There are $ n ( \ge 4 ) $ people and some people shaked hands each other. Two people can shake hands at most 1 time. For arbitrary four people $ A, B, C, D$ such that $ (A,B), (B,C), (C,D) $ shaked hands, then one of $ (A,C), (A,D), (B,D) $ shaked hand each other. Prove the following statements. (a) Prove that $ n $ people can be divided into two groups, $ X, Y ( \ne \emptyset )$ , such that for all $ (x,y) $ where $ x \in X $ and $ y \in Y $, $ x $ and $ y $ shaked hands or $ x $ and $ y $ didn't shake hands. (b) There exist two people $ A , B $ such that the set of people who are not $ A $ and $ B $ that shaked hands with $ A $ is same wiith the set of people who are not $ A $ and $ B $ that shaked hands with $ B $.

Let $n$ be a positive integer. Show that there is a prime $p$ and a sequence $\left(a_k\right)_{k\ge1}$ of positive integers such that the sequence $\left(p+na_k\right)_{k\ge1}$ consists of distinct primes.

There are $n$ students in a class, and some pairs of these students are friends. Among any six students, there are two of them that are not friends, and for any pair of students that are not friends there is a student among the remaining four that is friends with both of them. Find the maximum value of $n$.

Let $n$ be a natural number greater than 6. $X$ is a set such that $|X| = n$. $A_1, A_2, \ldots, A_m$ are distinct 5-element subsets of $X$. If $m > \frac{n(n - 1)(n - 2)(n - 3)(4n - 15)}{600}$, prove that there exists $A_{i_1}, A_{i_2}, \ldots, A_{i_6}$ $(1 \leq i_1 < i_2 < \cdots, i_6 \leq m)$, such that $\bigcup_{k = 1}^6 A_{i_k} = 6$.

If $ A $ is a subset of $ X $, then we take $ A^1 = A $, $ A^{-1} = X - A $. The subsets $ A_1, A_2, \ldots, A_k $ are called mutually independent if the product $ A_1^{\varepsilon_1} \cap A_2^{\varepsilon_2} \ldots A_k^{\varepsilon_k} $ is nonempty for every system of numbers $ \varepsilon_1 , \varepsilon_2, \ldots, \varepsilon_k $, such that $ |\varepsilon_2| = $1 for $ i = 1, 2, \ldots, k $. What is the maximum number of mutually independent subsets of a $2^n $-element set?

By a partition $\pi$ of an integer $n\ge 1$, we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4$, then the partitions $\pi$ are $1+1+1+1$, $1+1+2$, $1+3, 2+2$, and $4$). For any partition $\pi$, define $A(\pi)$ to be the number of $1$'s which appear in $\pi$, and define $B(\pi)$ to be the number of distinct integers which appear in $\pi$. (E.g., if $n=13$ and $\pi$ is the partition $1+1+2+2+2+5$, then $A(\pi)=2$ and $B(\pi) = 3$). Prove that, for any fixed $n$, the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of $B(\pi)$ over all partitions of $\pi$ of $n$.

The numbers $\frac{50}{1},\frac{50}{2},...\frac{50}{97},\frac{50}{98}$ are written on the board.In each step,two random numbers $a$ and $b$ are chosen and deleted.Then,the number $2ab-a-b+1$ is written instead.What will be the number remained on the board after the last step.

Let $S = \{1, \dots, n\}$. Given a bijection $f : S \to S$ an [i]orbit[/i] of $f$ is a set of the form $\{x, f(x), f(f(x)), \dots \}$ for some $x \in S$. We denote by $c(f)$ the number of distinct orbits of $f$. For example, if $n=3$ and $f(1)=2$, $f(2)=1$, $f(3)=3$, the two orbits are $\{1,2\}$ and $\{3\}$, hence $c(f)=2$. Given $k$ bijections $f_1$, $\ldots$, $f_k$ from $S$ to itself, prove that \[ c(f_1) + \dots + c(f_k) \le n(k-1) + c(f) \] where $f : S \to S$ is the composed function $f_1 \circ \dots \circ f_k$. [i]Proposed by Maria Monks Gillespie[/i]

Nancy shuffles a deck of $52$ cards and spreads the cards out in a circle face up, leaving one spot empty. Andy, who is in another room and does not see the cards, names a card. If this card is adjacent to the empty spot, Nancy moves the card to the empty spot, without telling Andy; otherwise nothing happens. Then Andy names another card and so on, as many times as he likes, until he says "stop." [list][b](a)[/b] Can Andy guarantee that after he says "stop," no card is in its initial spot? [b](b)[/b] Can Andy guarantee that after he says "stop," the Queen of Spades is not adjacent to the empty spot?[/list]

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that \[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\] Prove there exist positive integers $\ell \leq s$ and $N$, such that \[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\] [i]Proposed by Morteza Saghafiyan, Iran[/i]

A [i]lattice point[/i] in the plane is a point with integral coordinates. The[i] centroid[/i] of four points $(x_i,y_i )$, $i = 1, 2, 3, 4$, is the point $\left(\frac{x_1 +x_2 +x_3 +x_4}{4},\frac{y_1 +y_2 +y_3 +y_4 }{4}\right)$. Let $n$ be the largest natural number for which there are $n$ distinct lattice points in the plane such that the centroid of any four of them is not a lattice point. Prove that $n = 12$.

