Designing Multistage BJT Amplifier: 51dB Gain, Rin 55Ω, Rout 50kΩ, β=100, ±5V Supply

Hello everyone,as a part of analog circuits course i take, I have an assignment in which I need to design a multistage BJT amplifier.The requirements are as follow:Gain = 51 dB (or equivalently Vout/Vin = 355)Rin = 55 ohmsRout = 50 kohmsV Early is infinite, current gain = 100DC suppliers allowed: +/- 5 VI tried to design with a CB input stage in order to achieve low Rin and the gain needed, and a CB output stage, with a high value resistor- paralleled to the collector's leg, in order to achieve high Rout.The problem is that the circuit isn't operating propperly and operating point isn't obtained.A PSpice schematic of the innitial circuit is attached.I'd very much appreciate any kind of help!Thank you all

You need to tidy up your schematic, you have some  wires that are not clearly laid out. Additionally, in the interests of clarity, it is customary (conventional) to keep the flow from positive  at the top through to negative at the bottom of the schematic, and possibly more likely you will be using NPN BJT's. The DC biasing of the common base amplifier needs sorting out. With such precise low input and medium output impedance's and gains being required, it is customary to use feedback (from memory shunt-series) to control these parameters. This requires doing quite a bit of maths first, before checking the results in Spice. Where is your working, otherwise you end up randomly changing values in spice attempting to get the results you desire,  and there is an interaction between the three parameters.Cheers,Richard

Your schematic reminds me of the schematics of PNP-germanium-transistor radios in the 1950s and 1960s, drawn with the top rail negative and bottom rail ground.

And apart from anything else, the input and Capacitor C1 are effectively shorted by the 5V DC source, which will be very low impedance.