Problem Statement
Suppose that $a$, $b$, $c$, and $d$ are positive integers that satisfy the equations
\[ab + cd = 38\] \[ac + bd = 34\] \[ad + bc = 43\]
What is the value of $a$ + $b$ + $c$ + $d$?
Problem Link
Solution
$$a, b, c, d \in \mathbb{Z}, a,b,c,d > 0$$ $$ab + cd = 38 \text{ (1)}$$ $$ac + bd = 34 \text{ (2)}$$ $$ad + bc = 43 \text{ (3)}$$ $$\text{(2) + (3): } ac + bd + ad + bc = 77$$ $$(a + b)(c + d) = 77$$ $$\text{Since } a, b, c, d > 0 \implies a+b=7, b+d = 11$$ $$\text{(1) + (3): } ab + cd + ad + bc = 81$$ $$(a + c)(b + d) = 81$$ $$\text{Since } a, b, c, d > 0 \implies a+c=9, b+d = 9$$ $$\therefore (a+b) + (b+d) + (a+c) + (b+d) = 2(a+b+c+d) = 36$$ $$\therefore a+b+c+d = 18$$ 