A company with $2004$ workers celebrated its anniversary by inviting everyone to a lunch served at a round table. When the $2004$ workers sat around this table, they formed a circle of people and soon discovered that they all had salaries. different and also that the difference between the salaries of any two neighbors, at the round table, was $2000$ or $3000$ pesos. Calculate the maximum difference that can exist between the wages of these workers.

The circumcircle of triangle $ABC$ has centre $O$. $P$ is the midpoint of $\widehat{BAC}$ and $QP$ is the diameter. Let $I$ be the incentre of $\triangle ABC$ and let $D$ be the intersection of $PI$ and $BC$. The circumcircle of $\triangle AID$ and the extension of $PA$ meet at $F$. The point $E$ lies on the line segment $PD$ such that $DE=DQ$. Let $R,r$ be the radius of the inscribed circle and circumcircle of $\triangle ABC$, respectively. Show that if $\angle AEF=\angle APE$, then $\sin^2\angle BAC=\dfrac{2r}R$

Let $r > 0$ be a real number. All the interior points of the disc $D(r)$ of radius $r$ are colored with one of two colors, red or blue. [list][*]If $r > \frac{\pi}{\sqrt{3}}$, show that we can find two points $A$ and $B$ in the interior of the disc such that $AB = \pi$ and $A,B$ have the same color [*]Does the conclusion in (a) hold if $r > \frac{\pi}{2}$?[/list] [i]Proposed by S Muralidharan[/i]

Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ mintues. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m=a-b\sqrt{c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$

Define a function $f:\mathbb{N}\rightarrow\mathbb{N}$, \[f(1)=p+1,\] \[f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,\] where $p$ is a prime number. Find all $p$ such that there exists a natural number $k$ such that $f(k)$ is a perfect square.

$a,b,c,d$ are positive numbers such that $\sum_{cyc} \frac{1}{ab} =1$. Prove that : $abcd+16 \geq 8 \sqrt{(a+c)(\frac{1}{a} + \frac{1}{c})}+8\sqrt{(b+d)(\frac{1}{b}+\frac{1}{d})}$

Three friends have a total of $6$ identical pencils, and each one has at least one pencil. In how many ways can this happen? $\textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12$

Find all polynomials $P$ with integer coefficients such that $P (0)\ne 0$ and $$P^n(m)\cdot P^m(n)$$ is a square of an integer for all nonnegative integers $n, m$. [i]Remark:[/i] For a nonnegative integer $k$ and an integer $n$, $P^k(n)$ is defined as follows: $P^k(n) = n$ if $k = 0$ and $P^k(n)=P(P(^{k-1}(n))$ if $k >0$. Proposed by Adrian Beker.

Depei is imprisoned by an evil wizard and is coerced to play the following game. Every turn, Depei flips a fair coin. Then, the following events occur in this order: $\bullet$ The wizard computes the difference between the total number of heads and the total number of tails Depei has flipped. If that number is greater than or equal to $4$ or less than or equal to $-3$, then Depei is vaporized by the wizard. $\bullet$ The wizard determines if Depei has flipped at least $10$ heads or at least $10$ tails. If so, then the wizard releases Depei from the prison. The probability that Depei is released by the evil wizard equals $\frac{m}{2^k}$ , where $m, k$ are positive integers. Compute $m+k$.

Let $a,b$ positive real numbers. Prove that $$\frac{(a+b)^3}{a^2b}\ge \frac{27}{4}.$$ When does equality occur?

Evaluate $\lim_{x\to 1^-}\prod_{n=0}^{\infty}\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.

Prove the following two statements: (a) If a triangle is isosceles, then two of its bisectors are of equal length. (b) If two angle bisectors in a triangle are of equal length, then it is isosceles.

Let $P$ be a point in the interior of triangle $ABC$. Let $X$, $Y$, and $Z$ be points on sides $BC$, $AC$, and $AB$ respectively, such that $<PXC=<PYA=<PZB$. Let $U$, $V$, and $W$ be points on sides $BC$, $AC$, and $AB$, respectively, or on their extensions if necessary, with $X$ in between $B$ and $U$, $Y$ in between $C$ and $V$, and $Z$ in between $A$ and $W$, such that $PU=2PX$, $PV=2PY$, and $PW=2PZ$. If the area of triangle $XYZ$ is $1$, find the area of triangle $UVW$.

On the plane 1000 lines are drawn, among which there are no parallel lines. From any seven of these lines, some three pass through one point. But no more than 500 lines pass through each point. Prove that there are three points such that each line contains at least of of them.

Let a positive integer number $n$ has $k$ different prime divisors. Prove that there exists a positive integer number $x \in \left(1, \frac{n}{k}+1 \right)$ such that $x^2-x$ divides by $n$.

