If $n$ is positive integer and $p, q, r$ are primes solve the system: $pqr=n$ and $(p+1)(q+1)r=n+138$

Given is that $x, y, z$ are real numbers, different from 0, $x$ and $z$ are different, such that $(x+y) ^2+(2-xy)=9$ and $(y+z) ^2-(3+yz)=4$ Find the value of $A=(x/y+y^2/x^2+z^3/x^2y)(y/z+z^2/y^2+x^3/y^2z)(z/x+x^2/z^2+y^3/z^2x)=?$

Let $ABCD$ be a trapezoid ($AD//BC$) with $\angle A=\angle B= 90^o$ and $AD<BC$. Let $E$ be the intersection point of the non parallel sides $AB$ and $CD$, $Z$ be the symmetric point of $A$ wrt line $BC$ and $M$ be the midpoint of $EZ$. If it is given than line $CM$ is perpendicular on line $DZ$, then prove that line $ZC$ is perpendicular on line $EC$.

Find the number ot 6-tuples $(x_1, x_2,...,x_6)$, where $x_i=0,1 or 2$ and $x_1+x_2+...+x_6$ is even