CGMO 2015

Let $\triangle ABC$ be an acute-angled triangle with $AB>AC$ , $O$ be its circumcenter and $D$ the midpoint of side $BC$ . The circle with diameter $AD$ meets sides $AB,AC$ again at points $E,F$ respectively. The line passing through $D$ parallel to $AO$ meets $EF$ at $M$ . Show that $EM=MF$ .

Let $a\in(0,1)$ , $f(x)=ax^3+(1-4a)x^2+(5a-1)x-5a+3$ , $g(x)=(1-a)x^3-x^2+(2-a)x-3a-1$ . Prove that:For any real number $x$ ,at least one of $|f(x)|$ and $|g(x)|$ not less than $a+1$ .

In a $12\times 12$ grid, colour each unit square with either black or white, such that there is at least one black unit square in any $3\times 4$ and $4\times 3$ rectangle bounded by the grid lines. Determine, with proof, the minimum number of black unit squares.

Let $g(n)$ be the greatest common divisor of $n$ and $2015$ . Find the number of triples $(a,b,c)$ which satisfies the following two conditions: $1)$ $a,b,c \in$ { $1,2,...,2015$ }; $2)$ $g(a),g(b),g(c),g(a+b),g(b+c),g(c+a),g(a+b+c)$ are pairwise distinct.

Determine the number of distinct right-angled triangles such that its three sides are of integral lengths, and its area is $999$ times of its perimeter. (Congruent triangles are considered identical.)

Let $\Gamma_1$ and $\Gamma_2$ be two non-overlapping circles. $A,C$ are on $\Gamma_1$ and $B,D$ are on $\Gamma_2$ such that $AB$ is an external common tangent to the two circles, and $CD$ is an internal common tangent to the two circles. $AC$ and $BD$ meet at $E$ . $F$ is a point on $\Gamma_1$ , the tangent line to $\Gamma_1$ at $F$ meets the perpendicular bisector of $EF$ at $M$ . $MG$ is a line tangent to $\Gamma_2$ at $G$ . Prove that $MF=MG$ .

Let $x_1,x_2,\cdots,x_n \in(0,1)$ , $n\geq2$ . Prove that $$\frac{\sqrt{1-x_1}}{x_1}+\frac{\sqrt{1-x_2}}{x_2}+\cdots+\frac{\sqrt{1-x_n}}{x_n}<\frac{\sqrt{n-1}}{x_1 x_2 \cdots x_n}.$$

Let $n\geq 2$ be a given integer. Initially, we write $n$ sets on the blackboard and do a sequence of moves as follows: choose two sets $A$ and $B$ on the blackboard such that none of them is a subset of the other, and replace $A$ and $B$ by $A\cap B$ and $A\cup B$ . This is called a $\textit{move}$ . Find the maximum number of moves in a sequence for all possible initial sets.