Nigerian Olympiad

Question

Suppose $a,b,c,d$ are integers satisfying $ab + cd = 44,ad - bc = 9.$ Find the minimum possible value of $a² + b² + c² + d².$

Answer 1

$$(ab+cd)^2=44^2\implies a^2b^2+2abcd+c^2d^2=44^2$$

Similarly:

$$a^2d^2-2abcd+c^2b^2=9^2$$

So:

$$a^2b^2+a^2d^2+c^2d^2+c^2b^2=44^2+9^2=2017\implies(a^2+c^2)(b^2+d^2)=2017$$

$2017$ is prime so one of $a^2+c^2$ or $b^2+d^2$ is $1$ and the other is $2017$. Assume $a^2+c^2=1$. Then one of $a$ and $c$ is $0$ and the other is $\pm 1$. If $a$ is $0$, then $d=\pm 44$. Then, $b^2=2017-d^2=9^2$.

So $$a^2+b^2+c^2+d^2=0^2+9^2+(\pm 1)^2+44^2=2018$$

You can exchange $a^2+c^2$ and $b^2+d^2$ w/o l.o.g. (and $a$ and $c$ or $b$ and $d$).