Competition problem — Number theory

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Number theory Difficulty 5.4 AIME, harder Find the answer

4. Positive integers $a, b, c, d$ satisfy: $1<a<b<c<d<1000$ , and $a+d=b+c$ , $bc-ad=2004$ . Find the number of all such positive integer tuples $(a, b, c, d)$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

4. Let $b=a+x, c=a+y$ , then $x<y$ , and $d=a+x+y$ (this is obtained from $a+d=b+c$ ), thus
$b c-a d=(a+x)(a+y)-a(a+x+y)=x y$

That is
$x y=2004$

Combining $a+x+y<1000$ and $2004=2^{2} \times 3 \times 167$ , we know that $(x, y)=(3,668),(4,501),(6,334),(12,167)$ .

Accordingly, $1<a<329,1<a<495,1<a<660,1<a<821$ . Thus, the number of qualifying arrays is $327+493+658+819=2297$ (groups).

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.