Points $ A$ and $ B$ lie on the circle with center $ O.$ Let point $ C$ lies outside the circle; let $ CS$ and $ CT$ be tangents to the circle. $ M$ be the midpoint of minor arc $ AB$ of $ (O).$ $ MS,\,MT$ intersect $ AB$ at points $ E,\,F$ respectively. The lines passing through $ E,\,F$ perpendicular to $ AB$ cut $ OS,\,OT$ at $ X$ and $ Y$ respectively. A line passed through $ C$ intersect the circle $ (O)$ at $ P,\,Q$ ($ P$ lies on segment $ CQ$). Let $ R$ be the intersection of $ MP$ and $ AB,$ and let $ Z$ be the circumcentre of triangle $ PQR.$ Prove that: $ X,\,Y,\,Z$ are collinear.

A rational number $ x$ is called good if it satisfies: $ x=\frac{p}{q}>1$ with $ p$, $ q$ being positive integers, $ \gcd (p,q)=1$ and there exists constant numbers $ \alpha$, $ N$ such that for any integer $ n\geq N$, \[ |\{x^n\}-\alpha|\leq\dfrac{1}{2(p+q)}\] Find all the good numbers.

There are $ 63$ points arbitrarily on the circle $ \mathcal{C}$ with its diameter being $ 20$. Let $ S$ denote the number of triangles whose vertices are three of the $ 63$ points and the length of its sides is no less than $ 9$. Fine the maximum of $ S$.

Find all functions $ f: \mathbb{Q}^{+} \mapsto \mathbb{Q}^{+}$ such that: \[ f(x) + f(y) + 2xy f(xy) = \frac {f(xy)}{f(x+y)}.\]

Let $ x_1, \ldots, x_n$ be $ n>1$ real numbers satisfying $ A=\left |\sum^n_{i=1}x_i\right |\not =0$ and $ B=\max_{1\leq i<j\leq n}|x_j-x_i|\not =0$. Prove that for any $ n$ vectors $ \vec{\alpha_i}$ in the plane, there exists a permutation $ (k_1, \ldots, k_n)$ of the numbers $ (1, \ldots, n)$ such that \[ \left |\sum_{i=1}^nx_{k_i}\vec{\alpha_i}\right | \geq \dfrac{AB}{2A+B}\max_{1\leq i\leq n}|\alpha_i|.\]

Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} + 5$.

When all vertex angles of a convex polygon are equal, call it equiangular. Prove that $ p > 2$ is a prime number, if and only if the lengths of all sides of equiangular $ p$ polygon are rational numbers, it is a regular $ p$ polygon.

Let $ I$ be the incenter of triangle $ ABC.$ Let $ M,N$ be the midpoints of $ AB,AC,$ respectively. Points $ D,E$ lie on $ AB,AC$ respectively such that $ BD=CE=BC.$ The line perpendicular to $ IM$ through $ D$ intersects the line perpendicular to $ IN$ through $ E$ at $ P.$ Prove that $ AP\perp BC.$

Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) = 1$ and $ (x + 1)[f(x)]^2 - 1$ is an odd function.

$ u,v,w > 0$,such that $ u + v + w + \sqrt {uvw} = 4$ prove that $ \sqrt {\frac {uv}{w}} + \sqrt {\frac {vw}{u}} + \sqrt {\frac {wu}{v}}\geq u + v + w$

Find all positive integers $ n$ such that there exists sequence consisting of $ 1$ and $ - 1: a_{1},a_{2},\cdots,a_{n}$ satisfying $ a_{1}\cdot1^2 + a_{2}\cdot2^2 + \cdots + a_{n}\cdot n^2 = 0.$

Assume there are $ n\ge3$ points in the plane, Prove that there exist three points $ A,B,C$ satisfying $ 1\le\frac{AB}{AC}\le\frac{n+1}{n-1}.$

Let $ ABC$ be a triangle. Circle $ \omega$­ passes through points $ B$ and $ C.$ Circle $ \omega_{1}$ is tangent internally to $ \omega$­ and also to sides $ AB$ and $ AC$ at $ T,\, P,$ and $ Q,$ respectively. Let $ M$ be midpoint of arc $ BC\, ($containing $ T)$ of ­$ \omega.$ Prove that lines $ PQ,\,BC,$ and $ MT$ are concurrent.

