CAML, for californians Options

[*salcha*]

Does anyone else take it?
The California Mathematics League. It's a math contest that you take monthly (up to April) and it contains 6 problems every time.

The problems from the last CAML
1. What is the only value of x which satisfies (x-2005)^2 = (x+2005)^2?

2. In convex quadrilateral ABCD, the area of triangle ABC is 3, the area of triangleCDA is 4, and the area of triangleDAB is 5. What is the area of triangleBCD?

3. At the right, a grid of 25 dots is shown. The distance between adjacent horizontal points and between adjacent vertical points is 1. Two segments of length 5 have been drawn. Altogether, how many different pairs of these points are exaclty 5 units apart?


4. How many positive two-digit numbers have a perfect square as the product of their digits?

5. For what rational number c do the equations x^3 + cx^2 + 3 = 0 and x^2 + cx+ 1 = 0 have a common solution?

6. As my younger brother shouted out four consecutive integers, I divided each by my age (an integer), then added all four remainders to get 40. As he shouted out four other consecutive integerts, I divided each by his age (a different integer), addeed all four remainders, and again got 40. What is the sum of our ages?

I got the first three right. If enough people try these, I'll post answers at the end. You don't really take these for anything, it's just for fun. My teacher gives out extra credit, however :x