Given that $ 2^{2004}$ is a $ 604$-digit number whose first digit is $ 1$, how many elements of the set $ S \equal{} \{2^0,2^1,2^2, \ldots,2^{2003}\}$ have a first digit of $ 4$? $ \textbf{(A)}\ 194 \qquad \textbf{(B)}\ 195 \qquad \textbf{(C)}\ 196 \qquad \textbf{(D)}\ 197 \qquad \textbf{(E)}\ 198$

Find the greatest odd divisor of $160^3$.

Given is a triangle $ABC$, the inscribed circle $G$ of which has radius $r$. Let $r_a$ be the radius of the circle touching $AB$, $AC$ and $G$. [This circle lies inside triangle $ABC$.] Define $r_b$ and $r_c$ similarly. Prove that $r_a + r_b + r_c \geq r$ and find all cases in which equality occurs. [i]Bosnia - Herzegovina Mathematical Olympiad 2002[/i]

Let $m,n\in \mathbb{Z}_{\ge 1}$ and a rectangular board $m\times n$ sliced by parallel lines to the rectangle's sides into $mn$ unit squares. At moment $t=0$, there is an ant inside every square, positioned exactly in its centre, such that it is oriented towards one of the rectangle's sides. Every second, all the ants move exactly a unit following their current orientation; however, if two ants meet at the centre of a unit square, both of them turn $180^o$ around (the turn happens instantly, without any loss of time) and the next second they continue their motion following their new orientation. If two ants meet at the midpoint of a side of a unit square, they just continue moving, without changing their orientation.\\ \\ Prove that, after finitely many seconds, some ant must fall off the table.\\ \\ [i](Oliver Hayman)[/i]

a) Replace each letter in the following sum by a digit from $0$ to $9$, in such a way that the sum is correct. $\tab$ $\tab$ $ABC$ $\tab$ $\tab$ $DEF$ [u]$+GHI$[/u] $\tab$ $\tab$ $\tab$ $J J J$ Different letters must be replaced by different digits, and equal letters must be replaced by equal digits. Numbers $ABC$, $DEF$, $GHI$ and $JJJ$ cannot begin by $0$. b) Determine how many triples of numbers $(ABC,DEF,GHI)$ can be formed under the conditions given in a).

Find all pairs of positive integers $(x,y)$ such that $2^x + 3^y$ is a perfect square.

A finite graph $G(V,E)$ on $n$ points is drawn in the plane. For an edge $e$ of the graph, let $\chi(e)$ denote the number of edges that cross over edge $e$. Prove that \[\sum_{e\in E}\frac{1}{\chi(e)+1}\leq 3n-6.\][i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

The cities of countries $A$ and $B$ are marked on the map, which has the form of a square with vertices at points $(0, 0)$ , $ (0, 1)$ , $(1, 1)$ , $(1, 0)$ of the plane. According to the trade agreement, country $A$ must ensure the delivery of $n$ kg of wheat to $n$ cities of country $B$, located at the points of the square with coordinates $y_1,..., y_n$, $1$ kg each city. Currently, $n$ kg of wheat are distributed among $n$ cities of country $A$, located at the points of the square with coordinates $x_1,... , x_n$, $1$ kg in each city. From each city of country $A$ to each city of the country $A$ any amount of wheat can be transported (of course, not more than $1$ kg). Transportation cost is for $t$ kg of wheat from a city with coordinates $x_i$ to a city with coordinates $y_j$ is equal to $tl_{ij}$, where $l_{ij }$is the length of the segment connecting the points $x_i$ and $y_j$. The government of country A is going to implement the optimal one (i,e. the cheapest) transportation plan. (a) Is it possible to implement the optimal transportation plan so that from each city of country $A$ to transport wheat only to one city of country $B$? (b) Will the response change if country $A$ is to deliver $n+1$ kg of wheat, in city $x_1$ is $2$ kg of wheat, and $2$ kg should be delivered to city $y_1$ (when for other cities the conditions remain the same)? [hide=original wording] Мiста країн A та B позначенi на мапi, що має вигляд квадрату з вершинами в точках (0, 0), (0, 1), (1, 1), (1, 0) площини. Згiдно торгової угоди, країна A має забез- печити доставку n кг пшеницi в n мiст країни B, що розташованi в точках квадрату з координатами y1, . . . , yn, по 1 кг в кожне мiсто. Наразi n кг пшеницi розподiленi серед n мiст країни A, що розташованi в точках квадрату з координатами x1, . . . , xn, по 1 кг в кожному мiстi. З кожного мiста країни A в кожне мiсто країни B можна перевезти довiльну кiлькiсть пшеницi (звичайно, не бiльше 1 кг). Вартiсть переве- зення t кг пшеницi з мiста з координатами xi в мiсто з координатами yj дорiвнює tlij , де lij – довжина вiдрiзку, що сполучає точки xi та yj . Уряд країни A збирається реалiзувати оптимальний (тобто найдешевший) план перевезення. 1. Чи можна реалiзувати оптимальний план перевезення таким чином, щоби з кожного мiста країни A перевозити пшеницю тiльке в одне мiсто країни B? 2. Чи змiниться вiдповiдь, якщо країна A має забезпечити доставку n + 1 кг пше- ницi, в мiстi x1 знаходиться 2 кг пшеницi, i в мiсто y1 має бути доставлено 2 кг пшеницi (щодо iнших мiст умови лишаються такими ж)?[/hide]

