sump pump drain pipe size

We have a 10 year old house which we bought 6 years ago. We have not had any problems with the sump pump system until last week. On Wednesday, my wife went down into the basement and found that even though the sump pump was running, the water was not being pulled out of the sump. The water had backed up in the little trench around the edge of the basement and had overflowed. The 10 X 10 ft. area rug we keep the exercise equipment on had turned into a large sponge.
I determined the sump pump itself was fine, so I bought 30 feet of 1 1/4 flexible tubing, attached it to the sump pump and put the tubing out the window. With this arrangement, the sump drained in about 1/2 hour. I have had to do this several times more, since the original time. Since we live in upstate NY, when the pump kicks off, I have to go outside and bring the tubing back in so it will not freeze. Needless to say, this gets old pretty fast.
I think the problem is that the regular drain pipe is too small and/or is frozen. The pipe from the sump pump to the wall is 1 inch ID PVC. Where it goes through the wall to go outside, it connects to a 3/4 inch ID black plastic, somewhat like a big stiff garden hose. This pipe runs about 50 feet, where it empties into the storm sewer. Unfortunately, it protrudes about 2 feet into the catch basin. I think this is where it freezes. The catch basin was full of snow, so last night when the sump pump came on, I went out and dumped several buckets of hot water into the catch basin to melt the snow so I could see the end of the pipe. I think I must have also melted the ice in the end of the pipe, because water started flowing out of the sump. (I could not really see the end of my pipe, because it was sticking into pile of leaves in the catch basin.)
I think the problem is exacerbated by the small size of the pipe. I besides being inherently too small, I also am inclined to think it makes it more susceptible to freezing.
My sump pump is 1/3HP, rated at 2880 gallons per hour. I tried looking on the web for flow rate calculations to see if a 3/4 inch pipe could pass this much water, but I think there are too many unknowns to make this calculation. It seems like the pipe is too small. Does anyone have any experience for what the correct size pipe would be or a site which would allow me to calculate the maximum flow rate ? Since the pipe is about 3-4 feet underground, replacing it would be a big deal and I want to make sure this is the problem before I do anything.
Thanks.
Marc
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You might consider placing a Tee in the pipe at a high point to vent the pipe so that when the pump shuts off the pipe will drain to the storm sewer and not freeze. This may require replacing the 3/4 inch pipe with a pipe larger than the 1" pipe (if your pump can fill the 1" pipe.) If it doesn't fill it then just a 3/4 tee and a vertical stub a few feet high might do it.
Boden
M. Laymon wrote:

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Yep, too small. Should be 1.5" or larger.

Way too small.

I agree. The pipe should be cut shorter so that there is an air gap allowing water to drain from the pipe. Sticking out into the catch basin like that, water can't drain from it after the pump shuts off, and it sits there and freezes.
[snip]

Correct.
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