Let $X$ be a finite set of real numbers. For any $x,x' \in X$ with $x<x'$, define a function $f(x,x')$, then $f$ is called an ordered pair function on $X$. For any given ordered pair function $f$ on $X$, if there exist elements $x_1 <x_2 <\cdots<x_k$ in $X$ such that $f(x_1 ,x_2 ) \le f(x_2 ,x_3 ) \le \cdots \le f(x_{k-1} ,x_k )$, then $x_1 ,x_2 ,\cdots,x_k$ is called an $f$-ascending sequence of length $k$ in $X$. Similarly, define an $f$-descending sequence of length $l$ in $X$. For integers $k,l \ge 3$, let $h(k,l)$ denote the smallest positive integer such that for any set $X$ of $s$ real numbers and any ordered pair function $f$ on $X$, there either exists an $f$-ascending sequence of length $k$ in $X$ or an $f$-descending sequence of length $l$ in $X$ if $s \ge h(k,l)$. Prove: 1.For $k,l>3,h(k,l) \le h(k-1,l)+h(k,l-1)-1$; 2.$h(k,l) \le \binom{l-2}{k+l-4} +1$.

Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$; Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$. Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins. [i]Proposed by Hans Zantema, Netherlands[/i]

Basil has a melon in a shape of a ball, $20$ in diameter. Using a long knife, Basil makes three mutually perpendicular cuts. Each cut carves a circular segment in a plane of the cut, $h$ deep ($h$ is a height of the segment). Does it necessarily follow that the melon breaks into two or more pieces if (a) $h = 17$ ? [i](6 points)[/i] (b) $h = 18$ ? [i](6 points)[/i]

Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.

On a $55\times 55$ square grid, $500$ unit squares were cut out as well as $400$ L-shaped pieces consisting of 3 unit squares (each piece can be oriented in any way) [refer to the figure]. Prove that at least two of the cut out pieces bordered each other before they were cut out. [asy]size(2.013cm); draw ((0,0)--(0,1)); draw ((0,0)--(1,0)); draw ((0,1)--(.5,1)); draw ((.5,1)--(.5,0)); draw ((0,.5)--(1,.5)); draw ((1,.5)--(1,0)); draw ((1,.5)--(1,0)); [/asy]

Find number of terms when we expand $(a+b+c)^{99}$ (in the general case).

Given $1989$ points in the space, any three of them are not collinear. We divide these points into $30$ groups such that the numbers of points in these groups are different from each other. Consider those triangles whose vertices are points belong to three different groups among the $30$. Determine the numbers of points of each group such that the number of such triangles attains a maximum.

Is it possible to divide all non-empty subsets of a set of 10 elements into triples so that in each triple, two of the subsets do not intersect and in their union give the third? [i]Proposed by Vladislav Frank[/i]

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Let $ p$ be an odd prime number and $ n$ a natural number. Then $ n$ is called $ p\minus{}partitionable$ if $ T\equal{}\{1,2,...,n \}$ can be partitioned into (disjoint) subsets $ T_1,T_2,...,T_p$ with equal sums of elements. For example, $ 6$ is $ 3$-partitionable since we can take $ T_1\equal{}\{ 1,6 \}$, $ T_2\equal{}\{ 2,5 \}$ and $ T_3\equal{}\{ 3,4 \}$. $ (a)$ Suppose that $ n$ is $ p$-partitionable. Prove that $ p$ divides $ n$ or $ n\plus{}1$. $ (b)$ Suppose that $ n$ is divisible by $ 2p$. Prove that $ n$ is $ p$-partitionable.

Bugs Bunny plays a game in the Euclidean plane. At the $n$-th minute $(n \geq 1)$, Bugs Bunny hops a distance of $F_n$ in the North, South, East, or West direction, where $F_n$ is the $n$-th Fibonacci number (defined by $F_1 = F_2 =1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$). If the first two hops were perpendicular, prove that Bugs Bunny can never return to where he started. [i]Proposed by Dylan Toh[/i]

All the rationals are coloured with $n$ colours so that, if rationals $a$ and $b$ are colored with different colours then $\frac{a+b}2$ is coloured with a colour different from both $a$ and $b$. Prove that every rational is coloured with the same colour.

