To solve the problem of finding the number of positive integer pairs \((x, y)\) such that
\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{2007} \quad \text{and} \quad x < y,
\]
we can follow these steps:
### Step 1: Rewrite the equation
Starting from the given equation, we can find a common denominator:
\[
\frac{y + x}{xy} = \frac{1}{2007}.
\]
Cross-multiplying gives us:
\[
2007(x + y) = xy.
\]
### Step 2: Rearranging the equation
Rearranging the equation, we can write:
\[
xy - 2007x - 2007y = 0.
\]
### Step 3: Adding \(2007^2\) to both sides
To factor this equation, we add \(2007^2\) to both sides:
\[
xy - 2007x - 2007y + 2007^2 = 2007^2.
\]
### Step 4: Factoring the left-hand side
Now we can factor the left side:
\[
(x - 2007)(y - 2007) = 2007^2.
\]
### Step 5: Setting variables
Let \(a = x - 2007\) and \(b = y - 2007\). Thus, we have:
\[
ab = 2007^2.
\]
### Step 6: Finding the factors of \(2007^2\)
Next, we need to find the number of positive integer solutions \((a, b)\) such that \(ab = 2007^2\).
First, we find the prime factorization of \(2007\):
\[
2007 = 3^2 \times 223^1.
\]
Thus,
\[
2007^2 = (3^2 \times 223^1)^2 = 3^4 \times 223^2.
\]
### Step 7: Calculating the number of factors
To find the number of factors of \(2007^2\), we use the formula for the number of factors based on the prime factorization:
If \(n = p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k}\), then the number of factors \(d(n)\) is given by:
\[
d(n) = (e_1 + 1)(e_2 + 1) \ldots (e_k + 1).
\]
For \(2007^2\):
\[
d(2007^2) = (4 + 1)(2 + 1) = 5 \times 3 = 15.
\]
### Step 8: Considering the pairs \((a, b)\)
Each factor pair \((a, b)\) corresponds to a solution \((x, y)\) where \(x = a + 2007\) and \(y = b + 2007\).
Since \(x < y\), we only consider pairs where \(a < b\). Out of the 15 factors, one pair corresponds to \(a = b\) (which gives \(x = y\)), and the remaining 14 pairs can be divided into two groups: one where \(a < b\) and the other where \(a > b\).
Thus, the number of pairs where \(a < b\) is:
\[
\frac{14}{2} = 7.
\]
### Final Answer
Therefore, the number of positive integer pairs \((x, y)\) such that \(x < y\) is:
\[
\boxed{7}.
\]
