CN-SMO 2009

Find all pairs ( $x,y$ ) of integers such that $x^2-2xy+126y^2=2009$ .

In the convex pentagon $ABCDE$ we know that $AB=DE, BC=EA$ but $AB \neq EA$ . $B,C,D,E$ are concyclic . Prove that $A,B,C,D$ are concyclic if and only if $AC=AD.$

Let $x,y,z$ be positive reals such that $\sqrt{a}=x(y-z)^2$ , $\sqrt{b}=y(z-x)^2$ and $\sqrt{c}=z(x-y)^2$ . Prove that $$a^2+b^2+c^2 \geq 2(ab+bc+ca)$$

Given 12 red points on a circle , find the mininum value of $n$ such that there exists $n$ triangles whose vertex are the red points . Satisfies: every chord whose points are the red points is the edge of one of the $n$ triangles .

Let $X=(x_1,x_2,......,x_9)$ be a permutation of the set $\{1,2,\ldots,9\}$ and let $A$ be the set of all such $X$ . For any $X \in A$ , denote $f(X)=x_1+2x_2+\cdots+9x_9$ and $M=\{f(X)|X \in A \}$ . Find $|M|$ . ( $|S|$ denotes number of members of the set $S$ .)

Let $\odot O$ , $\odot I$ be the circumcircle and inscribed circles of triangle $ABC$ . Prove that : From every point $D$ on $\odot O$ ,we can construct a triangle $DEF$ such that $ABC$ and $DEF$ have the same circumcircle and inscribed circles

Let $x,y,z\geq0$ be real numbers such that $x+y+z=1$ Define $f(x,y,z)$ in this way : $$f(x,y,z)=\frac{x(2y-z)}{1+x+3y}+\frac{y(2z-x)}{1+y+3z}+\frac{z(2x-y)}{1+z+3x}$$ Find the minimum value and maximum value of $f(x,y,z)$ .

In an ${8}$ × ${8}$ squares chart , we dig out $n$ squares , then we cannot cut a "T"shaped-5-squares out of the surplus chart . Then find the mininum value of $n$ .