1) 2004 (Online Source)
2) 18 (Application of British Flag Theorem)
Solution (1):
Some of you may have guessed the answer correctly because of the pattern between the numerators and the denominators. But how do we solve it formally? Obviously, expansion is pointless. This leaves us with the 'substitution' method. This method is useful when we see complicated expressions and want to simplify it.
After a few guesses, one should be able to use the substitution of y=x-2004.
This simplifies the equation to:
Let the underlined part be n. Then,
Since
.
Solution (2):
Before we start on the solution, I would like to introduce a new theorem, the British Flag Theorem. It states that if a point P is chosen inside a rectangle ABCD then the following equality holds:
Refer to the diagram below for a clearer picture:
The proof is a fancy application of Pythagoras'.
Proof
Notice that:
Now to the problem:
Notice that the condition of the British Flag Theorem is quite similar to our problem's condition. We only need to find which combination of numbers makes x an integer. There are a total of 3 combinations:
Case 1:
.
After solving, we see that x is not an integer.
Case 2:
After solving, we get x=18.
Case 3:
.
After solving, we see that x is not an integer,
