Summing amplifier with one op amp
You can start by determining the input resistors in terms of an arbitrary parallel resistance R3||R4||R5. Once you have the ratios, determine the gain to get the absolute factors 2, 3, 5. You don't want a resistor to ground from the non-inverting input because that would just increase the required gain, and make the circuit more sensitive to Vos, TCVos, noise etc. In this case, the minimum (and thus optimal) gain is 10. 

Once you have that, you should balance the impedances seen by the inverting and non-inverting terminals, and then there will be an arbitrary scaling factor for the entire network. There are thus many solutions. Below is one that happens to end up with the resistors having reasonable integer values. 







simulate this circuit &ndash; Schematic created using CircuitLab