In a party, each person knew exactly $ 22$ other persons. For each two persons $ X$ and $ Y$, if $ X$ and $ Y$ knew each other, there is no other person who knew both of them, and if $ X$ and $ Y$ did not know each other, there are exactly $ 6$ persons who knew both of them. Assume that $ X$ knew $ Y$ iff $ Y$ knew $ X$. How many people did attend the party? [i]Yudi Satria, Jakarta[/i]

The $2001 \times 2001$ trees in a park form a square grid. What is the largest number of trees that can be cut so that no tree stump can be seen from any other? (Each tree has zero width.)

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]

2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010 are chosen. On the red slip of paper, each MOPper writes the remainder when the product of A and his or her number is divided by 2011. On the blue slip of paper, he or she writes the remainder when the product of B and his or her number is divided by 2011. The MOPpers may then perform either of the following two operations: [list] [*] Each MOPper gives his or her red slip to the MOPper whose number is written on his or her blue slip. [*] Each MOPper gives his or her blue slip to the MOPper whose number is written on his or her red slip.[/list] Show that it is always possible to perform some number of these operations such that each MOPper is holding a red slip with his or her number written on it. [i]Brian Hamrick.[/i]

$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win? [i]Proposed by A. Slinko & S. Marshall, New Zealand[/i]

There are three kinds of things, which are designated respectively by the words (stripped of all common meaning) [i]notes[/i], [i]staves [/i], and [i]heads[/i]. There can be a certain relationship between a note and a head, which is expressed by the saying: they match. Also, a note and a head can match and two different staves can match. Given are the following axioms: (a) If a note and a head each match the same stave, then they match, (b) If two different notes both match with stave B, and also both match with head V, then B and V match, (c) If two staves match, then there is a note that matches both, (d) If a note and a stave are given, then there is a head that matches both. Prove the following theorem, denoting the axiom you apply by its letter. If three staves that differ from each other, each one matches every other, and no note matches any of the three staves, then there is a head that matches all three staves. [hide=original wording] Er zijn drie soorten van dingen, die respectievelijk worden aangeduid met de (van alle gangbare betekenis ontdane) woorden noten, balken en vellen. Tussen een noot en een vel kan een zekere betrekking bestaan die uitgedrukt wordt door de zegswijze: zij passen bij elkaar. Ook kunnen een noot en een vel bij elkaar passen en twee verschillende balken kunnen bij elkaar passen. Gegeven zijn de volgende axioma’s: (a) Als een noot en een vel elk passen bij de zelfde balk, dan passen zij bij elkaar; (b) Als tw’ee verschillende noten beide passen bij balk b, en ook passen bij het vel v, dan passen b en v bij elkaar; (c) Als twee balken bij elkaar passen, dan is er een noot die bij beiden past; (d) Als een noot en een balk zijn gegeven, dan is er een vel dat bij beiden past. Bewijs de volgende stelling en geef daarbij telkens door zijn letter het axioma aan dat U toepast. Als van drie onderling verschillende balken elke past bij elke andere en er geen noot bij de drie balken past, dan is er een vel dat bij alle drie de balken past.[/hide]

Cities $A,B,C,D$ are positioned in such a way that $A$ is closer to $C$ than to $D$, and $B$ is closer to $C$ than to $D$. Prove that every point on the straight road from $A$ to $B$ is closer to $C$ than to $D$.

Each of the 36 line segments joining 9 distinct points on a circle is coloured either red or blue. Suppose that each triangle determined by 3 of the 9 points contains at least one red side. Prove that there are four points such that the 6 segments connecting them are all red.

Alice and Bob play a game. There are $100$ gold coins, $100$ silver coins, and $100$ bronze coins. Players take turns to take at least one coin, but they cannot take two or more coins of same kind at once. Alice goes first. The player who cannot take any coin loses. Who has a winning strategy?

