Let a,b, and c be positive reals. Prove: $\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\ge (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying \[f(x+f(y))=x+f(f(y))\] for all real numbers $x$ and $y$, with the additional constraint $f(2004)=2005$.

Prove for every irrational real number a, there are irrational numbers b and b' such that a+b and ab' are rational while a+b' and ab are irrational.

Let $n$ be a positive integer. Solve the system of equations \begin{align*}x_{1}+2x_{2}+\cdots+nx_{n}&= \frac{n(n+1)}{2}\\ x_{1}+x_{2}^{2}+\cdots+x_{n}^{n}&= n\end{align*} for $n$-tuples $(x_{1},x_{2},\ldots,x_{n})$ of nonnegative real numbers.

Let x, y be reals satisfying: sin x+cos y=1 sin y+cos x=-1 Prove cos 2x=cos 2y

Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can not use a needle to peer through the tiling? Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can use a needle to through the tiling? What if it is a $6 \times 6$ board?

Mykolka the numismatist possesses $241$ coins, each worth an integer number of turgiks. The total value of the coins is $360$ turgiks. Is it necessarily true that the coins can be divided into three groups of equal total value?

In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won?

The squares of an n*n chessboard (n >1) are filled with 1s and -1s. A series of steps are performed. For each step, the number in each square is replaced with the product of the numbers that were in the squares adjacent. Find all values of n for which, starting from an arbitrary collection of numbers, after finitely many steps one obtains a board filled with 1s.

Find all pairs of positive integers (m, n) for which it is possible to paint each unit square of an m*n chessboard either black or white in such way that, for any unit square of the board, the number of unit squares which are painted the same color as that square and which have at least one common vertex with it (including the square itself) is even.

$ ABC$ is an acute triangle. The points $ B'$ and $ C'$are the reflections of $ B$ and $ C$ across the lines $ AC$ and $ AB$ respectively. Suppose that the circumcircles of triangles$ ABB$' and $ ACC'$ meet at $ A$ and $ P$. Prove that the line $ PA$ passes through the circumcenter of triangle$ ABC.$

In triangle $ ABC$,$ \angle BAC = 120^o$. Let the angle bisectors of angles $ A;B$and $ C$ meet the opposite sides at $ D;E$ and$ F$ respectively. Prove that the circle on diameter $ EF$ passes through $ D.$

1.14. Let P and Q be interior points of triangle ABC such that \ACP = \BCQ and \CAP = \BAQ. Denote by D;E and F the feet of the perpendiculars from P to the lines BC, CA and AB, respectively. Prove that if \DEF = 90, then Q is the orthocenter of triangle BDF.

Let $ABC$ be a right triangle with$ \angle A = 90^o$. Point $D$ lies on side $BC$ such that $\angle BAD = \angle CAD$. Point $I_a$ is the excenter of the triangle opposite $A$. Prove that $\frac{AD}{DI_a } \le \sqrt{2} -1$

Show that among the vertices of any area $1$ convex polygon with $n > 3$ sides there exist four such that the quadrilateral formed by these four has area at least $1/2$.

Prove that for positive integers $x_{1},...,x_{n}$, we have $\prod_{1\leq i<j\leq n}(j-i)|\prod_{1\leq i<j\leq n}(x_{j}-x_{i})$

Determine all pairs of positive integers $(m,n)$ such that m is but divisible by every integer from $1$ to $n$ (inclusive), but not divisible by $n + 1, n + 2$, and $n + 3$.

Let $X=\{A_{1},...,A_{n}\}$ be a set of distinct 3-element subsets of the set $\{1,2,...,36\}$ such that (a) $A_{i},A_{j}$ have nonempty intersections for all $i,j$ (b) The intersection of all elements of $X$ is the empty set. Show that $n\leq 100$. Determine the number of such sets $X$ when $n=100$

Find all pairs $(a,b)$ of positive real numbers such that $\lfloor a \lfloor bn \rfloor \rfloor =n - 1$ for all positive integers $n$.

Let $n$ be a nonnegative integer, and let $p$ be a prime number that is congruent to $7$ modulo $8$. Prove that $$\sum_{k=1}^{p} \left\{ \frac{k^{2n}}{p} - \frac{1}{2} \right\} = \frac{p-1}{2}$$

Find all functions $f : N \to N$ such that $f(m)+f(n)$ divides $m+n$ for all positive integers $m$ and $n$.

