CN-LEAGUE 2014

Let $a,b,c$ be real numbers such that $a+b+c=1$ and $abc>0$ . Prove that$$bc+ca+ab<\frac{\sqrt{abc}}{2}+\frac{1}{4}.$$

Let $ABC$ be an acute triangle such that $\angle BAC \neq 60^\circ$ . Let $D,E$ be points such that $BD,CE$ are tangent to the circumcircle of $ABC$ and $BD=CE=BC$ ( $A$ is on one side of line $BC$ and $D,E$ are on the other side). Let $F,G$ be intersections of line $DE$ and lines $AB,AC$ . Let $M$ be intersection of $CF$ and $BD$ , and $N$ be intersection of $CE$ and $BG$ . Prove that $AM=AN$ .

Let $S=\{1,2,3,\cdots,100\}$ . Find the maximum value of integer $k$ , such that there exist $k$ different nonempty subsets of $S$ satisfying the condition: for any two of the $k$ subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.

Let $x_1,x_2,\dots,x_{2014}$ be integers among which no two are congurent modulo $2014$ . Let $y_1,y_2,\dots,y_{2014}$ be integers among which no two are congurent modulo $2014$ . Prove that one can rearrange $y_1,y_2,\dots,y_{2014}$ to $z_1,z_2,\dots,z_{2014}$ , so that among $$x_1+z_1,x_2+z_2,\dots,x_{2014}+z_{2014}$$ no two are congurent modulo $4028$ .