Problem 1223

Given the equations:
$$2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A$$

We need to find the value of $A$ for distinct prime numbers $p, q, r$.

1. Equating the first two expressions:
$$2pqr + 50pq = 7pqr + 55pr$$
Subtract $2pqr$ from both sides:
$$50pq = 5pqr + 55pr$$
Divide both sides by $5p$:
$$10q = qr + 11r$$
Rearrange to isolate $r$:
$$10q = r(q + 11)$$
$$r = \frac{10q}{q + 11}$$

2. Equating the first and third expressions:
$$2pqr + 50pq = 8pqr + 12qr$$
Subtract $2pqr$ from both sides:
$$50pq = 6pqr + 12qr$$
Divide both sides by $2q$:
$$25p = 3pr + 6r$$
Rearrange to isolate $r$:
$$25p = 3r(p + 2)$$
$$r = \frac{25p}{3(p + 2)}$$

3. **Equating the two expressions for $r$:**
$$\frac{10q}{q + 11} = \frac{25p}{3(p + 2)}$$
Cross-multiply to solve for $p$ and $q$:
$$30q(p + 2) = 25p(q + 11)$$
$$30pq + 60q = 25pq + 275p$$
Subtract $25pq$ from both sides:
$$5pq + 60q = 275p$$
Rearrange to isolate $p$:
$$5pq - 275p = -60q$$
Factor out common terms:
$$p(5q - 275) = -60q$$
$$p = \frac{-60q}{5q - 275}$$

4. **Solving for $p$ and $q$:**
Since $p$ and $q$ are prime numbers, we need to find values that satisfy the equation:
$$p = \frac{-60q}{5q - 275}$$
Simplify the denominator:
$$p = \frac{-60q}{5(q - 55)}$$
$$p = \frac{-12q}{q - 55}$$

For $p$ to be a prime number, the numerator must be divisible by the denominator. Testing small prime values for $q$:
- If $q = 11$:
$$p = \frac{-12 \cdot 11}{11 - 55} = \frac{-132}{-44} = 3$$
$p = 3$ is a prime number.

- If $p = 3$ and $q = 11$, solve for $r$:
$$r = \frac{10 \cdot 11}{11 + 11} = \frac{110}{22} = 5$$
$r = 5$ is a prime number.

5. **Finding $A$:**
Substitute $p = 3$, $q = 11$, and $r = 5$ into the original equation:
$$A = 2pqr + 50pq$$
$$A = 2 \cdot 3 \cdot 11 \cdot 5 + 50 \cdot 3 \cdot 11$$
$$A = 330 + 1650$$
$$A = 1980$$

The final answer is $\boxed{1980}$.