Let $n$ and $b$ be positive integers. We say $n$ is $b$-discerning if there exists a set consisting of $n$ different positive integers less than $b$ that has no two different subsets $U$ and $V$ such that the sum of all elements in $U$ equals the sum of all elements in $V$. (a) Prove that $8$ is $100$-discerning. (b) Prove that $9$ is not $100$-discerning. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

Six dice are strung on a rigid wire so that the wire passes through two opposite faces of each die. Each die can be rotated independently of the others. Prove that it is always possible to rotate the dice and then place the wire horizontally on a table so that the six-digit number formed by their top faces is divisible by $7$. (The faces of a die are numbered from $1$ to $6$, the sum of the numbers on opposite faces is always equal to $7$.) (G Galperin)

Let $N$ be the number of non-empty subsets $T$ of $S = \{1,2, 3,4,...,2020\}$ satisfying $max (T) >1000$. Compute the largest integer $k$ such that $3^k$ divides $N$.

In Abaliba country there are twenty cities and two airline companies, Blue Planes and Red Planes. The flights are planned as follows: $\bullet$ Given any two cities, one and only one of the two companies operates direct flights (in both directions and without stops) between the two cities. Furthermore: $\bullet$There are two cities A and B between which it is not possible to fly (with possible stops) using only Red Planes. Prove that, given any two cities, a passenger can travel from one to the other using only Blue Planes, making at most one stop in a third city.

There are $25$ IMO participants attending a party. Every two of them speak to each other in some language, and they use only one language even if they both know some other language as well. Among every three participants there is a person who uses the same language to speak to the other two (in that group of three). Prove that there is an IMO participant who speaks the same language to at least $10$ other participants

[asy] draw((0,1)--(4,1)--(4,2)--(0,2)--cycle); draw((2,0)--(3,0)--(3,3)--(2,3)--cycle); draw((1,1)--(1,2)); label("1",(0.5,1.5)); label("2",(1.5,1.5)); label("32",(2.5,1.5)); label("16",(3.5,1.5)); label("8",(2.5,0.5)); label("6",(2.5,2.5)); [/asy] The image above is a net of a unit cube. Let $n$ be a positive integer, and let $2n$ such cubes are placed to build a $1 \times 2 \times n$ cuboid which is placed on a floor. Let $S$ be the sum of all numbers on the block visible (not facing the floor). Find the minimum value of $n$ such that there exists such cuboid and its placement on the floor so $S > 2011$.

Let $G$ be a simple graph with 100 vertices such that for each vertice $u$, there exists a vertice $v \in N \left ( u \right )$ and $ N \left ( u \right ) \cap N \left ( v \right ) = \o $. Try to find the maximal possible number of edges in $G$. The $ N \left ( . \right )$ refers to the neighborhood.

An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells as shown in the figure below. Initially all the cells contain $0$. A [i]move[/i] consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by $1$ simultaneously. Determine all positive integers $n$ such that after performing several such moves one can obtain $9$ consecutive numbers $n,(n+1),\cdots ,(n+8)$ in some order. [asy] size(3cm); pair A=(0,0),D=(1,0),B,C,E,F,G,H,I; G=rotate(60,A)*D; B=(1/3)*D; C=2*B;I=(1/3)*G;H=2*I;E=C+I-A;F=H+B-A; draw(A--D--G--A^^B--F--H--C--E--I--B,black);[/asy]

Given $m,n\geq 2$.Paint each cell of a $m\times n$ board $S$ red or blue so that:for any two red cells in a row,one of the two columns they belong to is all red,and the other column has at least one blue cell in it.Find the number of ways to paint $S$ like this.

* Given a regular dodecahedron. Find how many ways are there to draw a plane through it so that its section of the dodecahedron is a regular hexagon?

An ordered quadruple $(P_1, P_2, P_3, P_4)$ of corners in a regular $2013$-gon is called [i]crossing [/i] if the four corners are all different, and the line segment from $P_1$ to $P_2$ intersects the line segment from $P_3$ to $P_4$. How many [i]crossing [/i] quadruples are there in the $2013$-gon?

Let $a, \ b \in \mathbb{Z_{+}}$. Denote $f(a, b)$ the number sequences $s_1, \ s_2, \ ..., \ s_a$, $s_i \in \mathbb{Z}$ such that $|s_1|+|s_2|+...+|s_a| \le b$. Show that $f(a, b)=f(b, a)$.

Manzi has $n$ stamps and an album with $10$ pages. He distributes the $n$ stamps in the album such that each page has a distinct number of stamps. He finds that, no matter how he does this, there is always a set of $4$ pages such that the total number of stamps in these $4$ pages is at least $\frac{n}{2}$. Determine the maximum possible value of $n$.

