Omar drinks 1/4 * 12 = 12/4 = <<12/4=3>>3 ounces on the way to work.
Omar drinks 1/2 * 12 = 12/2 = <<12/2=6>>6 ounces when he gets to his office.
When he drinks 1 ounce more, he will have drunk 3 + 6 + 1 = <<3+6+1=10>>10 ounces.
Thus, there will be 12 - 10 = <<12-10=2>>2 ounces left in the cup afterward.
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