Find the number of possible values of $a$

Question

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 $a.$

Obviously, factoring,

$$(a-b)(a+b) + (c-d)(c+d) = 2010$$ $$a + b + c + d = 2010$$

Substituting you get:

$$(a-b)(a+b) + (c-d)(c+d) - (a + b) - (c + d) = 0$$

$$\implies (a - b - 1)(a + b) + (c - d- 1)(c+d) = 0$$

But I dont see anything else.

Answer 1

Hint: examine the conditions you are given in the question - what can you say about the value of the expression in your last equation? Can you find a condition which makes it zero?

Answer 2

Denote $$b=a-x, \quad d=c-y. \qquad(x,y\in\mathbb{N}).\tag{1}$$

Then $$ a+b+c+d = 2a-x+2c-y = 2010,\\ a^2-b^2+c^2-d^2=(2a-x)x+(2c-y)y=2010.\tag{2} $$

Since $x,y\in\mathbb{N}$, then unique solution of $(2)$ has $$x=y=1.\tag{3}$$

So, solution has form $$ (a,\;a-1,\;c,\;c-1).\tag{4} $$

$(2),(4)\Rightarrow$ $$a+c-1=1005,$$ $$c=1006-a.$$

So, solution has form $$ (a,\;a-1,\;1006-a,\;1005-a).\tag{5} $$

Smallest possible $a$: $a_{min}=504 \rightarrow (a,b,c,d)=(504,503,502,501)$.

Largest possible $a$: $a_{max}=1004 \rightarrow (a,b,c,d)=(1004,1003,2,1)$.

Number of possible values of $a$: $501$.