China Girls Math Olympiad 2005

1. As shown in the following figure, point $ P$ lies on the circumcicle of triangle $ ABC.$ Lines $ AB$ and $ CP$ meet at $ E,$ and lines $ AC$ and $ BP$ meet at $ F.$ The perpendicular bisector of line segment $ AB$ meets line segment $ AC$ at $ K,$ and the perpendicular bisector of line segment $ AC$ meets line segment $ AB$ at $ J.$ Prove that \[ \left(\frac{CE}{BF} \right)^2 = \frac{AJ \cdot JE}{AK \cdot KF}.\]
2. Find all ordered triples $ (x, y, z)$ of real numbers such that $xy + yz + zy = 1$ and \[ 5 \left(x + \frac{1}{x} \right) = 12 \left(y + \frac{1}{y} \right) = 13 \left(z + \frac{1}{z} \right).\]
3. Determine if there exists a convex polyhedron such that
   a) it has 12 edges, 6 faces and 8 vertices;
   b) it has 4 faces with each pair of them sharing a common edge of the polyhedron.
4. Determine all positive real numbers $ a$ such that there exists a positive integer $ n$ and sets $ A_1, A_2, \ldots, A_n$ satisfying the following conditions
• every set $ A_i$ has infinitely many elements;
• every pair of distinct sets $ A_i$ and $ A_j$ do not share any common element
• the union of sets $ A_1, A_2, \ldots, A_n$ is the set of all integers;
• for every set $ A_i,$ the positive difference of any pair of elements in $ A_i$ is at least $ a^i.$
5. Let $ x$ and $ y$ be positive real numbers with $ x^3 + y^3 = x - y.$ Prove that \[ x^2 + 4y^2 < 1.\]
6. An integer $ n$ is called good if there are $ n \geq 3$ lattice points $ P_1, P_2, \ldots, P_n$ in the coordinate plane satisfying the following conditions: If line segment $ P_iP_j$ has a rational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have irrational lengths; and if line segment $ P_iP_j$ has an irrational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have rational lengths.
   a) Determine the minimum good number.
   b) Determine if $2005$ is a good number. (A point in the coordinate plane is a lattice point if both of its coordinate are integers.)
7. Let $ m$ and $ n$ be positive integers with $ m > n \geq 2.$ Set $ S = \{1, 2, \ldots, m\},$ and $ T = \{a_l, a_2, \ldots, a_n\}$ is a subset of S such that every number in $ S$ is not divisible by any two distinct numbers in $ T.$ Prove that \[ \sum^n_{i = 1} \frac {1}{a_i} < \frac {m + n}{m}. \]
8. Given an $ a \times b$ rectangle with $ a > b > 0$, determine the minimum side of a square that covers the rectangle. (A square covers the rectangle if each point in the rectangle lies inside the square.)