China Girls Math Olympiad 2011

Day 1

1. Find all positive integers $n$ such that equation: $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ has exactly 2011 positive solutions $(x. y)$ where $x \le y$

2. The diagonals $AC;BD$ of the quadrilateral $ABCD$ intersect at $E$ . Let $M;N$ be the midpoints of $AB;CD$ , respectively. Let the perpendicular bisectors of the segments $AB;CD$ meet at $F$ . Suppose that $EF$ meets $AD;BC$ at $P;Q$ respectively. If $MF.CD = NF.AB$ and $DQ.BP = AQ.CP$ , prove that $PQ \perp BC$ .

3. The positive reals $a, b, c, d$ satisfy $abcd = 1$ . Prove that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{9}{a + b +c +d} \ge \frac{25}{9}$

4. A tennis tournament has $n > 2$ players and any two players play one game against each other (ties are not allowed). After the game these players can be arranged in a circle, such that for any three players $A, B, C$ , if $A, B$ are adjacent on the circle, then at least one of $A, B$ won against $C$ . Find all possible values for $n$ .

Day 2

1. A real number $\alpha \ge 0$ is given. Find the smallest $\lambda(\alpha) > 0$ , such that for any complex numbers $z_1; z_2$ and $0 \le x \le 1$ , if $\mid$ $z_1$ $\mid$ $\le \alpha$ $\mid$ $z_1 - z_2$ $\mid$ , then $\mid$ $z_1 - xz_2$ $\mid$ $\le \lambda$ $\mid$ $z_1 - z_2$ $\mid$

2. Do there exist positive integers $m, n$ , such that $m^{20} + 11^n$ is a square number?

3. There are n boxes $B_1, B_2, ..., B_n$ from left to right, and there are n balls in these boxes. If
there is at least 1 ball in $B_1$ , we can move one to $B_2$ . If there is at least 1 ball in $B_n$ , we
can move one to $B_{n - 1}$ . If there are at least 2 balls in $B_k$ , $2 \le k \le n - 1$ we can move one to
$B_{k - 1}$ , and one to $B_{k + 1}$ . Prove that, for any arrangement of the $n$ balls, we can achieve that
each box has one ball in it.

4. The A-excircle $(O)$ of $\Delta ABC$ touches $BC$ at $M$ . The points $D,E$ lie on the sides $AB, AC$ respectively such that $DE || BC$ . The incircle $(O_1)$ of $\Delta ADE$ touches $DE$ at $N$ . If $BO_1 \bigcap DO = F$ and $CO_1 \bigcap EO = G$ , prove that the midpoint of $FG$ lies on $MN$ .