Prove that $\frac{a}{(a + 1)(b + 1)} +\frac{ b}{(b + 1)(c + 1)} + \frac{c}{(c + 1)(a + 1)} \ge \frac34$ where $a, b$ and $c$ are positive real numbers satisfying $abc = 1$.

Let $a_{1},a_{2},...,a_{n}$ be positive real numbers with $a_{1}\leq a_{2}\leq ... a_{n}$ such that the arithmetic mean of $a_{1}^{2},...,a_{n}^{2}$ is 1. If the arithmetic mean of $a_{1}, a_{2},...,a_{n}$ is $m$. Prove that if $a_{i}\leq$ m for some $1 \leq i \leq n$, then $n(m-a_{i})^2\leq n-i$

Assume that $f : [0,1)\to R$ is a function such that $f(x)-x^3$ and $f(x)-3x$ are both increasing functions. Determine if $f(x)-x^2-x$ is also an increasing function.

Let $a_1, a_2,...,a_{2005}, b_1, b_2,...,b_{2005}$ be real numbers such that $(a_ix - b_i)^2 \ge \sum_{j\ne i,j=1}^{2005} (a_jx - b_j)$ for all real numbers x and every integer $i$ with $1 \le i \le 2005$. What is maximal number of positive $a_i$'s and $b_i$'s?

Let $n$ be an integer greater than $3$. Prove that all the roots of the polynomial $P(x) = x^n - 5x^{n-1} + 12x^{n-2}- 15x^{n-3} + a_{n-4}x^{n-4} +...+ a_0$ cannot be both real and positive.

for real number $a,b,c$ in interval $ (0,1]$ prove that: $\frac{a}{bc+1}+\frac{b}{ac+1}+\frac{c}{ab+1} \leq 2$

In how many ways can the set $N ={1, 2, \cdots, n}$ be partitioned in the form $p(N) = A_{1}\cup A_{2}\cup \cdots \cup A_{k},$ where $A_{i}$ consists of elements forming arithmetic progressions, all with the same common positive difference $d_{p}$ and of length at least one? At least two?

Determine the number of subset $S$ of the set $T = {1, 2,..., 2005}$ such that the sum of elements in $s$ is congruent to 2006 modulo 2048.

Let $P_{n}$ denote the number of paths in the coordinate plane traveling from $(0, 0)$ to $(n, 0)$ with three kinds of moves: upstep $u = [1, 1]$, downstep $d = [1,-1]$, and flatstep $f = [1, 0]$ with the path always staying above the line $y = 0.$ Let $C_{n}= \frac{1}{n+1}\binom{2n}{n}$ be the $n^{th}$ Catalan number. Prove that $P_{n}= \sum_{i = 0}^\infty \binom{n}{2i}C_{i}$ and $C_{n}= \sum_{i = 0}^{2n}(-1)^{i}\binom{2n}{i}P_{2n-i}.$

For positive integers $t,a$, and $b$, Lucy and Windy play the $(t,a,b)$- game defined by the following rules. Initially, the number $t$ is written on a blackboard. On her turn, a player erases the number on the board and writes either the number $t - a$ or $t - b$ on the board. Lucy goes first and then the players alternate. The player who first reaches a negative losses the game. Prove that there exist infinitely many values of $t$ in which Lucy has a winning strategy for all pairs $(a, b)$ with $a + b = 2005$.

Smallville is populated by unmarried men and women, some of which are acquainted. The two City Matchmakers know who is acquainted with whom. One day, one of the matchmakers claimed: "I can arrange it so that every red haired man will marry a woman with who he is acquainted." The other matchmaker claimed: "I can arrange it so that every blonde woman will marry a man with whom she is acquainted." An amateur mathematician overheard this conversation and said: "Then it can be arranged so that every red haired man will marry a woman with whom he is acquainted and at the same time very blonde woman will marry a man with who she is acquainted." Is the mathematician right?

The transportation ministry has just decided to pay $80$ companies to repair $2400$ roads. These roads connects $100$ cities. Each road is between two cities and there is at most one road between any two cities. Each company must repair exactly $30$ roads, and each road is repaired by exactly one company. For a company to repair a road, it is necessary for the company to set up stations at the both cities on its endpoints. Prove that there are at least $8$ companies stationed at one city.

