The numbers 1 to 8 are written into the circles shown so that there is one number in each circle. Along each of the five straight arrows the three numbers in the circles are multiplied. Their product is written next to the tip of the arrow. How big is the sum of the numbers in the three circles on the lowest row of the diagram? (A) 11 (B) 12 (C) 15 (D) 17 (E) 19
Let the bottom row be $x,$ $y,$ and $z$ from left to right, and we use the given data to build the following system of equations\[(x+2)z = 2(x+y)\]\[(y+2)x = 2(y+z)\]\[(z+2)y = 2(y+x).\]We simplify the system to \[xz-2y+z=0\]\[xy-2z+x=0\]\[yz-2x+y=0.\]Summing all three equations, we have $2(xy+yz+zx)-2(x+y+z)=0.$ Rearranging, we get $xy+yz+zx=x+y+z.$ These three cyclic terms then represent a symmetric sum, which we write as\begin{align*}xy+yz+zx&=x+y+z \\ &= (x+y+z) \\ & = (2(x+y+z)-(x+y+z)) \\ & = 2(8)-15 \\ & = 1.\end{align*}Checking each given answer choice, the answer of $\boxed{\textbf{(C) } 15}$ is indeed correct.