The plane is cut up into equilateral triangles by three families of parallel lines. Is it possible to find $4$ vertices of these triangles which form a square?

$A_1, A_2,\dots A_k$ are different subsets of the set $\{1,2,\dots ,2016\}$. If $A_i\cap A_j$ forms an arithmetic sequence for all $1\le i <j\le k$, what is the maximum value of $k$?

A Princeton slot machine has $100$ pictures, each equally likely to occur. One is a picture of a tiger. Alice and Bob independently use the slot machine, and each repeatedly makes independent plays. Alice keeps playing until she sees a tiger, at which point she stops. Similarly, Bob keeps playing until he sees a tiger. Given that Bob plays twice as much as Alice, let the expected number of plays for Alice be $\tfrac{a}{b}$ with $a, b$ relatively prime positive integers. Find the remainder when $a + b$ is divided by $1000$.

Each two-digit is number is coloured in one of $k$ colours. What is the minimum value of $k$ such that, regardless of the colouring, there are three numbers $a$, $b$ and $c$ with different colours with $a$ and $b$ having the same units digit (second digit) and $b$ and $c$ having the same tens digit (first digit)?

Raul's class has $15$ students, all with different heights. The Mathematics teacher wants to place them in a queue so that, at the beginning of the queue, they are ordered in ascending order of heights, from then on, they are ordered in descending order and Raul, who He is the tallest in the class, he cannot be at the extremes. In how many different ways is it possible to form this queue?

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?

In a team of guards, each is assigned a different positive integer. For any two guards, the ratio of the two numbers assigned to them is at least $3:1$. A guard assigned the number $n$ is on duty for $n$ days in a row, off duty for $n$ days in a row, back on duty for $n$ days in a row, and so on. The guards need not start their duties on the same day. Is it possible that on any day, at least one in such a team of guards is on duty?

[b](a)[/b]Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity. [b](b)[/b]What is the smallest area possible of pentagons with integral coordinates. Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.

One Martian, one Venusian, and one Human reside on Pluton. One day they make the following conversation: [b]Martian [/b]: I have spent $1/12$ of my life on Pluton. [b]Human [/b]: I also have. [b]Venusian [/b]: Me too. [b]Martian [/b]: But Venusian and I have spend much more time here than you, Human. [b]Human [/b]: That is true. However, Venusian and I are of the same age. [b]Venusian [/b]: Yes, I have lived $300$ Earth years. [b]Martian [/b]: Venusian and I have been on Pluton for the past $13$ years. It is known that Human and Martian together have lived $104$ Earth years. Find the ages of Martian, Venusian, and Human.* [hide="*"][i]*: Note that the numbers in the problem are not necessarily in base $10.$[/i][/hide]

A country has 2025 cites, with some pairs of cities having bidirectional flight routes between them. For any pair of the cities, the flight route between them must be operated by one of the companies $X, Y$ or $Z$. To avoid unfairly favoring specific company, the regulation ensures that if there have three cities $A, B$ and $C$, with flight routes $A \leftrightarrow B$ and $A \leftrightarrow C$ operated by two different companies, then there must exist flight route $B \leftrightarrow C$ operated by the third company different from $A \leftrightarrow B$ and $A \leftrightarrow C$ . Let $n_X$, $n_Y$ and $n_Z$ denote the number of flight routes operated by companies $X, Y$ and $Z$, respectively. It is known that, starting from a city, we can arrive any other city through a series of flight routes (not necessary operated by the same company). Find the minimum possible value of $\max(n_X, n_Y , n_Z)$. [i] Proposed by usjl and YaWNeeT[/i]

Given an integer number $n \geq 3$, consider $n$ distinct points on a circle, labelled $1$ through $n$. Determine the maximum number of closed chords $[ij]$, $i \neq j$, having pairwise non-empty intersections. [i]János Pach[/i]

Determine the greatest number of figures congruent to [img]https://cdn.artofproblemsolving.com/attachments/c/6/343f9197bcebf6794460ed1a74ba83ec18a377.png[/img] that can be placed in a $9 \times 9$ grid (without overlapping), such that each figure covers exactly $4$ unit squares. The figures can be rotated and flipped over. For example, the picture below shows that at least $3$ such figures can be placed in a $4 \times4$ grid. [img]https://cdn.artofproblemsolving.com/attachments/1/e/d38fc34b650a1333742bb206c29985c94146aa.png[/img]

Prove that among any seven real numbers there exist two,$ a$ and $b$, such that $\sqrt3|a-b|\le |1+ab|$. Give an example of six real numbers not having this property.

Let \(a\), \(b\), and \(n\) be positive integers. A lemonade stand owns \(n\) cups, all of which are initially empty. The lemonade stand has a [i]filling machine[/i] and an [i]emptying machine[/i], which operate according to the following rules: [list] [*]If at any moment, \(a\) completely empty cups are available, the filling machine spends the next \(a\) minutes filling those \(a\) cups simultaneously and doing nothing else. [*]If at any moment, \(b\) completely full cups are available, the emptying machine spends the next \(b\) minutes emptying those \(b\) cups simultaneously and doing nothing else. [/list] Suppose that after a sufficiently long time has passed, both the filling machine and emptying machine work without pausing. Find, in terms of \(a\) and \(b\), the least possible value of \(n\). [i]Proposed by Raymond Feng[/i]