Let $ x_1,x_2,\dots,x_n$ be positive real numbers. Let $ m=\min\{x_1,x_2,\dots,x_n\}$, $ M=\max\{x_1,x_2,\dots,x_n\}$, $ A=\frac{1}{n}(x_1+x_2+\dots+x_n)$, and $ G=\sqrt[n]{x_1x_2 \dots x_n}$. Prove that \[ A-G \ge \frac{1}{n}(\sqrt{M}-\sqrt{m})^2.\]

Two cirlces $ C_1$ and $ C_2$, with center $ O_1$ and $ O_2$ respectively, intersect at $ A$ and $ B$. Let $ O_1$ lies on $ C_2$. A line $ l$ passes through $ O_1$ but does not pass through $ O_2$. Let $ P$ and $ Q$ be the projection of $ A$ and $ B$ respectively on the line $ l$ and let $ M$ be the midpoint of $ \overline{AB}$. Prove that $ MPQ$ is an isoceles triangle.

Let $ n \ge 2009$ be an integer and define the set: \[ S = \{2^x|7 \le x \le n, x \in \mathbb{N}\}. \] Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that \[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}. \]

Sixteen people for groups of four people such that each two groups have at most two members in common. Prove that there exists a set of six people in which every group is not properly contained in it.

Ati has $ 7$ pots of flower, ordered in $ P_1,P_2,P_3,P_4,P_5,P_6,P_7$. She wants to rearrange the position of those pots to $ B_1,B_2,B_2,B_3,B_4,B_5,B_6,B_7$ such that for every positive integer $ n<7$, $ B_1,B_2,\dots,B_n$ is not the permutation of $ P_1,P_2,\dots,P_7$. In how many ways can Ati do this?

Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x+\log_2 (2^x-3a)}{1+\log_2 a} >2.\]

Let $ C_1$ be a circle and $ P$ be a fixed point outside the circle $ C_1$. Quadrilateral $ ABCD$ lies on the circle $ C_1$ such that rays $ AB$ and $ CD$ intersect at $ P$. Let $ E$ be the intersection of $ AC$ and $ BD$. (a) Prove that the circumcircle of triangle $ ADE$ and the circumcircle of triangle $ BEC$ pass through a fixed point. (b) Find the the locus of point $ E$.

Given positive integer $ n > 1$ and define \[ S = \{1,2,\dots,n\}. \] Suppose \[ T = \{t \in S: \gcd(t,n) = 1\}. \] Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)

Find the smallest odd integer $ k$ such that: for every $ 3-$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.

Prove that there exists two different permutations $ (a_1,a_2,\dots,a_{2009})$ and $ (b_1,b_2,\dots,b_{2009})$ of $ (1,2,\dots,2009)$ such that \[ \sum_{i=1}^{2009}i^i a_i - \sum_{i=1}^{2009} i^i b_i\] is divisible by $ 2009!$.

Find all function $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(x + y)(f(x) - y) = xf(x) - yf(y) \] for all $ x,y \in \mathbb{R}$.

Given triangle $ ABC$ with $ AB>AC$. $ l$ is tangent line of the circumcircle of triangle $ ABC$ at $ A$. A circle with center $ A$ and radius $ AC$, intersect $ AB$ at $ D$ and $ l$ at $ E$ and $ F$. Prove that the lines $ DE$ and $ DF$ pass through the incenter and excenter of triangle $ ABC$.

Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i + 1}$ have different color for each $ i = 1,2,\dots,n$ where $ a_{n + 1}=a_1$. Find the number of ways to do such coloring.

For every positive integer $ n$, let $ \phi(n)$ denotes the number of positive integers less than $ n$ that is relatively prime to $ n$ and $ \tau(n)$ denote the sum of all positive divisors of $ n$. Let $ n$ be a positive integer such that $ \phi(n)|n-1$ and that $ n$ is not a prime number. Prove that $ \tau(n)>2009$.

Let $ ABC$ be an isoceles triangle with $ AC=BC$. A point $ P$ lies inside $ ABC$ such that \[ \angle PAB = \angle PBC, \angle PAC = \angle PCB.\] Let $ M$ be the midpoint of $ AB$ and $ K$ be the intersection of $ BP$ and $ AC$. Prove that $ AP$ and $ PK$ trisect $ \angle MPC$.

Let $ a$, $ b$, and $ c$ be positive real numbers such that $ ab + bc + ca = 3$. Prove the inequality \[ 3 + \sum_{\mathrm{\cyc}} (a - b)^2 \ge \frac {a + b^2c^2}{b + c} + \frac {b + c^2a^2}{c + a} + \frac {c + a^2b^2}{a + b} \ge 3. \]

Let $ ABC$ be a triangle. A circle $ P$ is internally tangent to the circumcircle of triangle $ ABC$ at $ A$ and tangent to $ BC$ at $ D$. Let $ AD$ meets the circumcircle of $ ABC$ agin at $ Q$. Let $ O$ be the circumcenter of triangle $ ABC$. If the line $ AO$ bisects $ \angle DAC$, prove that the circle centered at $ Q$ passing through $ B$, circle $ P$, and the perpendicular line of $ AD$ from $ B$, are all concurrent.

Find the formula to express the number of $ n-$series of letters which contain an even number of vocals (A,I,U,E,O).

