Let $ k$ be a positive integer. Set $ A \subseteq \mathbb{Z}$ is called a $ \textbf{k \minus{} set}$ if there exists $ x_1, x_2, \cdots, x_k \in \mathbb{Z}$ such that for any $ i \neq j$, $ (x_i \plus{} A) \cap (x_j \plus{} A) \equal{} \emptyset$, where $ x \plus{} A \equal{} \{ x \plus{} a \mid a \in A \}$. Prove that if $ A_i$ is $ \textbf{k}_i\textbf{ \minus{} set}$($ i \equal{} 1,2, \cdots, t$), and $ A_1 \cup A_2 \cup \cdots \cup A_t \equal{} \mathbb{Z}$, then $ \displaystyle \frac {1}{k_1} \plus{} \frac {1}{k_2} \plus{} \cdots \plus{} \frac {1}{k_t} \geq 1$.

Let $S_{n}=\{1,2,...,n\}$. How many functions $f:S_{n} \rightarrow S_{n}$ satisfy $f(k) \leq f(k+1)$ and $f(k)=f(f(k+1))$ for $k <n?$

Denote by $ \mathcal{C}$ the family of all configurations $ C$ of $ N > 1$ distinct points on the sphere $ S^2,$ and by $ \mathcal{H}$ the family of all closed hemispheres $ H$ of $ S^2.$ Compute: $ \displaystyle\max_{H\in\mathcal{H}}\displaystyle\min_{C\in\mathcal{C}}|H\cap C|$, $ \displaystyle\min_{H\in\mathcal{H}}\displaystyle\max_{C\in\mathcal{C}}|H\cap C|$ $ \displaystyle\max_{C\in\mathcal{C}}\displaystyle\min_{H\in\mathcal{H}}|H\cap C|$ and $ \displaystyle\min_{C\in\mathcal{C}}\displaystyle\max_{H\in\mathcal{H}}|H\cap C|.$

Consider a $n \times n$ grid divided into $n ^ 2$ squares of $1 \times 1$. Each of the $(n + 1) ^ 2 $ vertices of the grid is colored red or blue. Find the number of coloring such that each unit square has two red and two blue vertices.

Let $ S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\}$ be a $ (k \plus{} l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called [i]nice[/i] if \[ \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl}\] Prove that the number of nice subsets is at least $ \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}$. [i]Proposed by Andrey Badzyan, Russia[/i]

Given a simple number $p>3$. Consider the set $M$ of the pairs $(x,y)$ with the integer coordinates in the plane such that $0 \le x < p, 0 \le y < p$. Prove that it is possible to mark $p$ points of $M$ such that not a triple of marked points will belong to one line and there will be no parallelogram with the vertices in the marked points.

$n$ points are given on a plane where $n\ge4$. All pairs of points are connected with a segment. Find the maximal number of segments which don't intersect with any other segments in their interior.

For every natural number $ t$, $ f(t)$ is the probability that if a fair coin is tossed $ t$ times, the number of times we get heads is 2008 more than the number of tails. What is the value of $ t$ for which $ f(t)$ attains its maximum? (if there is more than one, describe all of them)

[b]7.1[/b] Given a convex $n$-gon all of whose angles are obtuse. Prove that the sum of the lengths of the diagonals in it is greater than the sum of the lengths of the sides. [b]7.2[/b] Find all integer values for $x$ and $y$ such that $x^4 + 4y^4$ is a prime number[b]. (typo corrected)[/b] [b]7.3.[/b] Given a triangle $ABC$. Parallelograms $ABKL$, $BCMN$ and $ACFG$ are constructed on the sides, Prove that the segments $KN$, $MF$ and $GL$ can form a triangle. [img]https://cdn.artofproblemsolving.com/attachments/a/f/7a0264b62754fafe4d559dea85c67c842011fc.png[/img] [b]7.4 / 6.2[/b] Prove that a $10 \times 10$ chessboard cannot be covered with $ 25$ figures like [img]https://cdn.artofproblemsolving.com/attachments/0/4/89aafe1194628332ec13ad1c713bb35cbefff7.png[/img]. [b]7.5[/b] Find the greatest number of different natural numbers, each of which is less than $50$, and every two of which are coprime. [b]7.6.[/b] Given a triangle $ABC$.$ D$ and $E$ are the midpoints of the sides $AB$ and $BC$. Point$ M$ lies on $AC$ , $ME > EC$. Prove that $MD < AD$. [img]https://cdn.artofproblemsolving.com/attachments/e/c/1dd901e0121e5c75a4039d21b954beb43dc547.png[/img] PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here[/url].

