For the nonzero numbers $ a$, $ b$, and $ c$, define \[(a,b,c)\equal{}\frac{abc}{a\plus{}b\plus{}c}.\] Find $(2,4,6)$. $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 24$

Find all functions $f : R \to R$ satisfying $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.

Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.

A square with side $2008$ is broken into regions that are all squares with side $1$. In every region, either $0$ or $1$ is written, and the number of $1$'s and $0$'s is the same. The border between two of the regions is removed, and the numbers in each of them are also removed, while in the new region, their arithmetic mean is recorded. After several of those operations, there is only one square left, which is the big square itself. Prove that it is possible to perform these operations in such a way, that the final number in the big square is less than $\frac{1}{2^{10^6}}$.

Find all integer solutions $(x,y,z)$ of the equation $xy+yz+zx-xyz = 2$.

Determine all possible values of $m+n$, where $m$ and $n$ are positive integers satisfying \[\operatorname{lcm}(m,n) - \gcd(m,n) = 103.\]

For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$. Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a [i]stump[/i] of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n=S(n)+9T(n)$.

Let the continuous functions $ f_n(x), \; n\equal{}1,2,3,...,$ be defined on the interval $ [a,b]$ such that every point of $ [a,b]$ is a root of $ f_n(x)\equal{}f_m(x)$ for some $ n \not\equal{} m$. Prove that there exists a subinterval of $ [a,b]$ on which two of the functions are equal.

Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.

In a movie theatre there are $6$ VIP chairs labelled from $1$ to $6$. We call a few consecutive vacant chairs a block. In the online VIP seat reservation process the reservation of a seat consists of two steps: in the first step we choose the block, in the second step we reserve the first, last or middle seat (in case of a block of size even this means the middle chair with the smaller number) of that block. (In the second step the online system offers the three possibilities even though they might mean the same seat.) Benedek reserved all seats at some screeining. In how many ways could he do it if we distinguish two reservation if there were a step when Benedek chose a different option? For instance, if the seats $ 1$ and $6$ are reserved, then there are two blocks, the first one consists of the seat $ 1$, the second block consists of the seats $3, 4$ and $5$. Two reservation orders are different if there is a chair that was reserved in a different step, or there is a chair that was reserved with different option (first, last or middle). So if there were $2$ VIP chairs, then the answer would have been $9$.

Some lattice points in the Cartesian coordinate system are colored red, the rest of the lattice points are colored blue. Such a coloring is called [i]finitely universal[/i], if for any finite, non-empty $A\subset \mathbb Z$ there exists $k\in\mathbb Z$ such that the point $(x,k)$ is colored red if and only if $x\in A$. $a)$ Does there exist a finitely universal coloring such that each row has finitely many lattice points colored red, each row is colored differently, and the set of lattice points colored red is connected? $b)$ Does there exist a finitely universal coloring such that each row has a finite number of lattice points colored red, and both the set of lattice points colored red and the set of lattice points colored blue are connected? A set $H$ of lattice points is called [i]connected[/i] if, for any $x,y\in H$, there exists a path along the grid lines that passes only through lattice points in $H$ and connects $x$ to $y$. [i]Submitted by Anett Kocsis, Budapest[/i]

Let $S$ be the set of all rational numbers whose denominators are powers of 3. Let $a$, $b$ and $c$ be given non-zero real numbers. Determine all real-valued functions $f$ that are defined for $x \in S$, satisfy \[ f(x) = af(3x) + bf(3x - 1) + cf(3x - 2) \textrm{ if }0 \leq x \leq 1, \] and are zero elsewhere.

Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers. (a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$. (b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.

Show that $\binom{n}{m},\binom{n}{m+1},\binom{n}{m+2}$ and $\binom{n}{m+3}$ cannot be in arithmetic progression, where $n,m>0$ and $n\geq m+3$.

The bisector $BL$ was drawn in the triangle $ABC$. Let the points $I_1$ and $I_2$ be centers of the circles inscribed in the triangles $ABL$ and $CBL$, and the points $J_1$ and $J_2$ be centers of the excircles of these triangles touching the side $BL$. Prove that the points $I_1$, $I_2$, $J_1$ and $J_2$ lie on the same circle.

Bicycle license plates in Flatville each contain three letters. The first is chosen from the set $\{C,H,L,P,R\}$, the second from $\{A,I,O\}$, and the third from $\{D,M,N,T\}$. When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What is the largest possible number of $\text{ADDITIONAL}$ license plates that can be made by adding two letters? $ \text{(A)}\ 24\qquad\text{(B)}\ 30\qquad\text{(C)}\ 36\qquad\text{(D)}\ 40\qquad\text{(E)}\ 60 $

