Competition Diophantine Equation I am looking for the number of integer solutions to this system of equations 

$$a^2+b^2+c^2+d^2=2500$$ and 

$$(a+50)(b+50)=cd$$ 

I tried moving terms around in the first equation and using the difference of two squares to try and gain some information, but came up empty. I was wondering what the best way to attack this problem is. Should I take the equations mod something, or should I try and place some bounds on the variables and try to derive the number of solutions from that? I ask that y'all only give me hints. Thanks
Hint: Use the fact $$2cd\leq c^2+d^2$$

and replace $cd$ and $c^2+d^2$ with a given constrains.
I solve Robert Israel's equation. This lead pytagorean equation.

$(a+50)^2+c^2=25^2$


  famous examples $3^2+4^2=5^2$,$7^2+24^2=25^2$,and trivial one $0^2+25^2=25^2$ are solutions. Final answers is $(a,b,c,d)=(-5,-45,±15,±15)(-10,-40,±15,±15)(-18,-32,±24,±24)(-1,-49,±7,±7)(0,-50,0,0)(-50,0,0,0)$