Thirty two pairs of identical twins are lined up in an $8\times 8$ formation. Prove that it is possible to choose $32 $ persons, one from each pair of twins, so that there is at least one chosen person in each row and in each column

What is the maximum number of coins can be arranged in cells of the table $n \times n$ (each cell is not more of the one coin) so that any coin was not simultaneously below and to the right than any other?

Let $n$ be a natural number divisible by $4$. Determine the number of bijections $f$ on the set $\{1,2,...,n\}$ such that $f (j )+f^{-1}(j ) = n+1$ for $j = 1,..., n.$

There is a connected network with $ 2008$ computers, in which any of the two cycles don't have any common vertex. A hacker and a administrator are playing a game in this network. On the $ 1st$ move hacker selects one computer and hacks it, on the $ 2nd$ move administrator selects another computer and protects it. Then on every $ 2k\plus{}1th$ move hacker hacks one more computer(if he can) which wasn't protected by the administrator and is directly connected (with an edge) to a computer which was hacked by the hacker before and on every $ 2k\plus{}2th$ move administrator protects one more computer(if he can) which wasn't hacked by the hacker and is directly connected (with an edge) to a computer which was protected by the administrator before for every $ k>0$. If both of them can't make move, the game ends. Determine the maximum number of computers which the hacker can guarantee to hack at the end of the game.

Determine the maximal size of a set of positive integers with the following properties: $1.$ The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$. $2.$ No digit occurs more than once in the same integer. $3.$ The digits in each integer are in increasing order. $4.$ Any two integers have at least one digit in common (possibly at different positions). $5.$ There is no digit which appears in all the integers.

Let $k>1$ be a positive integer. A set $S{}$ is called [i]good[/i] if there exists a colouring of the positive integers with $k{}$ colours, such that no element from $S{}$ can be written as the sum of two distinct positive integers having the same colour. Find the greatest positive integer $t{}$ (in terms of $k{}$) for which the set \[S=\{a+1,a+2,\ldots,a+t\}\]is good, for any positive integer $a{}$.