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

We have a chessboard and we call a $1\times1$ square a room. A robot is standing on one arbitrary vertex of the rooms. The robot starts to move and in every one movement, he moves one side of a room. This robot has $2$ memories $A,B$. At first, the values of $A,B$ are $0$. In each movement, if he goes up, $1$ unit is added to $A$, and if he goes down, $1$ unit is waned from $A$, and if he goes right, the value of $A$ is added to $B$, and if he goes left, the value of $A$ is waned from $B$. Suppose that the robot has traversed a traverse (!) which hasn’t intersected itself and finally, he has come back to its initial vertex. If $v(B)$ is the value of $B$ in the last of the traverse, prove that in this traverse, the interior surface of the shape that the robot has moved on its circumference is equal to $|v(B)|$.

Let $n > 1$ be a natural number. There are $n$ bulbs in a line, each of which can be on or off. Every minute, simultaneously, all the lit bulbs turn off and the unlit bulbs that were adjacent to exactly one lit bulb turn on. Determine for what values of $n$ there is an initial arrangement such that if this process is followed indefinitely, all the lights will never be off.

At a party with $n$ people, it is known that among any $4$ people, there are either $3$ people who all know one another or $3$ people none of which knows another. Show that the $n$ people can be separated into two rooms, so that everyone in one room knows one another and no two people in the other room know each other.

At a mathematical olympiad, eight problems were given to 30 contestants. In order to take the difficulty of each problem into account, the jury decided to assign weights to the problems as follows: a problem is worth $n$ points if it was not solved by exactly $n$ contestants. For example, if a problem was solved by all contestants, then it is worth no points. (It is assumed that there are no partial marks for a problem.) Ivan got less points than any other contestant. Find the greatest score he can have.

Let $n{}$ be a natural number. The playing field for a "Master Sudoku" is composed of the $n(n+1)/2$ cells located on or below the main diagonal of an $n\times n$ square. A teacher secretly selects $n{}$ cells of the playing field and tells his student [list] [*]the number of selected cells on each row, and [*]that there is one selected cell on each column. [/list]The teacher's selected cells form a Master Sudoku if his student can determine them with the given information. How many Master Sudokus are there? [i]Proposed by T. Amdeberkhan, M. Ruby and F. Petrov[/i]

A certain class of $32$ pupils is organised into $33$ clubs , so that each club contains $3$ pupils and no two clubs have the same composition. Prove that there are two clubs which have exactly one common member.

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$. Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.

Positive integers $1, 2, 3, ..., n$ are written on a blackboard ($n > 2$). Every minute two numbers are erased and the least prime divisor of their sum is written. In the end only the number $97$ remains. Find the least $n$ for which it is possible.

The fields of the $8\times 8$ chessboard are numbered from $1$ to $64$ in the following manner: For $i=1,2,\cdots,63$ the field numbered by $i+1$ can be reached from the field numbered by $i$ by one move of the knight. Let us choose positive real numbers $x_{1},x_{2},\cdots,x_{64}$. For each white field numbered by $i$ define the number $y_{i}=1+x_{i}^{2}-\sqrt[3]{x_{i-1}^{2}x_{i+1}}$ and for each black field numbered by $j$ define the number $y_{j}=1+x_{j}^{2}-\sqrt[3]{x_{j-1}x_{j+1}^{2}}$ where $x_{0}=x_{64}$ and $x_{1}=x_{65}$. Prove that \[\sum_{i=1}^{64}y_{i}\geq 48\]

(a) On a long pavement, a sequence of $999$ integers is written in chalk. The numbers need not be in increasing order and need not be distinct. Merlijn encircles $500$ of the numbers with red chalk. From left to right, the numbers circled in red are precisely the numbers $1, 2, 3, ...,499, 500$. Next, Jeroen encircles $500$ of the numbers with blue chalk. From left to right, the numbers circled in blue are precisely the numbers $500, 499, 498, ...,2,1$. Prove that the middle number in the sequence of $999$ numbers is circled both in red and in blue. (b) Merlijn and Jeroen cross the street and find another sequence of $999$ integers on the pavement. Again Merlijn circles $500$ of the numbers with red chalk. Again the numbers circled in red are precisely the numbers $1, 2, 3, ...,499, 500$ from left to right. Now Jeroen circles $500$ of the numbers, not necessarily the same as Merlijn, with green chalk. The numbers circled in green are also precisely the numbers $1, 2, 3, ...,499, 500$ from left to right. Prove: there is a number that is circled both in red and in green that is not the middle number of the sequence of $999$ numbers.

Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$. Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$. Can Alice avoid that? [i](Proposed by Richard Henner)[/i]

Prove that the for all $n>1000$, we can arrange the number $1,2,\dots, \binom{n}{2}$ on edges of a complete graph with $n$ vertices so that the sum of the numbers assigned to edges of any length three path (possibly closed) is not less than $3n-1000log_2log_2 n$.