Two concentric circles are divided by $n$ radii into $2n$ parts. Two parts are called neighbors (of each other) if they share either a common side or a common arc. Initially, there are $4n + 1$ frogs inside the parts. At each second, if there are three or more frogs inside one part, then three of the frogs in the part will jump to its neighbors, with one to each neighbor. Prove that in a finite amount of time, for any part either there are frogs in the part or there are frogs in each of its neighbors

Triangle $ABC$ is inscribed in circle $w$. Line $l_{1}$ bisects $\angle BAC$ and meets segments $BC$ and $w$ in $D$ and $M$,respectively. Let $y$ denote the circle centered at $M$ with radius $BM$. Line $l_{2}$ passes through $D$ and meets circle $y$ at $X$ and $Y$. Prove that line $l_{1}$ also bisects $\angle XAY$

Points $P$ and $Q$ lies inside triangle $ABC$ such that $\angle ACP =\angle BCQ$ and $\angle CAP = \angle BAQ$. Denote by $D,E$, and $F$ the feet of perpendiculars from $P$ to lines $BC,CA$, and $AB$, respectively. Prove that if $\angle DEF = 90^o$, then $Q$ is the orthocenter of triangle $BDF$.

Let $ABC$ be a triangle with $AB\neq AC$, and let $A_{1}B_{1}C_{1}$ be the image of triangle $ABC$ through a rotation $R$ centered at $C$. Let $M,E , F$ be the midpoints of the segments $BA_{1}, AC, BC_{1}$ respectively Given that $EM = FM$, compute $\angle EMF$.

Let $ABCD$ be a tetrahedron and let $H_{a},H_{b},H_{c},H_{d}$ be the orthocenters of triangles $BCD,CDA,DAB,ABC$, respectively. Prove that lines $AH_{a},BH_{b},CH_{c}, DH_{d}$ are concurrent if and only if $AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2$

Let $ABCD$ be a convex quadrilateral. Lines $AB$ and $CD$ meet at $P$, and lines $AD$ and $BC$ meet at $Q$. Let $O$ be a point in the interior of $ABCD$ such that $\angle BOP = \angle DOQ$. Prove that $\angle AOB +\angle COD = 180$.

In triangle $ABC, AB \ne AC$. Circle $\omega$ passes through $A$ and meets sides $AB$ and $AC$ at $M$ and $N$, respectively, and the side $BC$ at $P$ and $Q$ such that $Q$ lies in between $B$ and $P$. Suppose that $MP // AC, NQ // AB$, and $BP \cdot AC = CQ \cdot AB$. Find $\angle BAC$.

In acute triangle $ABC, CA \ne BC$. Let $I$ denote the incenter of triangle $ABC$. Points $A_1$ and $B_1$ lie on rays $CB$ and $CA$, respectively, such that $2CA_1 = 2CB_1 = AB + BC + CA$. Line $CI$ intersects the circumcircle of triangle $ABC$ again at $P$ (other than $C$). Point $Q$ lies on line $AB$ such that $PQ \perp CP$. Prove that $QI \perp A_1B_1$.

Let $S$ be a set of rational numbers with the following properties: (a) $\frac12$ is an element in $S$, (b) if $x$ is in $S$, then both $\frac{1}{x+1}$ and $\frac{x}{x+1}$ are in $S$. Prove that $S$ contains all rational numbers in the interval $(0, 1)$.

Determine all unordered triples $(x,y,z)$ of integers for which the number $\sqrt{\frac{2005}{x+y}}+\sqrt{\frac{2005}{y+z}}+\sqrt{\frac{2005}{z+x}}$ is an integer.

For positive integer $k$, let $p(k)$ denote the greatest odd divisor of $k$. Prove that for every positive integer $n$, $$\frac{2n}{3} < \frac{p(1)}{1}+ \frac{p(2)}{2}+... +\frac{ p(n)}{n}<\frac{2(n + 1)}{3}$$

Given a prime number $p > 2$. Find the least $n\in Z_+$, for which every set of $n$ perfect squares not divisible by $p$ contains nonempty subset with product of all it's elements equal to $1\ (\text{mod}\ p)$

Set $X$ has $56$ elements. Determine the least positive integer $n$ such that for any 15 subsets of $X$, if the union of any $7$ of the subsets has at least $n$ elements, then 3 of the subsets have nonempty intersection.

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$

Let $n$ be a given integer greater than two, and let $S = \{1, 2,...,n\}$. Suppose the function $f : S^k \to S$ has the property that $f(a) \ne f(b)$ for every pair $a$ and $b$ of elements in $S^k$ with $a$ and $b$ differ in all components. Prove that $f$ is a function of one of its elements.

