Since it is a fill-in-the-blank question, it should be solved in a specialized way to avoid cumbersome operations. Let the triangle ABC be an isosceles right triangle, a = 90, AB=AC= 1, take a as the origin, vectors AB and AC as the x axis, and the y axis as the positive direction, and establish a rectangular coordinate system, and let P(x0, y0), A (0 0,0), and B (1 0).
Pc = (-x0, 1-y0) and AB = (1-y0) are substituted into the known condition: PA+PB+2PC=3AB, and the equation can be obtained.
1-4x 0 = 3 2-4 y0 = 0 = = = & gt； x0 =- 1/2 y0 = 1/2 = = = & gt； The area of the triangle PAB is 1/4.
The area of triangle ABC is 1/2.
So the area ratio of triangle PAB to triangle ABC is 1/2.
4. Solution: The general term is:1-/[1+2+3+...+(n+1)] = n (n+3)/[(n+1) (n+2).
So the original formula = [1/2 * 2/3 * 3/4 * ... * n/(n+1)] * [4/3 * 5/4 * 6/5 * ... * (n+3)/(n)
When n=20 10, the original formula =1/201* 2013 = 671/201.
It is obvious that x is not 0. It is known that sin (xy) = (-x 2-16)/8x =-x/8+2/(-x).
Let the function f(x)=-x/8+2/(-x), then f(x) is odd function. If x
Then -x/8+2/(-x)>= 1 if and only if -x/8 = 2/(-x) (x
When x=-4, = = "sin (-4y) =1= = >; Sin4y=- 1, 0<= y & lt2PAI = = = & gtY=3pai/8/8 or 7pai/8 or1pai/8 or 15pai/8, when * * has four.
Similarly, x>0, f (x)
So, there are always eight groups of solutions.