CGMO 2005

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

$$\left(\frac{CE}{BF} \right)^2 = \frac{AJ \cdot JE}{AK \cdot KF}.$$

Attachments:

Find all ordered triples $(x, y, z)$ of real numbers such that

$$5 \left(x + \frac{1}{x} \right) = 12 \left(y + \frac{1}{y} \right) = 13 \left(z + \frac{1}{z} \right),$$

and $$xy + yz + zy = 1.$$

Determine if there exists a convex polyhedron such that

(1) it has 12 edges, 6 faces and 8 vertices;

(2) it has 4 faces with each pair of them sharing a common edge of the polyhedron.

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:

(1) every set $A_i$ has infinitely many elements; (2) every pair of distinct sets $A_i$ and $A_j$ do not share any common element (3) the union of sets $A_1, A_2, \ldots, A_n$ is the set of all integers; (4) for every set $A_i,$ the positive difference of any pair of elements in $A_i$ is at least $a^i.$

Let $x$ and $y$ be positive real numbers with $x^3 + y^3 = x - y.$ Prove that $$x^2 + 4y^2 < 1.$$

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.

(1) Determine the minimum good number.

(2) Determine if 2005 is a good number. (A point in the coordinate plane is a lattice point if both of its coordinate are integers.)

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 $$\sum^n_{i = 1} \frac {1}{a_i} < \frac {m + n}{m}.$$

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.)