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.

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