(1). If 1 company is selected, then this company needs to sign X- 1 contracts.

(2) After selecting 1 company from the remaining X- 1 companies, this company needs to sign an X- 1 contract with other companies, but one of them is the same as the contract in the process of (1), so the new contract signed by this company is X-2.

(3) The remaining X-2 companies choose 1 company to handle, and X-3 companies sign new contracts.

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Take this as an example. With three companies left, choose 1 company and sign two new contracts.

When there are only two companies left, the number of new contracts signed between them is 1.

So let's assume that this process is carried out y times, and we can get1+2+3+4+...+(y-1)+y = 45, so we can get Y=9, that is, after 9 times of treatment.

And the number of companies x-1= 9; So the number of companies X= 10.