There are $m$ identical rectangular chocolate bars and $n$ people. Each chocolate bar may be cut into two (possibly unequal) pieces at most once. For which $m$ and $n$ is it possible to split the chocolate evenly among all the people? [i]Selected from the Kvant Magazine (D. Bugaenko and N. Konstantinov)[/i]

Basil has an unrestricted supply of straight bricks $1 \times 1 \times 3$ and Γ-shape bricks made of three cubes $1\times 1\times 1$. Basil filled a whole box $m \times n \times k$ with these bricks, where $m, n$ and $k$ are integers greater than $1$. Prove that it was sufficient to use only Γ-shape bricks. (Mikhail Evdokimov)

In the following, a [i]word[/i] will mean a finite sequence of letters "$a$" and "$b$". The [i]length[/i] of a word will mean the number of the letters of the word. For instance, $abaab$ is a word of length $5$. There exists exactly one word of length $0$, namely the empty word. A word $w$ of length $\ell$ consisting of the letters $x_1$, $x_2$, ..., $x_{\ell}$ in this order is called a [i]palindrome[/i] if and only if $x_j=x_{\ell+1-j}$ holds for every $j$ such that $1\leq j\leq\ell$. For instance, $baaab$ is a palindrome; so is the empty word. For two words $w_1$ and $w_2$, let $w_1w_2$ denote the word formed by writing the word $w_2$ directly after the word $w_1$. For instance, if $w_1=baa$ and $w_2=bb$, then $w_1w_2=baabb$. Let $r$, $s$, $t$ be nonnegative integers satisfying $r + s = t + 2$. Prove that there exist palindromes $A$, $B$, $C$ with lengths $r$, $s$, $t$, respectively, such that $AB=Cab$, if and only if the integers $r + 2$ and $s - 2$ are coprime.

There are $ n \ge 3 $ puddings in a room. If a pudding $ A $ hates a pudding $ B $, then $ B $ hates $ A $ as well. Suppose the following two conditions holds: 1. Given any four puddings, there are two puddings who like each other. 2. For any positive integer $ m $, if there are $ m $ puddings who like each other, then there exists $ 3 $ puddings (from the other $ n-m $ puddings) that hate each other. Find the smallest possible value of $ n $.

Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$

Let a set $S$ of 2004 points in the plane be given, no three of which are collinear. Let ${\cal L}$ denote the set of all lines (extended indefinitely in both directions) determined by pairs of points from the set. Show that it is possible to colour the points of $S$ with at most two colours, such that for any points $p,q$ of $S$, the number of lines in ${\cal L}$ which separate $p$ from $q$ is odd if and only if $p$ and $q$ have the same colour. Note: A line $\ell$ separates two points $p$ and $q$ if $p$ and $q$ lie on opposite sides of $\ell$ with neither point on $\ell$.

In the classroom of at least four students the following holds: no matter which four of them take seats around a round table, there is always someone who either knows both of his neighbours, or does not know either of his neighbours. Prove that it is possible to divide the students into two groups such that in one of them, all students know one another, and in the other, none of the students know each other. (Note: if student A knows student B, then student B knows student A as well.)

The number devil has coloured the integer numbers: every integer is coloured either black or white. The number $1$ is coloured white. For every two white numbers $a$ and $b$ ($a$ and $b$ are allowed to be equal) the numbers $a-b$ and $a + $b have different colours. Prove that $2011$ is coloured white.

Gourmet Jan compared $n$ restaurants ($n$ is a positive integer). Each pair of restaurants was compared in two categories: tastiness of food and quality of service. For some pairs Jan couldn't tell which restaurant was better in one category, but never in two categories. Moreover, if Jan thought restaurant $A$ was better than restaurant $B$ in one category and restaurant $B$ was better than restaurant $C$ in the same category, then $A$ is also better than $C$ in that category. Prove there exists a restaurant $R$ such that every other restaurant is worse than $R$ in at least one category.

Let $N$ be a positive integer. Two persons play the following game. The first player writes a list of positive integers not greater than $25$, not necessarily different, such that their sum is at least $200$. The second player wins if he can select some of these numbers so that their sum $S$ satisfies the condition $200-N\le S\le 200+N$. What is the smallest value of $N$ for which the second player has a winning strategy?

Suppose that $2018$ numbers $1$ and $-1$ are written around a circle. For every two adjacent numbers, their product is taken. Suppose that the sum of all $2018$ products is negative. Find all possible values of sum of $2018$ given numbers.

