Prove that for positive $a,b,c,d$ \[\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a}\ge 4\]

Let a cyclic quadrilateral $ ABCD$, $ AC \cap BD \equal{} E$ and let a circle $ \Gamma$ internally tangent to the arch $ BC$ (that not contain $ D$) in $ T$ and tangent to $ BE$ and $ CE$. Call $ R$ the point where the angle bisector of $ \angle ABC$ meet the angle bisector of $ \angle BCD$ and $ S$ the incenter of $ BCE$. Prove that $ R$, $ S$ and $ T$ are collinear. [i](Gabriel Giorgieri)[/i]

We define the polynomial $f(x)$ in $\mathbb R[x]$ as follows: $f(x)=x^n+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+.....+a_1x+a_0$ Prove that there exists an $i$ in the set $\{1,....,n\}$ such that we have $|f(i)|\ge \frac{n!}{\dbinom{n}{i}}$. [i]proposed by Mohammadmahdi Yazdi[/i]

Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$. Remark: "Natural numbers" is the set of positive integers.

Two circles are tangent to each other internally at a point $\ T $. Let the chord $\ AB $ of the larger circle be tangent to the smaller circle at a point $\ P $. Prove that the line $\ TP $ bisects $\ \angle ATB $.

Prove that there are real numbers $ a_1, a_2, ..$ such that: i) For all real numbers x, the serie $ f(x) \equal{} \sum_{n \equal{} 1}^\infty a_nx^n$ converge; ii) f is a bijection of R to R; iii) f'(x) >0; iv) f(Q) = A, where Q is the set of rational numbers and A is the set of algebraic numbers.

Each point of a plane is painted in one of three colors. Show that there exists a triangle such that: $ (i)$ all three vertices of the triangle are of the same color; $ (ii)$ the radius of the circumcircle of the triangle is $ 2008$; $ (iii)$ one angle of the triangle is either two or three times greater than one of the other two angles.

Let $ ABC$ be a triangle. Squares $ AB_{c}B_{a}C$, $ CA_{b}A_{c}B$ and $ BC_{a}C_{b}A$ are outside the triangle. Square $ B_{c}B_{c}'B_{a}'B_{a}$ with center $ P$ is outside square $ AB_{c}B_{a}C$. Prove that $ BP,C_{a}B_{a}$ and $ A_{c}B_{c}$ are concurrent.

Which statement is not true for at least one prime $p$? $ \textbf{(A)}\ \text{If } x^2+x+3 \equiv 0 \pmod p \text{ has a solution, then } \\ \qquad x^2+x+25 \equiv 0 \pmod p \text{ has a solution.} \\ \\ \textbf{(B)}\ \text{If } x^2+x+3 \equiv 0 \pmod p \text{ does not have a solution, then} \\ \qquad x^2+x+25 \equiv 0 \pmod p \text{ has no solution} \\ \\ \qquad\textbf{(C)}\ \text{If } x^2+x+25 \equiv 0 \pmod p \text{ has a solution, then} \\ \qquad x^2+x+3 \equiv 0 \pmod p \text{ has a solution}. \\ \\ \qquad\textbf{(D)}\ \text{If } x^2+x+25 \equiv 0 \pmod p \text{ does not have a solution, then} \\ \qquad x^2+x+3 \equiv 0 \pmod p \text{ has no solution. } \\ \\ \qquad\textbf{(E)}\ \text{None} $

How many positive integers not exceeding $ 2001$ are multiples of $ 3$ or $ 4$ but not $ 5$? $ \textbf{(A)}\ 768 \qquad \textbf{(B)}\ 801 \qquad \textbf{(C)}\ 934 \qquad \textbf{(D)}\ 1067 \qquad \textbf{(E)}\ 1167$

If $ f$ is a nonnegative, continuous, concave function on the closed interval $ [0,1]$ such that $ f(0)=1$, then \[ \int_0^1 xf(x)dx \leq \frac 23 \left[ %Error. "diaplaymath" is a bad command. \int_0^1 f(x)dx \right]^2.\] [i]Z. Daroczy[/i]

Consider the multiplicative group $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$. Find the number of homomorphisms $f:A\to A$ that satisfy $f(f(x))=f(x)$ for all elements $x\in A$.

Let $ABC$ be an acute triangle. Determine the locus of points $M$ in the interior of the triangle such that $AB-FG=\frac{MF \cdot AG+MG \cdot BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to lines $BC$ and $AC$, respectively.

For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers? $ \textbf{(A)}\ 794\qquad \textbf{(B)}\ 796\qquad \textbf{(C)}\ 798\qquad \textbf{(D)}\ 800\qquad \textbf{(E)}\ 802$

Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. Examples of such sequences are $ AA$, $ B$, and $ AABAA$, while $ BBAB$ is not such a sequence. How many such sequences have length 14?

Prove that there exist infinitely many integers $n$ such that $n, n+1, n+2$ are each the sum of the squares of two integers.

Let $ P$ be a polynomial with integer coefficients such that there are two distinct integers at which $ P$ takes coprime values. Show that there exists an infinite set of integers, such that the values $ P$ takes at them are pairwise coprime.

Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.

A right circular cone has base radius $ r$ and height $ h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making $ 17$ complete rotations. The value of $ h/r$ can be written in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

show that $\frac1{15} < \frac12\cdot\frac34\cdots\frac{99}{100} < \frac1{10}$

Let $p$ be an odd prime. If $g_{1}, \cdots, g_{\phi(p-1)}$ are the primitive roots $\pmod{p}$ in the range $1<g \le p-1$, prove that \[\sum_{i=1}^{\phi(p-1)}g_{i}\equiv \mu(p-1) \pmod{p}.\]

Find the sum of all positive integers $n$ such that \[ \frac{2n+1}{n(n-1)} \] has a terminating decimal representation. [i]Proposed by Evan Chen[/i]

Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$, $x$ divides at most one other element in $A$. Prove that \[\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil. \]

Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.

Show that the center of gravity of a convex region in the plane halves at least three chords of the region. [Gy. Hajos]