CGMO 2007

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.

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 $\frac{BD}{DC}$ , $\frac{CE}{EA}$ , $\frac{AF}{FB}$ , $\frac{BF}{FA}$ , $\frac{AE}{EC}$ , $\frac{CD}{DB}$ are integers, then triangle $ABC$ is isosceles.

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 $$\frac{a_1}{a_2^2 + 1}+ \frac{a_2}{a^2_3 + 1}+ \cdots + \frac{a_n}{a^2_1 + 1}.$$

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.

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 $DE \perp EF$ .

For $a,b,c\geq 0$ with $a+b+c=1$ , prove that

$\sqrt{a+\frac{(b-c)^2}{4}}+\sqrt{b}+\sqrt{c}\leq \sqrt{3}$

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.