Let $n$ be a positive integer. Consider a triangular array of nonnegative integers as follows: \[ \begin{array}{rccccccccc} \text{Row } 1: &&&&& a_{0,1} &&&& \smallskip\\ \text{Row } 2: &&&& a_{0,2} && a_{1,2} &&& \smallskip\\ &&& \vdots && \vdots && \vdots && \smallskip\\ \text{Row } n-1: && a_{0,n-1} && a_{1,n-1} && \cdots && a_{n-2,n-1} & \smallskip\\ \text{Row } n: & a_{0,n} && a_{1,n} && a_{2,n} && \cdots && a_{n-1,n} \end{array} \] Call such a triangular array [i]stable[/i] if for every $0 \le i < j < k \le n$ we have \[ a_{i,j} + a_{j,k} \le a_{i,k} \le a_{i,j} + a_{j,k} + 1. \] For $s_1, \ldots s_n$ any nondecreasing sequence of nonnegative integers, prove that there exists a unique stable triangular array such that the sum of all of the entries in row $k$ is equal to $s_k$.

At a tennis tournament there were $2n$ boys and $n$ girls participating. Every player played every other player. The boys won $\frac 75$ times as many matches as the girls. It is knowns that there were no draws. Find $n$. [i]Serbia[/i]

A point in the cartesian plane with integer coordinates is called a lattice point. Consider the following one player game. A finite set of selected lattice points and finite set of selected segments is called a position in this game if the following hold: (i) The endpoints of each selected segment are lattice points; (ii) Each selected segment is parallel to a coordinate axis or to one of the lines $y = \pm x$, (iii) Each selected segment contains exactly five lattice points, all of which are selected, (iv) Every two selected segments have at most one common point. A move in this game consists of selecting a lattice point and a segment such that the new set of selected lattice points and segments is a position. Prove or disprove that there exists an initial position such that the game can have infinitely many moves.

$ S$ is the set of all sequences $ \{a_i| 1 \leq i \leq 7, a_i \equal{} 0 \text{ or } 1\}.$ The distance between two elements $ \{a_i\}$ and $ \{b_i\}$ of $ S$ is defined as \[ \sum^7_{i \equal{} 1} |a_i \minus{} b_i|. \] $ T$ is a subset of $ S$ in which any two elements have a distance apart greater than or equal to 3. Prove that $ T$ contains at most 16 elements. Give an example of such a subset with 16 elements.

Given three planes and a ball in space. In space, find the number of different ways of placing another ball so that it would be tangent the three given planes and the given ball. It is assumed that the balls can only touch externally.

Two grade seven students were allowed to enter a chess tournament otherwise composed of grade eight students. Each contestant played once with each other contestant and received one point for a win, one half point for a tie and zero for a loss. The two grade seven students together gained a total of eight points and each grade eight student scored the same number of points as his classmates. How many students for grade eight participated in the chess tournament? Is the solution unique?

Let $n > 1$ be an integer and $\Omega=\{1,2,...,2n-1,2n\}$ the set of all positive integers that are not larger than $2n$. A nonempty subset $S$ of $\Omega$ is called [i]sum-free[/i] if, for all elements $x, y$ belonging to $S, x + y$ does not belong to $S$. We allow $x = y$ in this condition. Prove that $\Omega$ has more than $2^n$ distinct [i]sum-free[/i] subsets.

A number is [i]additive[/i] if it has three digits, all of them are different and the sum of two of the digits is equal to the remaining one. (For example, $123 (1+2=3), 945 (4+5=9)$). Find the sum of all additive numbers.

Once upon a time, there was a tribe called the Goblin Tribe, and their regular game was ''The ATM Game (Level Giveaway)'' . The game stats with a number of Goblin standing in a circle. Then the Chieftain assigns a Level to each Goblin, which can be the same or different (Level is a number which is a non-negative integer). Start play by selecting a Goblin with Level $k$ ($k \not=). 0$) comes up. Let's assume Goblin $A$. Goblin $A$ explodes itself. Goblin A's Level becomes $0$. After that, Level of Goblin $k$ next to Goblin $A$ clockwise gets Level $1$. Prove that: 1.) If after that Goblin $k$ next to Goblin $A$ explodes itself and keep doing this, $k'$ next to that Goblin clockwise explodes itself. Prove that the level of each Goblin will be the same again. 2) 2.) If after that we can choose any Goblin whose level is not $0$ to explode itself. And keep doing this. Prove that no matter what the initial level is, we can make each level the way we want. But there is a condition that the sum of all Goblin's levels must be equal to the beginning. [i](gools)[/i]

Let $n$ be a positive integer. Find the number of sequences $x_{1},x_{2},\ldots x_{2n-1},x_{2n}$, where $x_{i}\in\{-1,1\}$ for each $i$, satisfying the following condition: for any integer $k$ and $m$ such that $1\le k\le m\le n$ then the following inequality holds \[\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2\]

In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country: [i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]

Laura has $2010$ lamps connected with $2010$ buttons in front of her. For each button, she wants to know the corresponding lamp. In order to do this, she observes which lamps are lit when Richard presses a selection of buttons. (Not pressing anything is also a possible selection.) Richard always presses the buttons simultaneously, so the lamps are lit simultaneously, too. a) If Richard chooses the buttons to be pressed, what is the maximum number of different combinations of buttons he can press until Laura can assign the buttons to the lamps correctly? b) Supposing that Laura will choose the combinations of buttons to be pressed, what is the minimum number of attempts she has to do until she is able to associate the buttons with the lamps in a correct way?