Different possible perimeters

Question

For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. 

How many different values of $p$ are possible?

EDIT: $2015\geq{p}$

This is a contest problem by the AMC 12. As of right now, I am completely lost on how to solve it. I tried to rewrite everything as a single variable and then sieving through the integer values generated. Unfortunately, I could not consider the right angle.

I would appreciate IF HINTS ARE ONLY GIVEN AT FIRST. Thank you.

Answer 1

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If $AD =CD= x$, then we have $BC= \sqrt{x^2-(2-x)^2} =2\sqrt{x-1}$. Thus, perimeter $$p = 2x +2\sqrt{x-1} +2$$ Now check what are the possible values $p$ can take.

Also, on a side note, is there any other condition specified?

Answer 2

Hint. $AC$ is the bisector of the angle at $A$. Call it $2\beta$. Then $$|AD|=|CD|=\frac{|AC|}{2\cos\beta}=\frac{|AC|}{2\frac{2}{|AC|}}=\frac{|AC|^2}{4}=\frac{4+|BC|^2}{4}=1+\frac{|BC|^2}{4}.$$ So the four sides have all positive integer lengths iff $|BC|=2k$ with $k$ a positive integer.

Therefore the perimeter is $$p=2+|BC|+|AD|+|CD|=4+|BC|+\frac{|BC|^2}{2}=4+2k+2k^2.$$

Is there any other hypothesis in your problem?