We define a [i]chessboard polygon[/i] to be a polygon whose sides are situated along lines of the form $ x \equal{} a$ or $ y \equal{} b$, where $ a$ and $ b$ are integers. These lines divide the interior into unit squares, which are shaded alternately grey and white so that adjacent squares have different colors. To tile a chessboard polygon by dominoes is to exactly cover the polygon by non-overlapping $ 1 \times 2$ rectangles. Finally, a [i]tasteful tiling[/i] is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a $ 3 \times 4$ rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner. [asy]size(300); pathpen = linewidth(2.5); void chessboard(int a, int b, pair P){ for(int i = 0; i < a; ++i) for(int j = 0; j < b; ++j) if((i+j) % 2 == 1) fill(shift(P.x+i,P.y+j)*unitsquare,rgb(0.6,0.6,0.6)); D(P--P+(a,0)--P+(a,b)--P+(0,b)--cycle); } chessboard(2,2,(2.5,0));fill(unitsquare,rgb(0.6,0.6,0.6));fill(shift(1,1)*unitsquare,rgb(0.6,0.6,0.6)); chessboard(4,3,(6,0)); chessboard(4,3,(11,0)); MP("\mathrm{Distasteful\ tilings}",(2.25,3),fontsize(12)); /* draw lines */ D((0,0)--(2,0)--(2,2)--(0,2)--cycle); D((1,0)--(1,2)); D((2.5,1)--(4.5,1)); D((7,0)--(7,2)--(6,2)--(10,2)--(9,2)--(9,0)--(9,1)--(7,1)); D((8,2)--(8,3)); D((12,0)--(12,2)--(11,2)--(13,2)); D((13,1)--(15,1)--(14,1)--(14,3)); D((13,0)--(13,3));[/asy] a) Prove that if a chessboard polygon can be tiled by dominoes, then it can be done so tastefully. b) Prove that such a tasteful tiling is unique.

Given an odd $n\in\mathbb N$. In an $n\times n$ chessboard, you may place many $2\times2$ squares. How many grids, at most, are covered by exactly one square?

The area of triangle $ XYZ$ is 8 square inches. Points $ A$ and $ B$ are midpoints of congruent segments $ \overline{XY}$ and $ \overline{XZ}$. Altitude $ \overline{XC}$ bisects $ \overline{YZ}$. What is the area (in square inches) of the shaded region? [asy]/* AMC8 2002 #20 Problem */ draw((0,0)--(10,0)--(5,4)--cycle); draw((2.5,2)--(7.5,2)); draw((5,4)--(5,0)); fill((0,0)--(2.5,2)--(5,2)--(5,0)--cycle, mediumgrey); label(scale(0.8)*"$X$", (5,4), N); label(scale(0.8)*"$Y$", (0,0), W); label(scale(0.8)*"$Z$", (10,0), E); label(scale(0.8)*"$A$", (2.5,2.2), W); label(scale(0.8)*"$B$", (7.5,2.2), E); label(scale(0.8)*"$C$", (5,0), S); fill((0,-.8)--(1,-.8)--(1,-.95)--cycle, white);[/asy] $ \textbf{(A)}\ 1\frac12\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 2\frac12\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 3\frac12$

On the sides $AB,BC,CD$ and $DA$ of a unit square $ABCD$ points $P,Q,R$ and $S$ are chosen respectively. It turned out that the perimeter of $PQRS$ is $2\sqrt{2}$. Find the sum of perpendiculars from $A,B,C,D$ to $SP,PQ,QR,RS$ respectively.

We consider the sequences strictely increasing $(a_0,a_1,...)$ of naturals which have the following property : For every natural $n$, there is exactly one representation of $n$ as $a_i+2a_j+4a_k$, where $i,j,k$ can be equal. Prove that there is exactly a such sequence and find $a_{2002}$

There are $n$ coins in the pile. Two players play a game by alternately performing a move. A move consists of taking $5,7$ or $11$ coins away from the pile. The player unable to perform a move loses the game. Which player - the one playing first or second - has the winning strategy if: (a) $n=2001$; (b) $n=5000$?

There are $5$ yellow pegs, $4$ red pegs, $3$ green pegs, $2$ blue pegs, and $1$ orange peg on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color? [asy] unitsize(20); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((0,3)); dot((1,3)); dot((0,4));[/asy] $\text{(A)}\ 0\qquad\text{(B)}\ 1\qquad\text{(C)}\ 5!\cdot4!\cdot3!\cdot2!\cdot1!\qquad\text{(D)}\ \frac{15!}{5!\cdot4!\cdot3!\cdot2!\cdot1!}\qquad\text{(E)}\ 15!$

Let $a, b, c \geq 1$. Prove that $$\frac{1}{1+a^2}+\frac{1}{1+b^2}+\frac{1}{1+c^2}\geq\frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ca}.$$

We are given the finite sets $ X$, $ A_1$, $ A_2$, $ \dots$, $ A_{n \minus{} 1}$ and the functions $ f_i: \ X\rightarrow A_i$. A vector $ (x_1,x_2,\dots,x_n)\in X^n$ is called [i]nice[/i], if $ f_i(x_i) \equal{} f_i(x_{i \plus{} 1})$, for each $ i \equal{} 1,2,\dots,n \minus{} 1$. Prove that the number of nice vectors is at least \[ \frac {|X|^n}{\prod\limits_{i \equal{} 1}^{n \minus{} 1} |A_i|}. \]