Cold Start Problems.

I have a 2001 Pontiac Grand Am SE 4cyl.

Hello, folks. I have a bit of a problem.

Ever since the cold weather began to kick in, I've been having some rather nasty cold start problems. On very chilly mornings, when attempting to crank, I get no response. Not even a solenoid click. I've been able to get jumps and start the car up from time to time (only when the amps coming off the jumping car are higher than mine, or the weather is warmer allowing a lower amount to provide the neccessary boost I need). When the weather, and hence the car, begins to warm up the problem vanishes and the car starts as normal.

Steps I've tried:
1. Thinking that it might be a bad connection, I used a terminal cleaner as well as a cleaning solution for that purpose. I've checked all of the connections that I could get to with no positive results.
2. I had the battery (820ca 700cca which was bought in early Sept) checked, and it came out fine. I also had the alternator and starter checked at Autozone, and they came out fine in the tests they ran.
3. Thinking it might be a faulty battery that was having troubles with the cold (even though the cold here doesn't seem sufficient to cause this) I bought a new battey last night and installed it. When the cold of the morning came, same results.

The only thing I can think of at this point, is that I perhaps have a bad connection somewhere, or the starter solenoid is in the process of buying the farm. I'm guessing I'll try replacing the battery cables tonight.

Any ideas?

Before you replace anything, lets do a little diagnosis.....when this happens do your headlights work?
if the starter will not click when you turn the key, check the purple wire at the solenoid for power with the key in the start position.
If you have power there, them most likely you have a bad solenoid....if no power then you need to start looking at the possibility of a starter relay or a bad switch

When this happens the headlights are working just fine and without any noticeable dimming. I tested the power at the solenoid with the key in the start position, as you advised, and got nothing.

So I'm looking at either a starter relay or ignition switch? Any advice on how to best go about further testing or replacement?

EDIT:

Where is the starter relay located on this model? I've looked in the engine compartment fuse box, and none of the references seem to say anything about the starter relay. Both of the kick panel fuse-box references fail to mention anything about it as well. The Haynes manual and the owner manual have both been somewhat useless in finding it too.

If I had a tone tool, I'd track it back, but mine seems to be busted.

Any ideas?

Nearly a year later, and the same problem has returned. I've tried another round of cleaning and tightening, but I haven't had any luck. I'm getting to the point that I may just take it to the dealership if I can't figure it out myself.

Does anyone know where I should look for the starter relay in this car? Most reference materials haven't been helpful.