Let $ ABC$ be a triangle. Point $ D$ lies on its sideline $ BC$ such that $ \angle CAD = \angle CBA.$ Circle $ (O)$ passing through $ B,D$ intersects $ AB,AD$ at $ E,F$, respectively. $ BF$ meets $ DE$ at $ G$.Denote by$ M$ the midpoint of $ AG.$ Show that $ CM\perp AO.$

Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 = a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i + 1} + a_{i - 1}),i = 1,2,\cdots,n - 1,$ then $ (\sum_{i = 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i = 1}^n{a_{i}^2}.$

Prove that for any odd prime number $ p,$ the number of positive integer $ n$ satisfying $ p|n! + 1$ is less than or equal to $ cp^\frac{2}{3}.$ where $ c$ is a constant independent of $ p.$

Let positive real numbers $ a,b$ satisfy $ b - a > 2.$ Prove that for any two distinct integers $ m,n$ belonging to $ [a,b),$ there always exists non-empty set $ S$ consisting of certain integers belonging to $ [ab,(a + 1)(b + 1))$ such that $ \frac {\displaystyle\prod_{x\in S}}{mn}$ is square of a rational number.

Let $ m > 1$ be an integer, $ n$ is an odd number satisfying $ 3\le n < 2m,$ number $ a_{i,j} (i,j\in N, 1\le i\le m, 1\le j\le n)$ satisfies $ (1)$ for any $ 1\le j\le n, a_{1,j},a_{2,j},\cdots,a_{m,j}$ is a permutation of $ 1,2,3,\cdots,m; (2)$ for any $ 1 < i\le m, 1\le j\le n - 1, |a_{i,j} - a_{i,{j + 1}}|\le 1$ holds. Find the minimal value of $ M$, where $ M = max_{1 < i < m}\sum_{j = 1}^n{a_{i,j}}.$

Determine whether there exists an arithimethical progression consisting of 40 terms and each of whose terms can be written in the form $ 2^m + 3^n$ or not. where $ m,n$ are nonnegative integers.

Given that circle $ \omega$ is tangent internally to circle $ \Gamma$ at $ S.$ $ \omega$ touches the chord $ AB$ of $ \Gamma$ at $ T$. Let $ O$ be the center of $ \omega.$ Point $ P$ lies on the line $ AO.$ Show that $ PB\perp AB$ if and only if $ PS\perp TS.$

Let $ n,k$ be given positive integers satisfying $ k\le 2n - 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m = f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.

Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X = \sum_{i = 1}^{m}x,Y = \sum_{j = 1}^{n}y.$ Prove that $ 2XY\sum_{i = 1}^{m}\sum_{j = 1}^{n}|x_{i} - y_{j}|\ge X^2\sum_{j = 1}^{n}\sum_{l = 1}^{n}|y_{i} - y_{l}| + Y^2\sum_{i = 1}^{m}\sum_{k = 1}^{m}|x_{i} - x_{k}|$

In convex pentagon $ ABCDE$, denote by $ AD\cap BE = F,BE\cap CA = G,CA\cap DB = H,DB\cap EC = I,EC\cap AD = J; AI\cap BE = A',BJ%Error. "capCA" is a bad command. = B',CF%Error. "capDB" is a bad command. = C',DG\cap EC = D',EH\cap AD = E'.$ Prove that $ \frac {AB'}{B'C}\cdot\frac {CD'}{D'E}\cdot\frac {EA'}{A'B}\cdot\frac {BC'}{C'D}\cdot\frac {DE'}{E'A} = 1$.

Find all the pairs of integers $ (a,b)$ satisfying $ ab(a - b)\not = 0$ such that there exists a subset $ Z_{0}$ of set of integers $ Z,$ for any integer $ n$, exactly one among three integers $ n,n + a,n + b$ belongs to $ Z_{0}$.

Consider function $ f: R\to R$ which satisfies the conditions for any mutually distinct real numbers $ a,b,c,d$ satisfying $ \frac {a - b}{b - c} + \frac {a - d}{d - c} = 0$, $ f(a),f(b),f(c),f(d)$ are mutully different and $ \frac {f(a) - f(b)}{f(b) - f(c)} + \frac {f(a) - f(d)}{f(d) - f(c)} = 0.$ Prove that function $ f$ is linear

Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$

In convex quadrilateral $ ABCD$, $ CB,DA$ are external angle bisectors of $ \angle DCA,\angle CDB$, respectively. Points $ E,F$ lie on the rays $ AC,BD$ respectively such that $ CEFD$ is cyclic quadrilateral. Point $ P$ lie in the plane of quadrilateral $ ABCD$ such that $ DA,CB$ are external angle bisectors of $ \angle PDE,\angle PCF$ respectively. $ AD$ intersects $ BC$ at $ Q.$ Prove that $ P$ lies on $ AB$ if and only if $ Q$ lies on segment $ EF$.

Let $ f(x)$ be a $ n -$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x = 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k + 1} - 1.$

Given that points $ D,E$ lie on the sidelines $ AB,BC$ of triangle $ ABC$, respectively, point $ P$ is in interior of triangle $ ABC$ such that $ PE = PC$ and $ \bigtriangleup DEP\sim \bigtriangleup PCA.$ Prove that $ BP$ is tangent of the circumcircle of triangle $ PAD.$

Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$

Let $ X$ be a set containing $ 2k$ elements, $ F$ is a set of subsets of $ X$ consisting of certain $ k$ elements such that any one subset of $ X$ consisting of $ k - 1$ elements is exactly contained in an element of $ F.$ Show that $ k + 1$ is a prime number.

Let $ n$ be a composite. Prove that there exists positive integer $ m$ satisfying $ m|n, m\le\sqrt {n},$ and $ d(n)\le d^3(m).$ Where $ d(k)$ denotes the number of positive divisors of positive integer $ k.$

In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$

Let nonnegative real numbers $ a_{1},a_{2},a_{3},a_{4}$ satisfy $ a_{1} + a_{2} + a_{3} + a_{4} = 1.$ Prove that $ max\{\sum_{1}^4{\sqrt {a_{i}^2 + a_{i}a_{i - 1} + a_{i - 1}^2 + a_{i - 1}a_{i - 2}}},\sum_{1}^4{\sqrt {a_{i}^2 + a_{i}a_{i + 1} + a_{i + 1}^2 + a_{i + 1}a_{i + 2}}}\}\ge 2.$ Where for all integers $ i, a_{i + 4} = a_{i}$ holds.

Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n-1,$ then $ a^b > \frac {3^n}{n}.$

Find all complex polynomial $ P(x)$ such that for any three integers $ a,b,c$ satisfying $ a + b + c\not = 0, \frac{P(a) + P(b) + P(c)}{a + b + c}$ is an integer.

Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$