Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$

Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$

Let $k \geq 2, 1 < n_1 < n_2 < \ldots < n_k$ are positive integers, $a,b \in \mathbb{Z}^+$ satisfy \[ \prod^k_{i=1} \left( 1 - \frac{1}{n_i} \right) \leq \frac{a}{b} < \prod^{k-1}_{i=1} \left( 1 - \frac{1}{n_i} \right) \] Prove that: \[ \prod^k_{i=1} n_i \geq (4 \cdot a)^{2^k - 1}. \]

Points $D,E,F$ are on the sides $BC, CA$ and $AB$, respectively which satisfy $EF || BC$, $D_1$ is a point on $BC,$ Make $D_1E_1 || D_E, D_1F_1 || DF$ which intersect $AC$ and $AB$ at $E_1$ and $F_1$, respectively. Make $\bigtriangleup PBC \sim \bigtriangleup DEF$ such that $P$ and $A$ are on the same side of $BC.$ Prove that $E, E_1F_1, PD_1$ are concurrent. [Edit by Darij: See my post #4 below for a possible correction of this problem. However, I am not sure that it is in fact the problem given at the TST... Does anyone have a reliable translation?]

Let $p_1, p_2, \ldots, p_{25}$ are primes which don’t exceed 2004. Find the largest integer $T$ such that every positive integer $\leq T$ can be expressed as sums of distinct divisors of $(p_1\cdot p_2 \cdot \ldots \cdot p_{25})^{2004}.$

Let $a, b, c$ be sides of a triangle whose perimeter does not exceed $2 \cdot \pi.$, Prove that $\sin a, \sin b, \sin c$ are sides of a triangle.

Using $ AB$ and $ AC$ as diameters, two semi-circles are constructed respectively outside the acute triangle $ ABC$. $ AH \perp BC$ at $ H$, $ D$ is any point on side $ BC$ ($ D$ is not coinside with $ B$ or $ C$ ), through $ D$, construct $ DE \parallel AC$ and $ DF \parallel AB$ with $ E$ and $ F$ on the two semi-circles respectively. Show that $ D$, $ E$, $ F$ and $ H$ are concyclic.

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

Find all positive integer $ n$ satisfying the following condition: There exist positive integers $ m$, $ a_1$, $ a_2$, $ \cdots$, $ a_{m-1}$, such that $ \displaystyle n = \sum_{i=1}^{m-1} a_i(m-a_i)$, where $ a_1$, $ a_2$, $ \cdots$, $ a_{m-1}$ may not distinct and $ 1 \leq a_i \leq m-1$.

Given integer $ n$ larger than $ 5$, solve the system of equations (assuming $x_i \geq 0$, for $ i=1,2, \dots n$): \[ \begin{cases} \displaystyle x_1+ \phantom{2^2} x_2+ \phantom{3^2} x_3 + \cdots + \phantom{n^2} x_n &= n+2, \\ x_1 + 2\phantom{^2}x_2 + 3\phantom{^2}x_3 + \cdots + n\phantom{^2}x_n &= 2n+2, \\ x_1 + 2^2x_2 + 3^2 x_3 + \cdots + n^2x_n &= n^2 + n +4, \\ x_1+ 2^3x_2 + 3^3x_3+ \cdots + n^3x_n &= n^3 + n + 8. \end{cases} \]

Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}=60^o$, $ BC=CD=1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter of quadrilateral $ ABCD$.

$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying: (1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)=\gcd(b,c)=1$. (2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$. Find the largest possible value of $ |S|$.

Let $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (may not distinct) and $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (may not distinct) be two groups of positive integers such that for any positive integer $ d$ larger than $ 1$, the numbers of which can be divided by $ d$ in group $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (including repeated numbers) are no less than that in group $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (including repeated numbers). Prove that $ \displaystyle \frac{m_1 \cdot m_2 \cdots m_r}{n_1 \cdot n_2 \cdots n_s}$ is integer.

Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$. Prove that $ M$, $ N$, $ O$ are collinear.

Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k+3$, where $ k$ is also an integer.

Find the largest value of the real number $ \lambda$, such that as long as point $ P$ lies in the acute triangle $ ABC$ satisfying $ \angle{PAB}=\angle{PBC}=\angle{PCA}$, and rays $ AP$, $ BP$, $ CP$ intersect the circumcircle of triangles $ PBC$, $ PCA$, $ PAB$ at points $ A_1$, $ B_1$, $ C_1$ respectively, then $ S_{A_1BC}+ S_{B_1CA}+ S_{C_1AB} \geq \lambda S_{ABC}$.

Find the largest positive real $ k$, such that for any positive reals $ a,b,c,d$, there is always: \[ (a+b+c) \left[ 3^4(a+b+c+d)^5 + 2^4(a+b+c+2d)^5 \right] \geq kabcd^3\]

Find all positive integer $ m$ if there exists prime number $ p$ such that $ n^m-m$ can not be divided by $ p$ for any integer $ n$.

Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) = af(x) + bx$ and $ \displaystyle f(x)f(y) = f(xy) + f \left( \frac {x}{y} \right)$.

Let $ k$ be a positive integer. Set $ A \subseteq \mathbb{Z}$ is called a $ \textbf{k - set}$ if there exists $ x_1, x_2, \cdots, x_k \in \mathbb{Z}$ such that for any $ i \neq j$, $ (x_i + A) \cap (x_j + A) = \emptyset$, where $ x + A = \{ x + a \mid a \in A \}$. Prove that if $ A_i$ is $ \textbf{k}_i\textbf{ - set}$($ i = 1,2, \cdots, t$), and $ A_1 \cup A_2 \cup \cdots \cup A_t = \mathbb{Z}$, then $ \displaystyle \frac {1}{k_1} + \frac {1}{k_2} + \cdots + \frac {1}{k_t} \geq 1$.

In convex quadrilateral $ ABCD$, $ AB=a$, $ BC=b$, $ CD=c$, $ DA=d$, $ AC=e$, $ BD=f$. If $ \max \{a,b,c,d,e,f \}=1$, then find the maximum value of $ abcd$.

Given sequence $ \{ c_n \}$ satisfying the conditions that $ c_0=1$, $ c_1=0$, $ c_2=2005$, and $ c_{n+2}=-3c_n - 4c_{n-1} +2008$, ($ n=1,2,3, \cdots$). Let $ \{ a_n \}$ be another sequence such that $ a_n=5(c_{n+1} - c_n) \cdot (502 - c_{n-1} - c_{n-2}) + 4^n \times 2004 \times 501$, ($ n=2,3, \cdots$). Is $ a_n$ a perfect square for every $ n > 2$?

There are $ n \geq 5$ pairwise different points in the plane. For every point, there are just four points whose distance from which is $ 1$. Find the maximum value of $ n$.

The largest one of numbers $ p_1^{\alpha_1}, p_2^{\alpha_2}, \cdots, p_t^{\alpha_t}$ is called a $ \textbf{Good Number}$ of positive integer $ n$, if $ \displaystyle n= p_1^{\alpha_1} \cdot p_2^{\alpha_2} \cdots p_t^{\alpha_t}$, where $ p_1$, $ p_2$, $ \cdots$, $ p_t$ are pairwisely different primes and $ \alpha_1, \alpha_2, \cdots, \alpha_t$ are positive integers. Let $ n_1, n_2, \cdots, n_{10000}$ be $ 10000$ distinct positive integers such that the $ \textbf{Good Numbers}$ of $ n_1, n_2, \cdots, n_{10000}$ are all equal. Prove that there exist integers $ a_1, a_2, \cdots, a_{10000}$ such that any two of the following $ 10000$ arithmetical progressions $ \{ a_i, a_i + n_i, a_i + 2n_i, a_i + 3n_i, \cdots \}$($ i=1,2, \cdots 10000$) have no common terms.