SMO 2005 Problem 3

Let $a,b,c$ be real numbers satisfying $a<b<c,a+b+c=6,ab+bc+ac=9$ . Prove that $0<a<1<b<3<c<4$

Since the $x$ -coordinate of the interception point between $y=x(x-3)^2$ and $y=4$ which is the tangent line at

local maximum point $(1,4)$ is a point $(4,4)$ ,Moving the line $y=k$ so that the two graphs $y=x(x-3)^2$ and $y=k$ have the distinct three interception points,we can find that the range of $a,b,c$ are $0<a<1,1<b<3,3<c<4$ ,we are done.