CWMO 2007

Let set $T = \{1,2,3,4,5,6,7,8\}$ . Find the number of all nonempty subsets $A$ of $T$ such that $3|S(A)$ and $5\nmid S(A)$ , where $S(A)$ is the sum of all the elements in $A$ .

Let $C$ and $D$ be two intersection points of circle $O_1$ and circle $O_2$ . A line, passing through $D$ , intersects the circle $O_1$ and the circle $O_2$ at the points $A$ and $B$ respectively. The points $P$ and $Q$ are on circles $O_1$ and $O_2$ respectively. The lines $PD$ and $AC$ intersect at $H$ , and the lines $QD$ and $BC$ intersect at $M$ . Suppose that $O$ is the circumcenter of the triangle $ABC$ . Prove that $OD\perp MH$ if and only if $P,Q,M$ and $H$ are concyclic.

Let $a,b,c$ be real numbers such that $a+b+c=3$ . Prove that $$\frac{1}{5a^2-4a+11}+\frac{1}{5b^2-4b+11}+\frac{1}{5c^2-4c+11}\leq\frac{1}{4}$$

Is there a triangle with sides of integer lengths such that the length of the shortest side is $2007$ and that the largest angle is twice the smallest?

Find all natural numbers $n$ such that there exist $x_1,x_2,\ldots,x_n,y\in\mathbb{Z}$ where $x_1,x_2,\ldots,x_n,y\neq 0$ satisfying: $$x_1 + x_2 + \ldots + x_n = 0$$ $$ny^2 = x_1^2 + x_2^2 + \ldots + x_n^2$$

Let $P$ be an interior point of an acute angled triangle $ABC$ . The lines $AP,BP,CP$ meet $BC,CA,AB$ at points $D,E,F$ respectively. Given that triangle $\triangle DEF$ and $\triangle ABC$ are similar, prove that $P$ is the centroid of $\triangle ABC$ .