34. There are 2012 students standing in a circle; they are numbered $1,2, \ldots, 2012$ clockwise. The counting starts from the first student (number 1) and proceeds around the circle clockwise, Alternate students will be eliminated from the circle in the following way: The first student stays in the circle while the second student leaves the circle. The third student stays while the fourth student leaves and so on. When the counting reaches number 2012, it goes back to number 1 and the elimination continues until the last student remains. What is the number of the last student?
34. Answer: 1976.
If there are $1024=2^{10}$ students, then the $1024^{\text {th }}$ student is the last one leaving the circle. Suppose $2012-1024=988$ students have left. Among the remaining 1024 students, the last student is $(2 \times 988-1)+1=1976$.