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Googling the problem leads us to a [similar Quora post](https://www.quora.com/How-do-you-find-the-positive-integer-solutions-to-frac-x-y+z-+-frac-y-z+x-+-frac-z-x+y-4/answer/Alon-Amit) on this challenge.

Modifying the script in the Quora post for N=10, we can solve for a,b, and c.
To make all numbers positive, we can brute force a multiple of P until all 3 numbers are positive.

Here's the script in Sage:

def orig(P,N):
x=P[0]
y=P[1]
a=(8*(N+3)-x+y)/(2*(N+3)*(4-x))
b=(8*(N+3)-x-y)/(2*(N+3)*(4-x))
c=(-4*(N+3)-(N+2)*x)/((N+3)*(4-x))
da=denominator(a)
db=denominator(b)
dc=denominator(c)
l=lcm(da,lcm(db,dc))
return [a*l,b*l,c*l]

ee = EllipticCurve([0,517,0,416,0])
P = ee(-416,4160)

for i in range(1,1000):
u = orig(P*i,10)
(a,b,c)=(u[0],u[1],u[2])
if a > 0 and b > 0 and c > 0:
print(a)
print(b)
print(c)
break