Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

In a class of $30$ students, each students knows exactly six other students. (Of course, knowing is a mutual relation, so if $A$ knows $B$, then $B$ knows $A$). A group of three students is balanced if either all three students know each other, or no one knows anyone else within that group. How many balanced groups exist?

An $ (n, k) \minus{}$ tournament is a contest with $ n$ players held in $ k$ rounds such that: $ (i)$ Each player plays in each round, and every two players meet at most once. $ (ii)$ If player $ A$ meets player $ B$ in round $ i$, player $ C$ meets player $ D$ in round $ i$, and player $ A$ meets player $ C$ in round $ j$, then player $ B$ meets player $ D$ in round $ j$. Determine all pairs $ (n, k)$ for which there exists an $ (n, k) \minus{}$ tournament. [i]Proposed by Carlos di Fiore, Argentina[/i]

We call a subset $S$ of vertices of graph $G$, $2$-dominating, if and only if for every vertex $v\notin S,v\in G$, $v$ has at least two neighbors in $S$. prove that every $r$-regular $(r\ge3)$ graph has a $2$-dominating set with size at most $\frac{n(1+\ln(r))}{r}$.(15 points) time of this exam was 3 hours

Fix an odd integer $n > 1$. For a permutation $p$ of the set $\{1,2,...,n\}$, let S be the number of pairs of indices $(i, j)$, $1 \le i \le j \le n$, for which $p_i +p_{i+1} +...+p_j$ is divisible by $n$. Determine the maximum possible value of $S$. Croatia

A given triangle is divided into $n$ triangles in such a way that any line segment which is a side of a tiling triangle is either a side of another tiling triangle or a side of the given triangle. Let $s$ be the total number of sides and $v$ be the total number of vertices of the tiling triangles (counted without multiplicity). (a) Show that if $n$ is odd then such divisions are possible, but each of them has the same number $v$ of vertices and the same number $s$ of sides. Express $v$ and $s$ as functions of $n$. (b) Show that, for $n$ even, no such tiling is possible

Every of $n$ guests invited to a dinner has got an invitation denoted by a number from $1$ to $n$. The guests will be sitting around a round table with $n$ seats. The waiter has decided to derve them according to the following rule. At first, he selects one guest and serves him/her at any place. Thereafter, he selects the guests one by one: having chosen a guest, he goes around the table for the number of seats equal to the preceeding guest's invitation number (starting from the seat of the preceeding guest), and serves the guest there. Find all $n$ for which he can select the guests in such an order to serve all the guests.

At a company, there are several workers, some of which are enemies. They go to their job with 100 buses, in such a way that there aren't any enemies in either bus. Having arrived at the job, their chief wants to assign them to brigades of at least two people, without assigning two enemies to the same brigade. Prove that the chief can split the workers in at most 100 brigades, or he cannot split them at all in any number of brigades.

Let $A$ and $B$ be the number of odd positive integers $n<1000$ for which the number formed by the last three digits of $n^{2015}$ is greater and smaller than $n$, respectively. Prove that $A=B$.

Let $n \ge 2$ be an integer. There are $n$ houses in a town. All distances between pairs of houses are different. Every house sends a visitor to the house closest to it. Find all possible values of $n$ (with full justification) for which we can design a town with $n$ houses where every house is visited.

Let $n$ be a positive integer, and let $[n]=\{1,2,\dots,n\}$. Find the maximum posible cardinality of a subset $S$ of $[n]$ with the property that there aren't any distinct $a,b,c\in S$ such that $a+b=c$.

Let $x_0 = x_{101} = 0$. The numbers $x_1, x_2,...,x_{100}$ are chosen at random from the interval $[0, 1]$ uniformly and independently. Compute the probability that $2x_i \ge x_{i-1} + x_{i+1}$ for all $i = 1, 2,..., 100.$