Suppose $H$ and $O$ are orthocenter and circumcenter of triangle $ABC$. $\omega$ is circumcircle of $ABC$. $AO$ intersects with $\omega$ at $A_1$. $A_1H$ intersects with $\omega$ at $A'$ and $A''$ is the intersection point of $\omega$ and $AH$. We define points $B',\ B'',\ C'$ and $C''$ similiarly. Prove that $A'A'',B'B''$ and $C'C''$ are concurrent in a point on the Euler line of triangle $ABC$.

The angles $B$ and $C$ of an acute-angled​ triangle $ABC$ are greater than $60^\circ$. Points $P,Q$ are chosen on the sides $AB,AC$ respectively so that the points $A,P,Q$ are concyclic with the orthocenter $H$ of the triangle $ABC$. Point $K$ is the midpoint of $PQ$. Prove that $\angle BKC > 90^\circ$. [i]Proposed by A. Mudgal[/i]

$N \ge 3$ different points are marked on the plane. It is known that among pairwise distances between marked points there are not more than $n$ different distances. Prove that $N \le (n + 1)^2$.

In the adjoining figure of a rectangular solid, $\angle DHG=45^\circ$ and $\angle FHB=60^\circ$. Find the cosine of $\angle BHD$. [asy] size(200); import three;defaultpen(linewidth(0.7)+fontsize(10)); currentprojection=orthographic(1/3+1/10,1-1/10,1/3); real r=sqrt(3); triple A=(0,0,r), B=(0,r,r), C=(1,r,r), D=(1,0,r), E=O, F=(0,r,0), G=(1,0,0), H=(1,r,0); draw(D--G--H--D--A--B--C--D--B--F--H--B^^C--H); draw(A--E^^G--E^^F--E, linetype("4 4")); label("$A$", A, N); label("$B$", B, dir(0)); label("$C$", C, N); label("$D$", D, W); label("$E$", E, NW); label("$F$", F, S); label("$G$", G, W); label("$H$", H, S); triple H45=(1,r-0.15,0.1), H60=(1-0.05, r, 0.07); label("$45^\circ$", H45, dir(125), fontsize(8)); label("$60^\circ$", H60, dir(25), fontsize(8));[/asy] $\textbf {(A) } \frac{\sqrt{3}}{6} \qquad \textbf {(B) } \frac{\sqrt{2}}{6} \qquad \textbf {(C) } \frac{\sqrt{6}}{3} \qquad \textbf {(D) } \frac{\sqrt{6}}{4} \qquad \textbf {(E) } \frac{\sqrt{6}-\sqrt{2}}{4}$

Let $n$ be an odd integer and $m=\phi(n)$ be the Euler's totient function. Call a set of residues $T=\{a_1, \cdots, a_k\} \pmod n$ to be [i]good[/i] if $\gcd(a_i, n) > 1$ $\forall i$, and $\gcd(a_i, a_j) = 1, \forall i \neq j$. Define the set $S_n$ consisting of the residues $$\sum_{i=1}^k a_i ^m\pmod{n}$$ over all possible residue sets $T=\{a_1,\cdots,a_k\}$ that is good. Determine $|S_n|$. [i]Proposed by Anzo Teh Zhao Yang[/i]

Let us denote by $s(n)= \sum_{d|n} d$ the sum of divisors of a positive integer $n$ ($1$ and $n$ included). If $n$ has at most $5$ distinct prime divisors, prove that $s(n) < \frac{77}{16} n.$ Also prove that there exists a natural number $n$ for which $s(n) < \frac{76}{16} n$ holds.

In the circle $\omega$, we draw a chord $BC$, which is not a diameter. Point $A$ moves in a circle $\omega$. $H$ is the orthocenter triangle $ABC$. Prove that for any location of point $A$, a circle constructed on $AH$ as on diameter, touches two fixed circles $\omega_1$ and $\omega_2$. (Dmitry Prokopenko)

An $m\times n(m,n\in \mathbb{Z}_+)$ rectangle is divided into some smaller squares. All sides of each square are parallel to the sides of the rectangle, and the length of each side is an integer. Determine the minimum value of the sum of the lengths of sides of these squares.

(a) Is $2005^{2004}$ the sum of two perfect squares? (b) Is $2004^{2005}$ the sum of two perfect squares?

Points $M$ and $N$ are the midpoints of sides $AB$ and $BC$ of the square $ABCD$. According to the fgure, we have drawn a regular hexagon and a regular $12$-gon. The points $P, Q$ and $R$ are the centers of these three polygons. Prove that $PQRS$ is a cyclic quadrilateral. [i]Proposed by Mahdi Etesamifard - Iran[/i]