[b]p1.[/b] Fleming has a list of 8 mutually distinct integers between $90$ to $99$, inclusive. Suppose that the list has median $94$, and that it contains an even number of odd integers. If Fleming reads the numbers in the list from smallest to largest, then determine the sixth number he reads. [b]p2.[/b] Find the number of ordered pairs $(x,y)$ of three digit base-$10$ positive integers such that $x-y$ is a positive integer, and there are no borrows in the subtraction $x-y$. For example, the subtraction on the left has a borrow at the tens digit but not at the units digit, whereas the subtraction on the right has no borrows. $$\begin{tabular}{ccccc} & 4 & 7 & 2 \\ - & 1 & 9 & 1\\ \hline & 2 & 8 & 1 \\ \end{tabular}\,\,\, \,\,\, \begin{tabular}{ccccc} & 3 & 7 & 9 \\ - & 2 & 6 & 3\\ \hline & 1 & 1 & 6 \\ \end{tabular}$$ [b]p3.[/b] Evaluate $$1 \cdot 2 \cdot 3-2 \cdot 3 \cdot 4+3 \cdot 4 \cdot 5- 4 \cdot 5 \cdot 6+ ... +2017 \cdot 2018 \cdot 2019 -2018 \cdot 2019 \cdot 2020+1010 \cdot 2019 \cdot 2021$$ [b]p4.[/b] Find the number of ordered pairs of integers $(a,b)$ such that $$\frac{ab+a+b}{a^2+b^2+1}$$ is an integer. [b]p5.[/b] Lin Lin has a $4\times 4$ chessboard in which every square is initially empty. Every minute, she chooses a random square $C$ on the chessboard, and places a pawn in $C$ if it is empty. Then, regardless of whether $C$ was previously empty or not, she then immediately places pawns in all empty squares a king’s move away from $C$. The expected number of minutes before the entire chessboard is occupied with pawns equals $\frac{m}{n}$ for relatively prime positive integers $m$,$n$. Find $m+n$. A king’s move, in chess, is one square in any direction on the chessboard: horizontally, vertically, or diagonally. [b]p6.[/b] Let $P(x) = x^5-3x^4+2x^3-6x^2+7x+3$ and $a_1,...,a_5$ be the roots of$ P(x)$. Compute $$\sum^5_{k=1}(a^3_k -4a^2_k +a_k +6).$$ [b]p7.[/b] Rectangle $AXCY$ with a longer length of $11$ and square $ABCD$ share the same diagonal $\overline{AC}$. Assume $B$,$X$ lie on the same side of $\overline{AC}$ such that triangle$ BXC$ and square $ABCD$ are non-overlapping. The maximum area of $BXC$ across all such configurations equals $\frac{m}{n}$ for relatively prime positive integers $m$,$n$. Compute $m+n$. [b]p8.[/b] Earl the electron is currently at $(0,0)$ on the Cartesian plane and trying to reach his house at point $(4,4)$. Each second, he can do one of three actions: move one unit to the right, move one unit up, or teleport to the point that is the reflection of its current position across the line $y=x$. Earl cannot teleport in two consecutive seconds, and he stops taking actions once he reaches his house. Earl visits a chronologically ordered sequence of distinct points $(0,0)$, $...$, $(4,4)$ due to his choice of actions. This is called an [i]Earl-path[/i]. How many possible such [i]Earl-paths[/i] are there? [b]p9.[/b] Let $P(x)$ be a degree-$2022$ polynomial with leading coefficient $1$ and roots $\cos \left( \frac{2\pi k}{2023} \right)$ for $k = 1$ , $...$,$2022$ (note $P(x)$ may have repeated roots). If $P(1) =\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, then find the remainder when $m+n$ is divided by $100$. [b]p10.[/b] A randomly shuffled standard deck of cards has $52$ cards, $13$ of each of the four suits. There are $4$ Aces and $4$ Kings, one of each of the four suits. One repeatedly draws cards from the deck until one draws an Ace. Given that the first King appears before the first Ace, the expected number of cards one draws after the first King and before the first Ace is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [b]p11.[/b] The following picture shows a beam of light (dashed line) reflecting off a mirror (solid line). The [i]angle of incidence[/i] is marked by the shaded angle; the[i] angle of reflection[/i] is marked by the unshaded angle. [img]https://cdn.artofproblemsolving.com/attachments/9/d/d58086e5cdef12fbc27d0053532bea76cc50fd.png[/img] The sides of a unit square $ABCD$ are magically distorted mirrors such that whenever a light beam hits any of the mirrors, the measure of the angle of incidence between the light beam and the mirror is a positive real constant $q$ degrees greater than the measure of the angle of reflection between the light beam and the mirror. A light beam emanating from $A$ strikes $\overline{CD}$ at $W_1$ such that $2DW_1 =CW_1$, reflects off of $\overline{CD}$ and then strikes $\overline{BC}$ at $W_2$ such that $2CW_2 = BW_2$, reflects off of $\overline{BC}$, etc. To this end, denote $W_i$ the $i$-th point at which the light beam strikes $ABCD$. As $i$ grows large, the area of $W_iW_{i+1}W_{i+2}W_{i+3}$ approaches $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [b]p12.[/b] For any positive integer $m$, define $\phi (m)$ the number of positive integers $k \le m$ such that $k$ and $m$ are relatively prime. Find the smallest positive integer $N$ such that $\sqrt{ \phi (n) }\ge 22$ for any integer $n \ge N$. [b]p13.[/b] Let $n$ be a fixed positive integer, and let $\{a_k\}$ and $\{b_k\}$ be sequences defined recursively by $$a_1 = b_1 = n^{-1}$$ $$a_j = j(n- j+1)a_{j-1}\,\,\, , \,\,\, j > 1$$ $$b_j = nj^2b_{j-1}+a_j\,\,\, , \,\,\, j > 1$$ When $n = 2021$, then $a_{2021} +b_{2021} = m \cdot 2017^2$ for some positive integer $m$. Find the remainder when $m$ is divided by $2017$. [b]p14.[/b] Consider the quadratic polynomial $g(x) = x^2 +x+1020100$. A positive odd integer $n$ is called $g$-[i]friendly[/i] if and only if there exists an integer $m$ such that $n$ divides $2 \cdot g(m)+2021$. Find the number of $g$-[i]friendly[/i] positive odd integers less than $100$. [b]p15.[/b] Let $ABC$ be a triangle with $AB < AC$, inscribed in a circle with radius $1$ and center $O$. Let $H$ be the intersection of the altitudes of $ABC$. Let lines $\overline{OH}$, $\overline{BC}$ intersect at $T$. Suppose there is a circle passing through $B$, $H$, $O$, $C$. Given $\cos (\angle ABC-\angle BCA) = \frac{11}{32}$ , then $TO = \frac{m\sqrt{p}}{n}$ for relatively prime positive integers $m$,$n$ and squarefree positive integer $p$. Find $m+n+ p$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Zé Roberto and Humberto are playing the Millenium Game! There are 30 empty boxes in a queue, and each box have a capacity of one blue stome. Each player takes a blue stone and places it in a box (and it is a [i]move[/i]). The winner is who, in its move, obtain three full consecutive boxes. If Zé Roberto is the first player, who has the winner strategy?

