Let $n \ge 1$ be an integer. For each subset $S \subset \{1, 2, \ldots , 3n\}$, let $f(S)$ be the sum of the elements of $S$, with $f(\emptyset) = 0$. Determine, as a function of $n$, the sum $$\sum_{\mathclap{\substack{S \subset \{1,2,\ldots,3n\}\\ 3 \mid f(S)}}} f(S)$$ where $S$ runs through all subsets of $\{1, 2,\ldots, 3n\}$ such that $f(S)$ is a multiple of $3$.

Find the greatest number $M$ with the following property: in each rearrangement of the $2006$ integer numbers $1,2,...2006$ there are $1010$ numbers located consecutively in that rearrangement whose sum is greater than or equal to $M$.

Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruit is the banana!'' The last fruit responds "No, one of the $\emph{last}$ forty fruit is the banana!'' The fruit in the middle yells "I'm the banana!'' In how many positions could the banana be?

Let $f(x,y)$ and $g(x,y)$ be real valued functions defined for every $x,y \in \{1,2,..,2000\}$. If there exist $X,Y \subset \{1,2,..,2000\}$ such that $s(X)=s(Y)=1000$ and $x\notin X$ and $y\notin Y$ implies that $f(x,y)=g(x,y)$ than, what is the maximum number of $(x,y)$ couples where $f(x,y)\neq g(x,y)$.

$n$ is a fixed natural number. Find the least $k$ such that for every set $A$ of $k$ natural numbers, there exists a subset of $A$ with an even number of elements which the sum of it's members is divisible by $n$.

Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$: (i) move the last digit of $a$ to the first position to obtain the numb er $b$; (ii) square $b$ to obtain the number $c$; (iii) move the first digit of $c$ to the end to obtain the number $d$. (All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.) Find all numbers $a$ for which $d\left( a\right) =a^2$. [i]Proposed by Zoran Sunic, USA[/i]

One needs to ffll the cells of an $n\times n$ table ($n > 1$) with distinct integers from $1$ to $n^2$ so that every two consecutive integers are placed in cells that share a side, while every two integers with the same remainder if divided by $n$ are placed in distinct rows and distinct columns. For which $n$ is this possible? (Alexandr Gribalko)

Each of the $48$ faces of eight $1\times 1\times 1$ cubes is randomly painted either blue or green. The probability that these eight cubes can then be assembled into a $2\times 2\times 2$ cube in a way so that its surface is solid green can be written $\frac{p^m}{q^n}$ , where $p$ and $q$ are prime numbers and $m$ and $n$ are positive integers. Find $p + q + m + n$.

Find the numbers of ordered array $(x_1,...,x_{100})$ that satisfies the following conditions: ($i$)$x_1,...,x_{100}\in\{1,2,..,2017\}$; ($ii$)$2017|x_1+...+x_{100}$; ($iii$)$2017|x_1^2+...+x_{100}^2$.

Let $n$ be a fixed natural number. The maze is a grid of dimensions $n \times n$, with a gate to the sky on one of the squares and some adjacent squares with partitions separated from each other so that it is still possible to move from one square to another. The program is in the UP, DOWN, RIGHT, LEFT final sequence, With each command, the Creature moves from its current square to the corresponding neighboring square, unless the partition or the outer boundary of the labyrinth prevents execution of the command (otherwise it does nothing), upon entering the gate, the Creature moves on to heaven. God creates a program, then Satan creates a labyrinth and places it on a square. Prove that God can make such a program that, independently of Satan's labyrinth and selected from the source square, the Creature always reaches heaven by following this program.

Sets $A_0, A_1, \dots, A_{2023}$ satisfy the following conditions: [list] [*] $A_0 = \{ 3 \}$ [*] $A_n = \{ x + 2 \mid x \in A_{n - 1} \} \ \cup \{x(x+1) / 2 \mid x \in A_{n - 1} \}$ for each $n = 1, 2, \dots, 2023$. [/list] Find $|A_{2023}|$.