Given an integer $ k > 1.$ We call a $ k -$digits decimal integer $ a_{1}a_{2}\cdots a_{k}$ is $ p -$monotonic, if for each of integers $ i$ satisfying $ 1\le i\le k - 1,$ when $ a_{i}$ is an odd number, $ a_{i} > a_{i + 1};$ when $ a_{i}$ is an even number, $ a_{i}<a_{i + 1}.$ Find the number of $ p -$monotonic $ k -$digits integers.

Show that there exists a positive integer $ k$ such that $ k \cdot 2^{n} + 1$ is composite for all $ n \in \mathbb{N}_{0}$.

Let $ a_{1},a_{2},\cdots,a_{n}$ be positive real numbers satisfying $ a_{1} + a_{2} + \cdots + a_{n} = 1$. Prove that \[\left(a_{1}a_{2} + a_{2}a_{3} + \cdots + a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 + a_{2}} + \frac {a_{2}}{a_{3}^2 + a_{3}} + \cdots + \frac {a_{n}}{a_{1}^2 + a_{1}}\right)\ge\frac {n}{n + 1}\]

After multiplying out and simplifying polynomial $ (x - 1)(x^2 - 1)(x^3 - 1)\cdots(x^{2007} - 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) = (1 - x)(1 - x^{2})...(1 - x^{2007})$ $ (mod$ $ x^{2008}).$

Let $ n$ be positive integer, $ A,B\subseteq[0,n]$ are sets of integers satisfying $ \mid A\mid + \mid B\mid\ge n + 2.$ Prove that there exist $ a\in A, b\in B$ such that $ a + b$ is a power of $ 2.$

Let convex quadrilateral $ ABCD$ be inscribed in a circle centers at $ O.$ The opposite sides $ BA,CD$ meet at $ H$, the diagonals $ AC,BD$ meet at $ G.$ Let $ O_{1},O_{2}$ be the circumcenters of triangles $ AGD,BGC.$ $ O_{1}O_{2}$ intersects $ OG$ at $ N.$ The line $ HG$ cuts the circumcircles of triangles $ AGD,BGC$ at $ P,Q$, respectively. Denote by $ M$ the midpoint of $ PQ.$ Prove that $ NO = NM.$

Given $ n$ points arbitrarily in the plane $ P_{1},P_{2},\ldots,P_{n},$ among them no three points are collinear. Each of $ P_{i}$ ($1\le i\le n$) is colored red or blue arbitrarily. Let $ S$ be the set of triangles having $ \{P_{1},P_{2},\ldots,P_{n}\}$ as vertices, and having the following property: for any two segments $ P_{i}P_{j}$ and $ P_{u}P_{v},$ the number of triangles having $ P_{i}P_{j}$ as side and the number of triangles having $ P_{u}P_{v}$ as side are the same in $ S.$ Find the least $ n$ such that in $ S$ there exist two triangles, the vertices of each triangle having the same color.

Find the smallest constant $ k$ such that $ \frac {x}{\sqrt {x + y}} + \frac {y}{\sqrt {y + z}} + \frac {z}{\sqrt {z + x}}\leq k\sqrt {x + y + z}$ for all positive $ x$, $ y$, $ z$.

Find all the pairs of positive integers $ (a,b)$ such that $ a^2 + b - 1$ is a power of prime number $ ; a^2 + b + 1$ can divide $ b^2 - a^3 - 1,$ but it can't divide $ (a + b - 1)^2.$

Let $ ABCD$ be the inscribed quadrilateral with the circumcircle $ \omega$.Let $ \zeta$ be another circle that internally tangent to $ \omega$ and to the lines $ BC$ and $ AD$ at points $ M,N$ respectively.Let $ I_1,I_2$ be the incenters of the $ \triangle ABC$ and $ \triangle ABD$.Prove that $ M,I_1,I_2,N$ are collinear.

Consider a $ 7\times 7$ numbers table $ a_{ij} = (i^2 + j)(i + j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.