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 $f(a, b)=f(b, a)$.

The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final number?

In a school, there are totally $n$ students, with $n \ge 2$. The students take part in $m$ clubs and in each club, there are at least $2$ members (a student may take part in more than $1$ club). Eventually, the Principal notices that: If $2$ clubs share at least $2$ common members then they have different numbers of members. Prove that $$m \le (n - 1)^2$$

Let $ n$ be positive integer, $ A,B\subseteq[0,n]$ are sets of integers satisfying $ \mid A\mid \plus{} \mid B\mid\ge n \plus{} 2.$ Prove that there exist $ a\in A, b\in B$ such that $ a \plus{} b$ is a power of $ 2.$

You have a collection of at least two tokens where each one has a number less than or equal to 10 written on it. The sum of the numbers on the tokens is \( S \). Find all possible values of \( S \) that guarantee that the tokens can be separated into two groups such that the sum of each group does not exceed 80.

Consider an alphabetized list of all the arrangements of the letters in the word BETWEEN. Then BEEENTW would be in position $1$ in the list, BEEENWT would be in position $2$ in the list, and so forth. Find the position that BETWEEN would be in the list.

In a wagon, every $m \geq 3$ people have exactly one common friend. (When $A$ is $B$'s friend, $B$ is also $A$'s friend. No one was considered as his own friend.) Find the number of friends of the person who has the most friends.

The keyword that Ana Viso chose for her computer has the $7$ characters of her name: A, N, A, V, I, S, O. Sorting all the different words alphabetically formed by all these $7$ characters, Ana's keyword appears in the $881$st position. What it is Ana's keyword?