The square $ABCD$ is divided into $n^2$ equal small (elementary) squares by parallel lines to its sides, (see the figure for the case $n = 4$). A spider starts from point$ A$ and moving only to the right and up tries to arrive at point $C$. Every ” movement” of the spider consists of: ”$k$ steps to the right and $m$ steps up” or ”$m$ steps to the right and $k$ steps up” (which can be performed in any way). The spider first makes $l$ ”movements” and in then, moves to the right or up without any restriction. If $n = m \cdot l$, find all possible ways the spider can approach the point $C$, where $n, m, k, l$ are positive integers with $k < m$. [img]https://cdn.artofproblemsolving.com/attachments/2/d/4fb71086beb844ca7c492a30c7d333fa08d381.png[/img]

There are $3$ clubs $A,B,C$ with non-empty members. For any triplet of members $(a, b, c)$ with $a \in A, b \in B, c \in C$, two of them are friend and two of them are not friend (here the friend relationship is bidirectional). Prove that one of these statements must be true 1. There exist one student from $A$ that knows all students from $B$ 2. There exist one student from $B$ that knows all students from $C$ 3. There exist one student from $C$ that knows all students from $A$

Given $n>4$ points in the plane, no three collinear. Prove that there are at least $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals with vertices amongst the $n$ points.

Given two distinct points $A_1,A_2$ in the plane, determine all possible positions of a point $A_3$ with the following property: There exists an array of (not necessarily distinct) points $P_1,P_2,...,P_n$ for some $n \ge 3$ such that the segments $P_1P_2,P_2P_3,...,P_nP_1$ have equal lengths and their midpoints are $A_1, A_2, A_3, A_1, A_2, A_3, ...$ in this order.

An architect designed a hexagonal column with $37$ metal tubes of equal thickness. The figure shows the cross-section of this column. Is it possible to build a similar column whose number of tubes ends in $2003$? [img]https://cdn.artofproblemsolving.com/attachments/6/a/eb5714d2324aac8b78042d1f48f03b74ab0d78.png[/img]

Maria have $14$ days to train for an olympiad. The only conditions are that she cannot train by $3$ consecutive days and she cannot rest by $3$ consecutive days. Determine how many configurations of days(in training) she can reach her goal.

Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|, |b - d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1, 1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_0, y_0)= (1, 1)$, $(x_1, y_1), (x_2, y_2)$, $\ldots, (x_n, y_n)$ such that, for all $i = 1, 2, \ldots, n$, $1 \leq x_i , y_i \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_i, y_i)$. In this case, each cell $(x_i, y_i)$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.

Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

Patty is standing on a line of planks playing a game. Define a block to be a sequence of adjacent planks, such that both ends are not adjacent to any planks. Every minute, a plank chosen uniformly at random from the block that Patty is standing on disappears, and if Patty is standing on the plank, the game is over. Otherwise, Patty moves to a plank chosen uniformly at random within the block she is in; note that she could end up at the same plank from which she started. If the line of planks begins with $n$ planks, then for sufficiently large n, the expected number of minutes Patty lasts until the game ends (where the first plank disappears a minute after the game starts) can be written as $P(1/n)f(n) + Q(1/n)$, where $P,Q$ are polynomials and $f(n) =\sum^n_{i=1}\frac{1}{i}$ . Find $P(2023) + Q(2023)$.

A domino has two numbers (which may be equal) between $0$ and $6$, one at each end. The domino may be turned around. There is one domino of each type, so $28$ in all. We want to form a chain in the usual way, so that adjacent dominos have the same number at the adjacent ends. Dominos can be added to the chain at either end. We want to form the chain so that after each domino has been added the total of all the numbers is odd. For example, we could place first the domino $(3,4)$, total $3 + 4 = 7$. Then $(1,3)$, total $1 + 3 + 3 + 4 = 11$, then $(4,4)$, total $11 + 4 + 4 = 19$. What is the largest number of dominos that can be placed in this way? How many maximum-length chains are there?

At a big New Year's Eve party, each guest receives two hats: one red and one blue. At the beginning of the party, all the guests put on the red hat. Several times throughout the evening, the announcer announces the name of one of the guests and, at that moment, the named and each of his friends change the hat they are wearing for the other color. Show that the announcer can make it so that all the guests are wearing the blue hat when the party is over. Note: All guests remain at the party from start to finish.

Nine fair coins are flipped independently and placed in the cells of a $3$ by $3$ square grid. Let $p$ be the probability that no row has all its coins showing heads and no column has all its coins showing tails. If $p=\frac ab$ for relatively prime positive integers $a$ and $b$, compute $100a+b$.

Determine the values of $n$ for which an $n\times n$ square can be tiled with pieces of the type [img]http://oi53.tinypic.com/v3pqoh.jpg[/img].

Given $100$ points on the plane. Prove that you can cover them with a family of circles with the sum of their diameters less than $100$ and the distance between any two of the circles more than one.