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 $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

For each subset $T$ of $S = \{1, 2, ... , 7\}$, the result $r(T)$ of T is computed as follows: the elements of $T$ are written, largest to smallest, and alternating signs $(+, -)$ starting with $+$ are put in front of each number. The value of the resulting expression is$ r(T)$. (For example, for $T =\{2, 4, 7\}$, we have $r(T) = +7 - 4 + 2 = 5$.) Compute the sum of $r(T)$ as $T$ ranges over all subsets of $S$.

Let $n$ be positive integer, set $M = \{ 1, 2, \ldots, 2n \}$. Find the minimum positive integer $k$ such that for any subset $A$ (with $k$ elements) of set $M$, there exist four pairwise distinct elements in $A$ whose sum is $4n + 1$.

Find the number of ways to partition a set of $10$ elements, $S = \{1, 2, 3, . . . , 10\}$ into two parts; that is, the number of unordered pairs $\{P, Q\}$ such that $P \cup Q = S$ and $P \cap Q = \emptyset$.

A set $ S$ of points in space satisfies the property that all pairwise distances between points in $ S$ are distinct. Given that all points in $ S$ have integer coordinates $ (x,y,z)$ where $ 1 \leq x,y, z \leq n,$ show that the number of points in $ S$ is less than $ \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$ [i]Dan Schwarz, Romania[/i]

Let $p_n(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^nk p_n(k)=n!$.

Let $S$ be a set of $n$ points in the plane such that no four points are collinear. Let $\{d_1,d_2,\cdots ,d_k\}$ be the set of distances between pairs of distinct points in $S$, and let $m_i$ be the multiplicity of $d_i$, i.e. the number of unordered pairs $\{P,Q\}\subseteq S$ with $|PQ|=d_i$. Prove that $\sum_{i=1}^k m_i^2\leq n^3-n^2$.

Determine all natural numbers $n \ge 2$ with the property that there are two permutations $(a_1, a_2,... , a_n) $ and $(b_1, b_2,... , b_n)$ of the numbers $1, 2,..., n$ such that $(a_1 + b_1, a_2 +b_2,..., a_n + b_n)$ are consecutive natural numbers. [i](Walther Janous)[/i]

In a square $10^{2019} \times 10^{2019}, 10^{4038}$ points are marked. Prove that there is such a rectangle with sides parallel to the sides of a square whose area differs from the number of points located in it by at least $6$.

The fields of an $n\times n$ chess board are colored black and white, such that in every small $2\times 2$-square both colors should be the same number. How many there possibilities are for this?

Is it possible for some natural number $k$ to divide all natural numbers from $1$ to $k$ into two groups and write down the numbers in each group in a row in some order so that you get two the same numbers? [hide=original wording beacuse it doesn't make much sense]Можно ли при каком-то натуральном k разбить все натуральные числа от 1 до k на две группы и выписать числа в каждой группе подряд в некотором порядке так, чтобы получились два одинаковых числа?[/hide]

Let $X$ be a set with $n$ elements. Find the largest number of subsets of $X$, each with $3$ elements, so that no two of them are disjoint.

$6$ points are selected on the plane, none of which $3$ lie on one straight line, and all pairwise segments connecting these points are plotted. Some of the sections are plotted in red and others in blue. Prove that any three of the given points are the vertices of a triangle with sides of the same color.

there are some identical squares with sides parallel, in a plane. Among any $k+1$ of them, there are two with a point in common. Prove they can be divided into $2k-1$ sets, such that all the squares in one set aint pairwise disjoint.

In a tennis club, each member has exactly $k > 0$ friends, and a tournament is organized in rounds such that each pair of friends faces each other in matches exactly once. Rounds are played in simultaneous matches, choosing pairs until they cannot choose any more (that is, among the unchosen people, there is not a pair of friends which has its match pending). Determine the maximum number of rounds the tournament can have, depending on $k$.

A circle with radius $4$ and $251$ distinct points inside the circle are given. Show that it is possible to draw a circle with radius $1$ and containing at least $11$ of these points.

How many ways are there to tile a $3 \times 3$ square with $4$ dominoes of size $1 \times 2$ and $1$ domino of size $1 \times 1$? Tilings that can be obtained from each other by rotating the square are considered different. Dominoes of the same size are completely identical

Find the maximal number of discs which can be disposed on the plane so that each two of them have a common point and no three have it

Prove that the following statement holds for all odd integers $n \ge 3$: If a quadrilateral $ABCD$ can be partitioned by lines into $n$ cyclic quadrilaterals, then $ABCD$ is itself cyclic.

Determine the number of distinct right-angled triangles such that its three sides are of integral lengths, and its area is $999$ times of its perimeter. (Congruent triangles are considered identical.)

An economist and a statistician play a game on a calculator which does only one operation. The calculator displays only positive integers and it is used in the following way: Denote by $n$ an integer that is shown on the calculator. A person types an integer, $m$, chosen from the set $\{ 1, 2, . . . , 99 \}$ of the first $99$ positive integers, and if $m\%$ of the number $n$ is again a positive integer, then the calculator displays $m\%$ of $n$. Otherwise, the calculator shows an error message and this operation is not allowed. The game consists of doing alternatively these operations and the player that cannot do the operation looses. How many numbers from $\{1, 2, . . . , 2019\}$ guarantee the winning strategy for the statistician, who plays second? For example, if the calculator displays $1200$, the economist can type $50$, giving the number $600$ on the calculator, then the statistician can type $25$ giving the number $150$. Now, for instance, the economist cannot type $75$ as $75\%$ of $150$ is not a positive integer, but can choose $40$ and the game continues until one of them cannot type an allowed number [i]Proposed by Serbia [/i]

In equality $$1 * 2 * 3 * 4 * 5 * ... * 60 * 61 * 62 = 2023$$ Instead of each asterisk, you need to put one of the signs “+” (plus), “-” (minus), “•” (multiply) so that the equality becomes true. What is the smallest number of "•" characters that can be used?

A lattice point in the Carthesian plane is a point with both coordinates integers. A rectangle minor (respectively a square minor) is a set of lattice points lying inside or on the boundaries of a rectangle (respectively square) with vertices lattice points. We assign to each lattice point a real number, such that the sum of all the numbers in any square minor is less than $1$ in absolute value. Prove that the sum of all the numbers in any rectangle minor is less than $4$ in absolute value.

A permutation $\sigma: \{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ is called [i]straight[/i] if and only if for each integer $k$, $1\leq k\leq n-1$ the following inequality is fulfilled \[ |\sigma(k)-\sigma(k+1)|\leq 2. \] Find the smallest positive integer $n$ for which there exist at least 2003 straight permutations. [i]Valentin Vornicu[/i]