Let [i]n[/i] $\ge$ 3 be an integer and let ([i]$p_1$[/i], [i]$p_2$[/i], [i]$p_3$[/i], $\dots$, [i]$p_n$[/i]) be a permutation of {1, 2, 3, $\dots$ [i]n[/i]}. For this permutation we say that [i]$p_t$[/i] is a [i]turning point[/i] if 2$\le$ [i]t[/i] $\le$ [i]n[/i]-1 and ([i]$p_t$[/i] - [i]$p_{t-1}$[/i])([i]$p_t$[/i] - [i]$p_{t+1}$[/i]) > 0 For example, for [i]n[/i] = 8, the permutation (2, 4, 6, 7, 5, 1, 3, 8) has two turning points: [i]$p_4$[/i] = 7 and [i]$p_6$[/i] = 1. For fixed [i]n[/i], let [i]q[/i]([i]n)[/i] denote the number of permutations of {1, 2, 3, $\dots$ [i]n[/i]} with exactly one turning point. Find all [i]n[/i] $\ge$ 3 for which [i]q[/i]([i]n)[/i] is a perfect square.

In a class with $23$ students, each pair of students have watched a movie together. Let the set of movies watched by a student be his [i]movie collection[/i]. If every student has watched every movie at most once, at least how many different movie collections can these students have?

The subsets $A_1,A_2,\ldots ,A_{2000}$ of a finite set $M$ satisfy $|A_i|>\frac{2}{3}|M|$ for each $i=1,2,\ldots ,2000$. Prove that there exists $m\in M$ which belongs to at least $1334$ of the subsets $A_i$.

Let $X$ and $Y$ be two finite subsets of the half-open interval $[0, 1)$ such that $0 \in X \cap Y$ and $x + y = 1$ for no $x \in X$ and no $y \in Y$. Prove that the set $\{x + y - \lfloor x + y \rfloor : x \in X \textrm{ and } y \in Y\}$ has at least $|X| + |Y| - 1$ elements. [i]***[/i]

In a basketball tournament every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$, and there are no draws. A single team wins the tournament with $26$ points and exactly two teams share the last position with $20$ points. How many teams participated in the tournament?

On a circle there are $2n+1$ points, dividing it into equal arcs ($n\ge 2$). Two players take turns to erase one point. If after one player's turn, it turned out that all the triangles formed by the remaining points on the circle were obtuse, then the player wins and the game ends. Who has a winning strategy: the starting player or his opponent?

Determine the least positive integer $n$ with the following property – for every 3-coloring of numbers $1,2,\ldots,n$ there are two (different) numbers $a,b$ of the same color such that $|a-b|$ is a perfect square.

Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.

Let $m$ and $n$ be positive integers. Determine the number of ways to fill each cell of an $m \times n $ table with a number from $\{-2, -1, 1, 2\}$ so that the product of the numbers written in each row and column is $-2$.

Let $(m,n,N)$ be a triple of positive integers. Bruce and Duncan play a game on an m\times n array, where the entries are all initially zeroes. The game has the following rules. $\bullet$ The players alternate turns, with Bruce going first. $\bullet$ On Bruce's turn, he picks a row and either adds $1$ to all of the entries in the row or subtracts $1$ from all the entries in the row. $\bullet$ On Duncan's turn, he picks a column and either adds $1$ to all of the entries in the column or subtracts $1$ from all of the entries in the column. $\bullet$ Bruce wins if at some point there is an entry $x$ with $|x|\ge N$. Find all triples $(m, n,N)$ such that no matter how Duncan plays, Bruce has a winning strategy.

On a $(4n + 2)\times (4n + 2)$ square grid, a turtle can move between squares sharing a side.The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of $n$, what is the largest positive integer $k$ such that there must be a row or column that the turtle has entered at least $k$ distinct times?

When preparing for a competition with more than two participating teams two of them play against each other at most once. When looking at the game plan it turns out: (1) If two teams play against each other, there are no more team playing against both of them. (2) If two teams do not play against each other, then there is always exactly two other teams playing against them both. Prove that all teams play the same number of games.

On an island live $n$ ($n \ge 2$) $xyz$s. Any two $xyz$s are either friends or enemies. Every $xyz$ wears a necklace made of colored beads such that any two $xyz$s that are befriended have at least one bead of the same color and any two $xyz$s that are enemies do not have any common colors in their necklaces. It is also possible for some necklaces not to have any beads. What is the minimum number of colors of beads that is sufficient to manufacture such necklaces regardless on the relationship between the $xyz$s?