“the 1st and 2nd students” or “the 1st and 2nd student”?

Solution 1:

When you compare first and second children with first and second child, you see that in published books, the general principle people most commonly follow in this would seem to be that:

• The plural is used when there are both several first children and several second children.

• The singular is used when there is just one of each: a single first child and a single second child.

Therefore if is just one first student and one second student who will be given the award, you would award the first and second student.

Solution 2:

Just to rephrase what @tchrist said: You are attempting to write a contracted version of the sentence without being sure of the explicit version first. In the given sentence:

I will give an award to the 1st and 2nd students.

The words 1st and 2nd both apply to students, so if we take your sentence literally, the following sentence also applies:

I will give an award to the 1st students and 2nd students.

Is that what you meant to say?

It helps to be sure of the explicit version before creating a contracted version. If you meant to say that there will only be one 1st student and one 2nd student, then you have to say:

I will give an award to the 1st student and the 2nd student.

or

I will give an award to the 1st and 2nd student.

This version and your original version have subtly different meanings.

Sometimes rephrasing a sentence more explicitly helps to determine how contracted versions should be phrased.