STEP I 2004 question 5 solution

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Sum of a term in B, and a term in C:

This is the term of E.

The square of a term in B:

Hence:

Hence, it is expressible as:

Which is in D.

Now consider the square of a term in C:

This is again a term in D.

Assume that for some positive integers "x", and "y": .

Let "x" represent some term in B, hence is a term in D.

Hence, as , , hence, there should be a term in D for which the equation is true, hence:

As "n" is a positive integer this is not possible for a term in B, or C (as the square is in D).

Now consider where the squares of the other sequences go:

Consider the square of a term in D:

.

This is a term in A.

Consider the square of a term in E:

This is a term in E.

Consider the square of a term in A:

This is a term in A.

Hence the following can be written (note that means the square of a term in "A", etc.):

There are no terms in A, or E such that the statement is true, and therefore the statement is false.

The fourth power of a term in B, or a term in C is in A.

Consider where a term in A is when two times another term in A is added to it:

.

Hence, a term in A which has had two times another term in A added to it is in C.

Hence, consider:

.

Now, it is possible to say that for a term in B, or C to fit the values of "x", or "y":

Hence there is no term in B, or C.

Now consider what happens to the other terms using the notation again:

Hence, for all of the numbers that are not in E, the above argument is sufficient; now consider terms in E added together (one term added to two times another):

This is in "E", and therefore if there existed a number for which the equation (aforementioned) was true, there would have to be (due to elimination) a term in "E" that satisfied it:

.

There is no such term.

Now consider that adding a term from E to any other term does not change the sequence it is in, hence:

.

Now consider the possibility that "x" is from "E", and "y" is from elsewhere.

Consider 2 times a term in A:

.

Hence:

.

The term of would correspond to a term in E plus two times a term in A; overall a term in B; hence:

Hence there is no such term, and no such numbers "x", and "y".

Consider that two times a term in E is also in E, and that this plus a term in A is also in A, therefore for any integer solution to exist, the sum () must be in A, hence:

.

As no such term exists there are no integers "x", or "y" for the equation to be true.