ABCD is a cyclic quadrilateral. AB and DC are produced to meet at P. AD and BC are produced to meet at Q.

If $\angle AQB=2x^\circ$, $\angle QAB=3x^\circ$ and $\angle BPC=x^\circ$, then what is the value of $x$?

Let the minor arcs $AB,BC,CD,DA$ be $a,b,c,d$ degrees respectively.

Using the inscribed-angle theorem and the external-secant angle theorem:

From the relations, $a=c+4x$ and $d=b+2x$.

The four arcs total $360^\circ$:

$a+b+c+d=(c+4x)+b+c+(b+2x)=2(b+c)+6x$

Using $b+c=6x$:

$360=12x+6x=18x$

$x=20$.