A seemingly simple question

Number Theory Level 3

Positive integers $a, b, c,$ and $d$ satisfy $a > b > c > d$ , $a + b + c + d = 2010$ , and $a^2 - b^2 + c^2 - d^2 = 2010$ . Find the number of possible values of $a$ .

499 500 501 498

1 solution

Alan Yan
Sep 22, 2015

Notice that you can factor $a^2 - b^2 + c^2 - d^2 = (a+b)(a-b) + (c+d)(c-d) = 2010$ .

Since $a + b + c + d = 2010$ , we know that $a - b = 1 , c - d = 1$ .

Substituting back into the original identity, we have that $a + c = 1006$ .

The only values of $a$ that work are $1004, 1003, ..., 504$ which is $\boxed{501}$ values.

2010 AIME I Problem 5

A seemingly simple question

1 solution

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