回複：回複：歐洲奧賽題：做三角形（3星）

Strange, wxc ate some of the text I sent. I guess the backslash mess up the text. Below, I will use intsec(A,B) and union(A,B) to represent the intersection and union of two sets A and B.

We can find M>0, such that each of the three sets have elements in (0,M], to simplify our notation, let X,Y,Z be the intersection of A,B,C with (0,M] respectively, and let x,y,z be the corresponding supremums. Clearly, (0,M] = union(X,Y,Z)

Without loss of generality, we may assump x<=y<=z=M
case 1. x so, for any x1 in X, we can find y1 in Y1 and z1 in Z1, such that |y1-z1|=x1, z1>=x1, so x1,y1,z1 can form a triangle.
case 2. x=y, then inf(Z1) = x, so we can find x1,y1,z1 from X,Y,Z respecively, so that they are close to x, with maximum difference less than x/9. Then it can be easily shown that x1,y1,z1 can be the lengths of the edges of a triangle.