{ - 1}}r$ are in $A.P.$ then $p, q, r$ are satisfies the relation

The number of triples $(x, y, z)$ of real numbers satisfying the equation $x^4+y^4+z^4+1=4 x y z$ is

If $A . M$. and $G M$. of two positive numbers $a$ and $b$ are $10$ and $8 , $ respectively, find the numbers.

If the ratio of $A.M.$ between two positive real numbers $a$ and $b$ to their $H.M.$ is $m:n$, then $a:b$ is