Solved! Let's start by assuming we don't need R8. Then we can proceed as follows:
Step 1:
Choose R1, R2 and R3 arbitrarily, but in the correct proportions relative to each other.
e.g. We can have R1=18K, R2=15K and R3=10K
because 18K*0.5 = 15K*0.6 = 10K*0.5 = 9K
Step 2:
Choose R4 and R5 arbitrarily, but in the correct proportions relative to each other.
e.g. We can have R4=18K and R5=16K
because 18K*0.8 = 16K*0.9 = 14.4K
Step 3:
Calculate the feedback resistor. This is simple if we think in terms of only one input at a time. With V1 = V2 = V3 = 0, it is a simple inverting amplifier as far as the other two inputs are concerned.
For V4 we need R6 = 18K * 0.8 = 14.4K
For V5 we need R6 = 16K * 0.9 = 14.4K
Step 4:
Finally, we can calculate R7. Again, think in terms of only one input at a time. With V4 = V5 = 0, it is a simple non-inverting amplifier (with a voltage divider at the input) as far as the other three inputs are concerned.
For example, if V1 = 1V and all other inputs are grounded, there will be 0.25V at the non-inverting input to the opamp. We want 0.5V at the output so the amp needs to have a gain of two. Double-checking with the V2 or V3 gives the same result.
R6 has already been set to 14.4K so for a gain of 2, the parallel combination of R4,R5 and R7 must also be 14.4K. Oops! That won't work because the parallel combination of R4 and R5 is already lower than 14.4K.
We're going to need R8 after all, to reduce the input to the non-inverting input, so the amp can have a higher gain. If we do that then we don't need R7.
Step 5(oops):
Leaving out R7, we can calculate R8. With the given values of R4, R5 and R6, the gain of the amplifier = 2.7. Given the required attenuation and the values of R1, R2 and R3, we can work out that R8 = 12.857143K.
That works but it's a nasty value. We can either try a parallel combination of two resistors, or change R1, R2 and R3. e.g. If we multiply all of them by 7, we get R1=126K, R2=105K, R3=70K and R8=90K exactly.