Define a sequence $\{a_n\}$ by\[S_1=1,\ S_{n+1}=\frac{(2+S_n)^2}{ 4+S_n} (n=1,\ 2,\ 3,\ \cdots).\] Where $S_n$ the sum of first $n$ terms of sequence $\{a_n\}$. For any positive integer $n$ ,prove that\[a_{n}\ge \frac{4}{\sqrt{9n+7}}.\]

Jorge chooses $6$ different positive integers and writes one on each face of a cube. He threw his bucket three times. The first time his cube showed the number $5$ facing up and also the sum of the numbers on the faces sides was $20$. The second time his cube showed the number $7$ facing up and also the sum of the numbers on the faces sides was $17$. The third time his cube showed the number $4$ up, plus all the numbers on the side faces. They turned out to be primes. What are the numbers that Jorge chose and how did he distribute them on the faces of the cube? Analyze all odds. Remember that $1$ is not prime.

Let $\vartriangle ABC$ be a triangle with $AB = 4$ and $AC = \frac72$ . Let $\omega$ denote the $A$-excircle of $\vartriangle ABC$. Let $\omega$ touch lines $AB$, $AC$ at the points $D$, $E$, respectively. Let $\Omega$ denote the circumcircle of $\vartriangle ADE$. Consider the line $\ell$ parallel to $BC$ such that $\ell$ is tangent to $\omega$ at a point $F$ and such that $\ell$ does not intersect $\Omega$. Let $\ell$ intersect lines $AB$, $AC$ at the points $X$, $Y$ , respectively, with $XY = 18$ and $AX = 16$. Let the perpendicular bisector of $XY$ meet the circumcircle of $\vartriangle AXY$ at $P$, $Q$, where the distance from $P$ to $F$ is smaller than the distance from $Q$ to$ F$. Let ray $\overrightarrow {PF}$ meet $\Omega$ for the first time at the point $Z$. If $PZ^2 = \frac{m}{n}$ for relatively prime positive integers $m$, $n$, find $m + n$.

In the triangle $ABC$, $\angle{B} = 120^{\circ}$, point $M$ is the midpoint of side $AC$. On the sides $AB$ and $BC$, the points $K$ and $L$ are chosen such that $KL \parallel AC$. Prove that $MK + ML \geq MA$.

Find all pairs of nonnegative integers $ (x, y)$ satisfying $ (14y)^x \plus{} y^{x\plus{}y} \equal{} 2007$.

Andy and Harry are trying to make an O for the MOAA logo. Andy starts with a circular piece of leather with radius 3 feet and cuts out a circle with radius 2 feet from the middle. Harry starts with a square piece of leather with side length 3 feet and cuts out a square with side length 2 feet from the middle. In square feet, what is the positive difference in area between Andy and Harry's final product to the nearest integer? [i]Proposed by Andy Xu[/i]

$p$ is a prime. Find the all $(m,n,p)$ positive integer triples satisfy $m^3+7p^2=2^n$.

Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?

Circle $\omega$ has radius $5$ and is centered at $O$. Point $A$ lies outside $\omega$ such that $OA=13$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$. Compute $AB+AC$ given that $BC=7$.

Find all polynomials $ P (x) $ with real coefficients that satisfy the identity $ P (x ^ 2) = P (x) P (x + 1) $.

How many different sets of three points in this equilateral triangular grid are the vertices of an equilateral triangle? Justify your answer. [center][img]http://i.imgur.com/S6RXkYY.png[/img][/center]

$a,b,c$ are non-negative integers. Solve: $a!+5^b=7^c$ [i]Proposed by Serbia[/i]

Suppose each element $i\in S=\{1,2,\ldots,n\}$ is assigned a nonempty set $S_i\subseteq S$ so that the following conditions are fulfilled: (i) for any $i,j\in S$, if $j\in S_i$ then $i\in S_j$; (ii) for any $i,j\in S$, if $|S_i|=|S_j|$ then $S_i\cap S_j=\emptyset$. Prove that there exists $k\in S$ for which $|S_k|=1$.