Let $a,b,c{}$ be positive real numbers. Prove that \[108\cdot(ab+bc+ca)\leqslant(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})^4.\]

An $n$-stick is a connected figure consisting of $n$ matches of length $1$ which are placed horizontally or vertically and no two touch each other at points other than their ends. Two shapes that can be transformed into each other by moving, rotating or flipping are considered the same. An $n$-mino is a shape which is built by connecting $n$ squares of side length 1 on their sides such that there's a path on the squares between each two squares of the $n$-mino. Let $S_n$ be the number of $n$-sticks and $M_n$ the number of $n$-minos, e.g. $S_3=5$ And $M_3=2$. (a) Prove that for any natural $n$, $S_n \geq M_{n+1}$. (b) Prove that for large enough $n$ we have $(2.4)^n \leq S_n \leq (16)^n$. A [b]grid segment[/b] is a segment on the plane of length 1 which it's both ends are integer points. A polystick is called [b]wise[/b] if using it and it's rotations or flips we can cover all grid segments without overlapping, otherwise it's called [b]unwise[/b]. (c) Prove that there are at least $2^{n-6}$ different unwise $n$-sticks. (d) Prove that any polystick which is in form of a path only going up and right is wise. (e) Extra points: Prove that for large enough $n$ we have $3^n \leq S_n \leq 12^n$ Time allowed for this exam was 2 hours.

Each of $n$ black squares and $n$ white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to $n$ squares. [i](V. Dolnikov)[/i]

Let $ a, b, c$ be integers satisfying $ 0 < a < c \minus{} 1$ and $ 1 < b < c$. For each $ k$, $ 0\leq k \leq a$, Let $ r_k,0 \leq r_k < c$ be the remainder of $ kb$ when divided by $ c$. Prove that the two sets $ \{r_0, r_1, r_2, \cdots , r_a\}$ and $ \{0, 1, 2, \cdots , a\}$ are different.

Find the strictly increasing functions $f : \{1,2,\ldots,10\} \to \{ 1,2,\ldots,100 \}$ such that $x+y$ divides $x f(x) + y f(y)$ for all $x,y \in \{ 1,2,\ldots,10 \}$. [i]Cristinel Mortici[/i]

Justin throws a standard six-sided die three times in a row and notes the number of dots on the top face after each roll. How many different sequences of outcomes could he get?

Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer. [i]Proposed by fattypiggy123[/i]

Determine the range of the following function defined for integer $k$, \[f(k)=(k)_3+(2k)_5+(3k)_7-6k\] where $(k)_{2n+1}$ denotes the multiple of $2n+1$ closest to $k$

Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? $ \textbf{(A)}\ 60 \qquad\textbf{(B)}\ 170 \qquad\textbf{(C)}\ 290 \qquad\textbf{(D)}\ 320 \qquad\textbf{(E)}\ 660 $

Let $A$, $B$, and $C$ be three points such that $B$ is the midpoint of segment $AC$ and let $P$ be a point such that $<PBC=60$. Equilateral triangle $PCQ$ is constructed such that $B$ and $Q$ are on different half=planes with respect to $PC$, and the equilateral triangle $APR$ is constructed in such a way that $B$ and $R$ are in the same half-plane with respect to $AP$. Let $X$ be the point of intersection of the lines $BQ$ and $PC$, and let $Y$ be the point of intersection of the lines $BR$ and $AP$. Prove that $XY$ and $AC$ are parallel.

A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Find all pairs $(a, b)$ of positive rational numbers such that $$\sqrt{a}+\sqrt{b} = \sqrt{2+\sqrt{3}}.$$

Let $AH_1$, $BH_2$ be two altitudes of an acute-angled triangle $ABC$ , $D$ be the projection of $H_1$ to $AC$, $E$ be the projection of $D$ to $AB$, $F$ be the common point of $ED$ and $AH_1$. Prove that $H_2F \parallel BC$. [i](Proposed by E.Diomidov)[/i]

Let $ABC$ be a given triangle. Let $A_1$ and $A_2$ be points on the side $BC$. Let $B_1$ and $B_2$ be points on the side $CA$. Let $C_1$ and $C_2$ be points on the side $AB$. Suppose that the points $A_1,A_2,B_1,B_2,C_1$ and $C_2$ lie on a circle. Prove that the lines $AA_1, BB_1$ and $CC_1$ are concurrent if and only if $AA_2, BB_2$ and $CC_2$ are concurrent.

Let the sequences $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ satisfy $a_0 = b_0 = 1, a_n = 9a_{n-1} -2b_{n-1}$ and $b_n = 2a_{n-1} + 4b_{n-1}$ for all positive integers $n$. Let $c_n = a_n + b_n$ for all positive integers $n$. Prove that there do not exist positive integers $k, r, m$ such that $c^2_r = c_kc_m$.

Determine all functions $ f$ defined on the natural numbers that take values among the natural numbers for which \[ (f(n))^p \equiv n\quad {\rm mod}\; f(p) \] for all $ n \in {\bf N}$ and all prime numbers $ p$.