Determine all positive real numbers $a$ such that there exists a positive integer $n$ and partition $A_1$, $A_2$, ..., $A_n$ of infinity sets of the set of the integers satisfying the following condition: for every set $A_i$, the positive difference of any pair of elements in $A_i$ is at least $a^i$.

Let $a, b_1, b_2, \dots, b_n, c_1, c_2, \dots, c_n$ be real numbers such that \[x^{2n} + ax^{2n - 1} + ax^{2n - 2} + \dots + ax + 1 = \prod_{i = 1}^{n}{(x^2 + b_ix + c_i)}\] Prove that $c_1 = c_2 = \dots = c_n = 1$. As a consequence, all complex zeroes of this polynomial must lie on the unit circle.

Find the number of all infinite sequences $a_1$, $a_2$, ... of positive integers such that $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2005$ for all positive integers $n$.

Determine if there exists a strictly increasing sequence of positive integers $a_1$, $a_2$, ... such that $a_n \le n^3$ for every positive integer $n$ and that every positive integer can be written uniquely as the difference of two terms in the sequence.

Let $\{a_n\}^{\inf}_{n=1}$ and $\{b_n\}^{\inf}_{n=1}$ be two sequences of real numbers such that $a_{n+1}=2b_n-a_n$ and $b_{n+1}=2a_n-b_n$ for every positive integer $n$. Prove that $a_n>0$ for all $n$, then $a_1=b_1$.

Let $\mathbb{R}*$ denote the set of nonzero real numbers. Find all functions $f:\mathbb{R}* \rightarrow \mathbb{R}*$ such that $f(x^2+y)=f(f(x))+\frac{f(xy)}{f(x)}$ for every pair of nonzero real numbers $x$ and $y$ with $x^2+y \neq 0$.

Let $S$ denote the set of rational numbers in the interval $(0,1)$. Determine, with proof, if there exists a subset $T$ of $S$ such that every element in $S$ can be uniquely written as the sum of finitely many distinct elements in $T$.

In isosceles triangle $ABC$, $AB=AC$. Extend segment $BC$ through $C$ to $P$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, such that $PX \parallel AC$ and $PY \parallel AB$. Point $T$ lies on the circumcircle of triangle $ABC$ such that $PT \perp XY$. Prove that $\angle BAT = \angle CAT$.

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.

There are $n$ distinct points in the plane. Given a circle in the plane containing at least one of the points in its interior. At each step one moves the center of the circle to the barycenter of all the points in the interior of the circle. Prove that this moving process terminates in the finite number of steps. what does barycenter of n distinct points mean?

Let $ABC$ be a triangle with circumcenter $O$. Let $A_1$ be the midpoint of side $BC$. Ray $AA_1$ meet the circumcircle of triangle $ABC$ again at $A_2$ (other than A). Let $Q_a$ be the foot of the perpendicular from $A_1$ to line $AO$. Point $P_a$ lies on line $Q_aA_1$ such that $P_aA_2 \perp A_2O$. Define points $P_b$ and $P_c$ analogously. Prove that points $P_a$, P_b$, and $P_c$ lie on a line.

Let $ABC$ be an acute triangle with $AC \neq BC$. Points $H$ and $I$ are the orthocenter and incenter of the triangle, respectively. Line $CH$ and $CI$ meet the circumcircle of triangle $ABC$ again at $D$ and $L$ (other than $C$), respectively. Prove that $\angle CIH=90^{\circ}$ if and only if $\angle IDL=90^{\circ}$.

Let $P$ be a convex polygon in the plane. A real number is assigned to each point in the plane so that the sum of the numbers assigned to the vertices of any polygon similar to $P$ is equal to $0$. Prove that all the assigned numbers are equal to $0$.

Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $T$. Let $X$ be a point on circle $\omega_1$. Line $l_1$ is tangent to circle $\omega_1$ and $X$, and line $l$ intersects circle $\omega_2$ at $A$ and $B$. Line $XT$ meets circle $\omega$ at $S$. Point $C$ lies on arc $TS$ (of circle $\omega_2$, not containing points $A$ and $B$). Point $Y$ lies on circle $\omega_1$ and line $YC$ is tangent to circle $\omega_1$. Let $I$ be the intersection of lines $XY$ ad $SC$. Prove that... a) points $C$, $T$, $Y$, $I$ lie on a circle (B) $I$ is an excenter of triangle $ABC$.