Let $a$, $b$ and $c$ be real numbers larger than $1$. Prove the inequality $$\frac{ab}{c-1}+\frac{bc}{a - 1}+\frac{ca}{b -1} \ge 12.$$ When does equality hold? [i](Karl Czakler)[/i]

It is intended to make a map locating $30$ different cities on it. For this, all the distances between these cities are available as data (each of these distances is considered as a “data”). Three of these cities are already laid out on the map, and they turn out to be non-collinear. How much data must be used as a minimum to complete the map?

Let $n\ge3$ be an integer. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.

In an acute angle $ \vartriangle ABC $, let $ AD, BE, CF $ be their altitudes (with $ D, E, F $ lying on $ BC, CA, AB $, respectively). Let's call $ O, H $ the circumcenter and orthocenter of $ \vartriangle ABC $, respectively. Let $ P = CF \cap AO $. Suppose the following two conditions are true: $\bullet$ $ FP = EH $ $\bullet$ There is a circle that passes through points $ A, O, H, C $ Prove that the $ \vartriangle ABC $ is equilateral.

Let be a natural number $ n\ge 2. $ Find the real numbers $ a $ that satisfy the equation $$ \lfloor nx \rfloor =\sum_{k=1}^{n} \lfloor x+(k-1)a \rfloor , $$ for any real numbers $ x. $ [i]Marius Perianu[/i]

Write the number $2004_{(5)}$ [ $2004$ base $5$ ] as a number in base $6$.

Estimate the number of positive integers less than or equal to $1,000,000$ that can be expressed as the sum of two nonnegative integer squares. Your estimate must be an integer, or you will receive a zero.

Evaluate $\int_0^{\frac{\pi}{4}} \frac{1}{1-\sin x}\sqrt{\frac{\cos x}{1+\cos x+\sin x}}dx.$ Own

Let $a$ and $b$ be parallel lines with $50$ distinct points marked on $a$ and $50$ distinct points marked on $b$. Find the greatest possible number of acute-angled triangles all of whose vertices are marked.

Determine for which values of $n$ it is possible to tile a square of side $n$ with figures of the type shown in the picture [asy] unitsize(0.4 cm); draw((0,0)--(5,0)); draw((0,1)--(5,1)); draw((1,2)--(4,2)); draw((2,3)--(3,3)); draw((0,0)--(0,1)); draw((1,0)--(1,2)); draw((2,0)--(2,3)); draw((3,0)--(3,3)); draw((4,0)--(4,2)); draw((5,0)--(5,1)); [/asy]

Solve the diophantine equation \[x^{2008}- y^{2008} = 2^{2009}.\]

Lavidópolis is a city with 2024 neighborhoods. Lavi Dopes was elected mayor, and since he saw that there were no roads in the city, he asked Gil Bento, the monster engineer, to design the city's roads according to the following rules: 1. Any two neighborhoods are connected by at most one two-way road; 2. For any two neighborhoods, there is exactly one route from one neighborhood to another, which may pass through some intermediate neighborhoods, but never passes through the same neighborhood more than once. Mayor Lavi Dopes wants to try for re-election, but since he knows nothing about the city and only shows up during campaign times (he spent all this time stealing... I mean, thinking about math problems), he wants to find a pair of neighborhoods such that the number of roads that are part of the route connecting them is maximized among all pairs of neighborhoods. To do this, he starts asking Gil Bento various questions, all in the following manner: he chooses two of the 2024 neighborhoods, say A and B, and asks: "Given neighborhoods A and B, how many roads are part of the route connecting A to B?" Knowing that Gil Bento always answers correctly to each question, determine the minimum number of questions that Lavi Dopes needs to ask to achieve his goal, regardless of how Gil Bento has designed the roads of Lavidópolis.