SMO 2005

An integer is square-free if it is not divisible by $a^2$ for any integer $a>1$ . Let $S$ be the set of positive square-free integers. Determine, with justification, the value of$$\sum_{k\epsilon S}\left[\sqrt{\frac{10^{10}}{k}}\right]$$where $[x]$ denote the greatest integer less than or equal to $x$

Let $G$ be the centroid of triangle $ABC$ . Through $G$ draw a line parallel to $BC$ and intersecting the sides $AB$ and $AC$ at $P$ and $Q$ respectively. Let $BQ$ intersect $GC$ at $E$ and $CP$ intersect $GB$ at $F$ . If $D$ is midpoint of $BC$ , prove that triangles $ABC$ and $DEF$ are similar

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.