Spring 2014 USA Online Math Open Qn 4

Spring 2014 USA OMO Qn 4. The integers 1, 2, \dots, n are written in order on a long slip of paper. The slip is then cut into five pieces, so that each piece consists of some (nonempty) consecutive set of integers. The averages of the numbers on the five slips are 1234, 345, 128, 19, and 9.5 in some order. Compute n.

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