## China Girls Math Olympiad 2007

1. A positive integer $m$ is called good if there is a positive integer $n$ such that $m$ is the quotient of $n$ by the number of positive integer divisors of $n$ (including $1$ and $n$ itself). Prove that $1, 2, \ldots, 17$ are good numbers and that $18$ is not a good number.
2. Let $ABC$ be an acute triangle. Points $D$, $E$, and $F$ lie on segments $BC$, $CA$, and $AB$, respectively, and each of the three segments $AD$, $BE$, and $CF$ contains the circumcenter of $ABC$. Prove that if any two of the ratios $$\dfrac{BD}{DC},\, \dfrac{CE}{EA},\, \dfrac{AF}{FB},\, \dfrac{BF}{FA},\, \dfrac{AE}{EC},\, \dfrac{CD}{DB}$$ are integers, then triangle $ABC$ is isosceles.
3. Let $n$ be an integer greater than $3$, and let $a_1, a_2, \cdots, a_n$ be non-negative real numbers with $a_1 + a_2 + \cdots + a_n = 2$. Determine the minimum value of $\frac{a_1}{a_2^2 + 1}+ \frac{a_2}{a^2_3 + 1}+ \cdots + \frac{a_n}{a^2_1 + 1}.$
4. The set $S$ consists of $n > 2$ points in the plane. The set $P$ consists of $m$ lines in the plane such that every line in $P$ is an axis of symmetry for $S$. Prove that $m\leq n$, and determine when equality holds.
5. Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.
6. Let $a,b,c\geq 0$ with $a+b+c=1$. Prove that $$\sqrt{a+\frac{(b-c)^2}{4}}+\sqrt{b}+\sqrt{c}\leq \sqrt{3}$$
7. Let $a$, $b$, $c$ be integers each with absolute value less than or equal to $10$. The cubic polynomial $f(x) = x^3 + ax^2 + bx + c$ satisfies the property $\Big|f\left(2 + \sqrt 3\right)\Big| < 0.0001.$ Determine if $2 + \sqrt 3$ is a root of $f$.
8. In a round robin chess tournament each player plays every other player exactly once. The winner of each game gets $1$ point and the loser gets $0$ points. If the game is tied, each player gets $0.5$ points. Given a positive integer $m$, a tournament is said to have property $P(m)$ if the following holds: among every set $S$ of $m$ players, there is one player who won all her games against the other $m-1$ players in $S$ and one player who lost all her games against the other $m - 1$ players in $S$. For a given integer $m \ge 4$, determine the minimum value of $n$ (as a function of $m$) such that the following holds: in every $n$-player round robin chess tournament with property $P(m)$, the final scores of the $n$ players are all distinct.