Every evening uncle Chernomor (see Pushkin's tales) appoints either $9$ or $10$ of his 33 "knights" in the "night guard". When it can happen, for the first time, that every knight has been on duty the same number of times?

There are many cities in penguin's land. A road runs between some of them, which either can be one or two lanes. When two streets meet outside of a city, it becomes prevent traffic chaos by building a bridge and avoiding any junctions. Now the penguin parliament has passed a new law, according to which every street is only a one-way street may be used. The Minister of Transport now liked the direction of each street stipulate that in each city at most one lane more or less leads in and out. He also knows that the streets of every city have odd number of tracks. Show that he can succeed in his endeavor.

[b]p1.[/b] Suppose that $20\%$ of a number is $17$. Find $20\%$ of $17\%$ of the number. [b]p2.[/b] Let $A, B, C, D$ represent the numbers $1$ through $4$ in some order, with $A \ne 1$. Find the maximum possible value of $\frac{\log_A B}{C +D}$. Here, $\log_A B$ is the unique real number $X$ such that $A^X = B$. [b]p3. [/b]There are six points in a plane, no four of which are collinear. A line is formed connecting every pair of points. Find the smallest possible number of distinct lines formed. [b]p4.[/b] Let $a,b,c$ be real numbers which satisfy $$\frac{2017}{a}= a(b +c), \frac{2017}{b}= b(a +c), \frac{2017}{c}= c(a +b).$$ Find the sum of all possible values of $abc$. [b]p5.[/b] Let $a$ and $b$ be complex numbers such that $ab + a +b = (a +b +1)(a +b +3)$. Find all possible values of $\frac{a+1}{b+1}$. [b]p6.[/b] Let $\vartriangle ABC$ be a triangle. Let $X,Y,Z$ be points on lines $BC$, $CA$, and $AB$, respectively, such that $X$ lies on segment $BC$, $B$ lies on segment $AY$ , and $C$ lies on segment $AZ$. Suppose that the circumcircle of $\vartriangle XYZ$ is tangent to lines $AB$, $BC$, and $CA$ with center $I_A$. If $AB = 20$ and $I_AC = AC = 17$ then compute the length of segment $BC$. [b]p7. [/b]An ant makes $4034$ moves on a coordinate plane, beginning at the point $(0, 0)$ and ending at $(2017, 2017)$. Each move consists of moving one unit in a direction parallel to one of the axes. Suppose that the ant stays within the region $|x - y| \le 2$. Let N be the number of paths the ant can take. Find the remainder when $N$ is divided by $1000$. [b]p8.[/b] A $10$ digit positive integer $\overline{a_9a_8a_7...a_1a_0}$ with $a_9$ nonzero is called [i]deceptive [/i] if there exist distinct indices $i > j$ such that $\overline{a_i a_j} = 37$. Find the number of deceptive positive integers. [b]p9.[/b] A circle passing through the points $(2, 0)$ and $(1, 7)$ is tangent to the $y$-axis at $(0, r )$. Find all possible values of $ r$. [b]p10.[/b] An ellipse with major and minor axes $20$ and $17$, respectively, is inscribed in a square whose diagonals coincide with the axes of the ellipse. Find the area of the square. PS. You had better use hide for answers.

There are $n$ passengers in a line, waiting to board a plane with $n$ seats. For $1 \le k \le n$, the $k^{th}$ passenger in line has a ticket for the $k^{th}$ seat. However, the rst passenger ignores his ticket, and decides to sit in a seat at random. Thereafter, each passenger sits as follows: If his/her assigned is empty, then he/she sits in it. Otherwise, he/she sits in an empty seat at random. How many different ways can all $n$ passengers be seated?

Let $P$ be a point on a circle $C$ and let $\phi$ be a given angle incommensurable with $2\pi$. For each $n \in N, P_n$ denotes the image of $P$ under the rotation about the center $O$ of $C$ by the angle $\alpha_n = n \phi$. Prove that the set $M = \{P_n | n \ge 0\}$ is dense in $C$.