Let's assume that the mathematician works for x hours a day and can solve y problems per hour. Also, the mathematician drinks some coffee and discovers that he can now solve z problems per hour. So, the mathematician works for n hours that day. We are given that:x*y = number of problems solved in a dayz * n = number of problems solved on the day he drank coffee

Then, we can write the equations:x*y = n * 2*z (he still solves twice as many problems as he would in a normal day)andx = n (he only works for n hours that day)Now, we need to simplify these equations to solve for the number of problems solved on the day he drank coffee. Here is how to do it:\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)Since x, y, n, and z are all positive integers, we can say that the expression 2*n*z/x is also a positive integer. Therefore, we can write:\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)where k is a positive integer.

Finally, the number of problems solved on the day he drank coffee is: **y = 2k** Therefore, the answer is that the mathematician solved 2k problems on the day he drank coffee.

**Answer: 2000 problems**

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