There are $2005$ players in a chess tournament played a game. Every pair of players played a game against each other. At the end of the tournament, it turned out that if two players $A$ and $B$ drew the game between them, then every other player either lost to $A$ or to $B$. Suppose that there are at least two draws in the tournament. Prove that all players can be lined up in a single file from left to right in the such a way that every play won the game against the person immediately to his right.

Let m be a positive integer, and let $S=\{a_1=1, a_2, ..., a_m\}$ be a set of positive integers. Prove that there exists a positive integer $n$ with $n\le m$ and a set $T={b_1, b_2, ..., b_n}$ of positive integers such that (a) all the subsets of $T$ have distinct sums of elements; (b) each of the numbers $a_1$, $a_2$, ..., $a_m$ is the sum of the elements of a subset of $T$.

There are $b$ boys and $g$ girls, with $g \ge 2b-1$, at presence at a party. Each boy invites a girl for the first dance. Prove that this can be done in such a way that either a boy is dancing with a girl he knows or all the girls he knows are not dancing.

A $k$-coloring of a graph $G$ is a coloring of its vertices using $k$ possible colors such that the end points of any edge have different colors. We say a graph $G$ is uniquely $k$-colorable if one hand it has a $k$-coloring, on the other hand there do not exist vertices $u$ and $v$ such that $u$ and $v$ have the same color in one $k$-coloring and $u$ and $v$ have different colors in another $k$-coloring. Prove that if a graph $G$ with $n$ vertices $(n \ge 3)$ is uniquely $3$-colorable, then it has at least $2n-3$ edges.

For a triple $(m,n,r)$ of integers with $0 \le r \le n \le m-2$, define $p(m,n,r)=\sum^r_{k=0} (-1)^k \dbinom{m+n-2(k+1)}{n} \dbinom{r}{k}$. Prove that $p(m,n,r)$ is positive and that $\sum^n_{r=0} p(m,n,r)=\dbinom{m+n}{n}$.

Suppose there are $18$ light houses on the Mexican gulf. Each of the lighthouses lightens an angle with size $20$ degrees. Prove that we can choose the directions of the lighthouses such that the whole gulf is lightened.

Let $A_{n,k}$ denote the set of lattice paths in the coordinate plane of upsteps $u=[1,1]$, downsteps $d=[1,-1]$, and flatsteps $f=[1,0]$ that contain $n$ steps, $k$ of which are slanted ($u$ or $d$). A sharp turn is a consecutive pair of $ud$ or $du$. Let $B_{n,k}$ denote the set of paths in $A_{n,k}$ with no upsteps among the first $k-1$ steps, and let $C_{n,k}$ denote the set of paths in $A_{n,k}$ with no sharps anywhere. For example, $fdu$ is in $B_{3,2}$ but not in $C_{3,2}$, while $ufd$ is in $C_{3,2}$ but not $B_{3,2}$. For $1 \le k \le n$, prove that the sets $B_{n,k}$ and $C_{n,k}$ contains the same number of elements.

Let $n$ be an integer greater than $1$, and let $a_1$, $a_2$, ..., $a_n$ be not all identical positive integers. Prove that there are infinitely many primes $p$ such that $p$ divides $a_1^k+a_2^k+...+a_n^k$ for some positive integer $k$.

Let $c$ be a fixed positive integer, and let ${a_n}^{\inf}_{n=1}$ be a sequence of positive integers such that $a_n < a_{n+1} < a_n+c$ for every positive integer $n$. Let $s$ denote the infinite string of digits obtained by writing the terms in the sequence consecutively from left to right, starting from the first term. For every positive integer $k$, let $s_k$ denote the number whose decimal representation is identical to the $k$ most left digits of $s$. Prove that for every positive integer $m$ there exists a positive integer $k$ such that $s_k$ is divisible by $m$.

Prove that the following inequality holds with the exception of finitely many positive integers $n$: $\sum^{n}_{i=1}\sum^{n}_{j=1}gcd(i,j)>4n^2$.

Let $n$ be a positive integer, and let $p$ be a prime number. Prove that if $p^p | n!$, then $p^{p+1} | n!$.

Let $a$, $b$, and $c$ be positive integers such that the product $ab$ divides the product $c(c^2-c+1)$ and the sum $a+b$ is divisible the number $c^2+1$. Prove that the sets ${a,b}$ and ${c,c^2-c+1}$ coincide.

Find all integers $n$ for which there exists an equiangular $n$-gon whose side lengths are distinct rational numbers.