Determine p, q, r, s if p+q+r+s=12, pqrs=27+pq+pr+ps+qr+qs+rs.

This is one very delicious problem that shows how inequality could come into the picture and help in solving system of equations.

Given that $p, q, r, s$ are all positive real numbers and they satisfy the system

$p+q+r+s=12$

$pqrs=27+pq+pr+ps+qr+qs+rs$

Determine $p, q, r$ and $s$.

Okay, our intuition tells us the logic doesn't follow for this problem because we all know very well solving a system of two nonlinear equations in four variables sounds pretty much like a mission impossible.

It seems like the usual solving the system of equations by substitution or elimination method won't work, but when we see that we are told the variables are all positive real, the idea of using AM-GM inequality pops up!

By applying the AM-GM inequality for both $p, q, r, s$ and $pq,pr,ps,qr,qs,rs$, we get:

1.

$\dfrac{p+q+r+s}{4} \ge \sqrt[4]{pqrs}$ which then gives $(\dfrac{12}{4})^4 \ge pqrs$ or [MATH]\color{yellow}\bbox[5px,purple]{pqrs \le 81}[/MATH].

2.

$\large{\dfrac{pq+pr+ps+qr+qs+rs}{6} \ge \sqrt[6]{(pqrs)^3}}$

which then gives $(\dfrac{pqrs-27}{6})^2 \ge pqrs$

$(pqrs)^2-54pqrs+729 \ge 36pqrs$

$(pqrs)^2-90pqrs+729 \ge 0$

$(pqrs-9)(pqrs-81) \ge 0$

[MATH]\color{yellow}\bbox[5px,red]{pqrs \le 9}[/MATH] or [MATH]\color{yellow}\bbox[5px,green]{pqrs \ge 81}[/MATH].

Now, we have to wisely combine what we're told and what we just showed in order to get one step closer in solving the problem.

We're told that $pqrs=27+pq+pr+ps+qr+qs+rs$, implicitly it tells us $\color{yellow}\bbox[5px,blue]{pqrs\geqslant27}$ since all $p, q, r, s$ are all positive real numbers. That rules out the possibility that [MATH]\color{yellow}\bbox[5px,red]{pqrs \le 9}[/MATH] safely.

What governs the current situation now is the second possibility in which [MATH]\color{yellow}\bbox[5px,green]{pqrs \ge 81}[/MATH].

But from the previous working, we got [MATH]\color{yellow}\bbox[5px,purple]{pqrs \le 81}[/MATH]. These two ([MATH]\color{yellow}\bbox[5px,green]{pqrs \ge 81}[/MATH] and [MATH]\color{yellow}\bbox[5px,purple]{pqrs \le 81}[/MATH]) suggests that $pqrs=81$.

That implies that equality occurs in the AM-GM inequality, and that only happens when all four quantities are equal. So $p=q=r=s=3$.