Given an inequation with P,Q,R all integers, $P \cdot R \cdot b + P \cdot Q \cdot c - Q \cdot R \cdot a \geq 0$
how many positive integer solutions of $(a, b, c)$ ? Here $a \leq P, b \leq Q, c \leq R$.
I cannot see any effective method except brute force in which three layers of loops are used to test each $(a,b,c)$. Any better method to solve this problem?
On
Given $P \cdot R \cdot b + P \cdot Q \cdot c - Q \cdot R \cdot a \geq 0$ with $a \leq P, b \leq Q, c \leq R$. Now set, $P \cdot Q \cdot c - Q \cdot R \cdot a \geq 0$, so, $P \cdot c \geq R \cdot a$ or $a \leq (P \cdot c)/R$ but $a \leq P$, hence, $a \leq (P \cdot c)/R \leq P$ and from the last inequality we have $c \leq R$, hence, for any $c$ satisfying this, the given inequality is automatically satisfied for any positive $b$. But since $a$, $b$ and $c$ are constrained and all their positive values satisfies the inequality, so the number of solutions is $P \cdot Q \cdot R$
Well, to begin, let's note that we can count the number of solutions using only two layers of loops instead of three layers of loops (i.e., we can compute the answer in time $O(QR)$ instead of $O(PQR)$). To do this, note that we can rearrange the inequality as follows:
$$a \leq \dfrac{P}{QR}(Rb+Qc)$$
It follows that the number of solutions $N$ to this inequality (that also satisfy $a \leq P$, $b\leq Q$, and $c\leq R$) is just:
$$N= \sum_{b=1}^{Q}\sum_{c=1}^{R} \min\left(\left\lfloor\dfrac{P}{QR}(Rb+Qc)\right\rfloor, P\right)$$
Of course, this solution is a bit unsatisfactory, since what we'd really like is a (relatively) closed form for $N$ in terms of $P$, $Q$, and $R$. I was unfortunately unable to find such a closed form. However, it should be noted that this problem is essentially the problem of counting the number of lattice points in a convex polyhedron (namely the polyhedron defined by the inequalities), and there is a huge body of literature devoted to this problem. See for example, this survey.
Here's one simple thing you can prove using these methods, for example. Let $N(P,Q,R)$ be the number of solutions as a function of $P$, $Q$, and $R$. Fix positive integers $\alpha$, $\beta$, and $\gamma$. Then $N(\alpha n, \beta n, \gamma n)$ is a quasi-polynomial in $n$.