Find all triples $ (x,y,z)$ of positive real numbers which satisfy $ 2x^3 = 2y(x^2 + 1) - (z^2 + 1)$; $ 2y^4 = 3z(y^2 + 1) - 2(x^2 + 1)$; $ 2z^5 = 4x(z^2 + 1) - 3(y^2 + 1)$.

Let $ n>1$ be an odd integer and define: \[ N=\{-n,-(n-1),\dots,-1,0,1,\dots,(n-1),n\}.\] A subset $ P$ of $ N$ is called basis if we can express every element of $ N$ as the sum of $ n$ different elements of $ P$. Find the smallest positive integer $ k$ such that every $ k-$elements subset of $ N$ is basis.

Prove that for all odd $ n > 1$, we have $ 8n + 4|C^{4n}_{2n}$.

Let $ f(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\cdots+a_1x+a_0$, with $ a_i=a_{2n-1}$ for all $ i=1,2,\ldots,n$ and $ a_{2n}\ne0$. Prove that there exists a polynomial $ g(x)$ of degree $ n$ such that $ g\left(x+\frac1x\right)x^n=f(x)$.

In how many ways we can choose 3 non empty and non intersecting subsets from $ (1,2,\ldots,2008)$.

Given triangle $ ABC$. Let the tangent lines of the circumcircle of $ AB$ at $ B$ and $ C$ meet at $ A_0$. Define $ B_0$ and $ C_0$ similarly. a) Prove that $ AA_0,BB_0,CC_0$ are concurrent. b) Let $ K$ be the point of concurrency. Prove that $ KG\parallel BC$ if and only if $ 2a^2=b^2+c^2$.

Given an $ n\times n$ chessboard. a) Find the number of rectangles on the chessboard. b) Assume there exists an $ r\times r$ square (label $ B$) with $ r<n$ which is located on the upper left corner of the board. Define "inner border" of $ A$ as the border of $ A$ which is not the border of the chessboard. How many rectangles in $ B$ that touch exactly one inner border of $ B$?

Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that \[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. \]

Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2+y^2+z^2$ if $ x^3+y^3+z^3-3xyz=1$.

Prove that there exist infinitely many positive integers $ n$ such that $ n!$ is not divisible by $ n^2+1$.

a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

Consider the following array: \[ 3, 5\\3, 8, 5\\3, 11, 13, 5\\3, 14, 24, 18, 5\\3, 17, 38, 42, 23, 5\\ \ldots \] Find the 5-th number on the $ n$-th row with $ n>5$.

Let $ ABC$ be an acute triangle with $ \angle BAC=60^{\circ}$. Let $ P$ be a point in triangle $ ABC$ with $ \angle APB=\angle BPC=\angle CPA=120^{\circ}$. The foots of perpendicular from $ P$ to $ BC,CA,AB$ are $ X,Y,Z$, respectively. Let $ M$ be the midpoint of $ YZ$. a) Prove that $ \angle YXZ=60^{\circ}$ b) Prove that $ X,P,M$ are collinear.

Let $ S$ be the set of nonnegative real numbers. Find all functions $ f: S\rightarrow S$ which satisfy $ f(x+y-z)+f(2\sqrt{xz})+f(2\sqrt{yz})=f(x+y+z)$ for all nonnegative $ x,y,z$ with $ x+y\ge z$.

Let $ [a]$ be the integer such that $ [a]\le a<[a]+1$. Find all real numbers $ (a,b,c)$ such that \[ \{a\}+[b]+\{c\}=2.9\\\{b\}+[c]+\{a\}=5.3\\\{c\}+[a]+\{b\}=4.0.\]

Let $ ABC$ be a triangle with $ \angle BAC=60^{\circ}$. The incircle of $ ABC$ is tangent to $ AB$ at $ D$. Construct a circle with radius $ DA$ and cut the incircle of $ ABC$ at $ E$. If $ AF$ is an altitude, prove that $ AE\ge AF$.

Find integer $ n$ with $ 8001 < n < 8200$ such that $ 2^n - 1$ divides $ 2^{k(n - 1)! + k^n} - 1$ for all integers $ k > n$.

2008 boys and 2008 girls sit on 4016 chairs around a round table. Each boy brings a garland and each girl brings a chocolate. In an "activity", each person gives his/her goods to the nearest person on the left. After some activities, it turns out that all boys get chocolates and all girls get garlands. Find the number of possible arrangements.

2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.

Let $ x_1,x_2,\ldots,x_n$ be real numbers greater than 1. Show that \[ \frac{x_1x_2}{x_3}+\frac{x_2x_3}{x_4}+\cdots+\frac{x_nx_1}{x_2}\ge4n\] and determine when the equality holds.

Let $ S=\{1,2,\ldots,n\}$. Let $ A$ be a subset of $ S$ such that for $ x,y\in A$, we have $ x+y\in A$ or $ x+y-n\in A$. Show that the number of elements of $ A$ divides $ n$.

Let $ ABCD$ be a convex quadrilateral. Let $ M,N$ be the midpoints of $ AB,AD$ respectively. The foot of perpendicular from $ M$ to $ CD$ is $ K$, the foot of perpendicular from $ N$ to $ BC$ is $ L$. Show that if $ AC,BD,MK,NL$ are concurrent, then $ KLMN$ is a cyclic quadrilateral.