Let $n$ be a natural number. There are $n$ boys and $n$ girls standing in a line, in any arbitrary order. A student $X$ will be eligible for receiving $m$ candies, if we can choose two students of opposite sex with $X$ standing on either side of $X$ in $m$ ways. Show that the total number of candies does not exceed $\frac 13n(n^2-1).$

Let $ABCDEF$ be a regular hexagon of side length $2$. Let us construct parallels to its sides passing through its vertices and midpoints, which divide the hexagon into $24$ congruent equilateral triangles, whose vertices are called nodes. For each node $X$, we define its trio as the figure formed by three adjacent triangles with vertex $X$, such that their intersection is only $X$ and they are not congruent in pairs. a) Determine the maximum possible area of a trio. b) Show that there exists a node whose trios can cover the entire hexagon, and a node whose trios cannot cover the entire hexagon. c) Determine the total number of triangles associated with the hexagon.

Ivan is playing Lego with $4n^2$ $1 \times 2$ blocks. First, he places $2n^2$ $1 \times 2$ blocks to fit a $2n \times 2n$ square as the bottom layer. Then he builds the top layer on top of the bottom layer using the remaining $2n^2$ $1 \times 2$ blocks. Note that the blocks in the bottom layer are connected to the blocks above it in the top layer, just like real Lego blocks. He wants the whole two-layered building to be connected and not in seperate pieces. Prove that if he can do so, then the four $1\times 2$ blocks connecting the four corners of the bottom layer, must be all placed horizontally or all vertically. [i]Proposed by Ivan Chan Kai Chin[/i]

A confused cockroach is initially at vertex $A$ of a cube $ABCDEFGH$ with edges measuring $1$ meter, as shown in the figure. Every second, the cockroach moves $1$ meter, always choosing to go to one of the three adjacent vertices to its current position. For example, after $1$ second, the cockroach could stop at vertex $B$, $D$, or $E$. (a) In how many ways can the cockroach stop at vertex $G$ after $3$ seconds? (b) Is it possible for the cockroach to stop at vertex A after exactly $2023$ seconds? (c) In how many ways can the cockroach stop at A after exactly $2024$ seconds? Note: One way for the cockroach to stop at a vertex after a certain number of seconds differs from another way if, at some point, the cockroach is at different vertices in the trajectory. For example, there are $2$ ways for the cockroach to stop at $C$ after $2$ seconds: one of them passes through $A$, $B$, $C$, and the other through $A$, $D$, $C$. [img]https://cdn.discordapp.com/attachments/954427908359876608/1299721377124847616/Screenshot_2024-10-16_173123.png?ex=671e3b5b&is=671ce9db&hm=76962ee2949d8324c2f7022ef63f8b7d3c6fe3aabf4ecf526f44249439f204ac&[/img]

Two children take turns breaking chocolate bar that is 5*10 squares. They can only break the bar using the divisions between squares and can only do 1 break at a time.. The first player that when breaking the chocolate bar breaks off only a single square wins. Is there a winning strategy for any player?

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 car rides along a circular track in the clockwise direction. At noon Peter and Paul took their positions at two different points of the track. Some moment later they simultaneously ended their duties and compared their notes. The car passed each of them at least $30$ times. Peter noticed that each circle was passed by the car $1$ second faster than the preceding one while Paul’s observation was opposite: each circle was passed $1$ second slower than the preceding one. Prove that their duty was at least an hour and a half long.

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]

Adam is playing Minesweeper on a $9\times9$ grid of squares, where exactly $\frac13$ (or 27) of the squares are mines (generated uniformly at random over all such boards). Every time he clicks on a square, it is either a mine, in which case he loses, or it shows a number saying how many of the (up to eight) adjacent squares are mines. First, he clicks the square directly above the center square, which shows the number $4$. Next, he clicks the square directly below the center square, which shows the number $1$. What is the probability that the center square is a mine? [i]Proposed by Adam Bertelli[/i]

[b]p1.[/b] At $8$ AM Black Widow and Hawkeye began to move towards each other from two cities. They were planning to meet at the midpoint between two cities, but because Black Widow was driving $100$ mi/h faster than Hawkeye, they met at the point that is located $120$ miles from the midpoint. When they met Black Widow said ”If I knew that you drive so slow I would have started one hour later, and then we would have met exactly at the midpoint”. Find the distance between cities. [b]p2.[/b] Solve the inequality: $\frac{x-1}{x-2} \le \frac{x-2}{x-1}$. [b]p3.[/b] Solve the equation: $(x - y - z)^2 + (2x - 3y + 2z + 4)^2 + (x + y + z - 8)^2 = 0$. [b]p4.[/b] Three camps are located in the vertices of an equilateral triangle. The roads connecting camps are along the sides of the triangle. Captain America is inside the triangle and he needs to know the distances between camps. Being able to see the roads he has found that the sum of the shortest distances from his location to the roads is $50$ miles. Can you help Captain America to evaluate the distances between the camps. [b]p5.[/b] $N$ regions are located in the plane, every pair of them have a nonempty overlap. Each region is a connected set, that means every two points inside the region can be connected by a curve all points of which belong to the region. Iron Man has one charge remaining to make a laser shot. Is it possible for him to make the shot that goes through all $N$ regions? [b]p6.[/b] Numbers $1, 2, . . . , 100$ are randomly divided in two groups $50$ numbers in each. In the first group the numbers are written in increasing order and denoted $a_1$, $a_2$, $...$ , $a_{50}$. In the second group the numbers are written in decreasing order and denoted $b_1$, $b_2$, $...$, $b_{50}$. Thus, $a_1 < a_2 < ... < a_{50}$ and $b_1 > b2_ > ... > b_{50}$. Evaluate $|a_1 - b_1| + |a_2 - b_2| + ... + |a_{50} - b_{50}|$. PS. You should use hide for answers.

p1. Is there any natural number n such that $n^2 + 5n + 1$ is divisible by $49$ ? Explain. p2. It is known that the parabola $y = ax^2 + bx + c$ passes through the points $(-3,4)$ and $(3,16)$, and does not cut the $x$-axis. Find all possible abscissa values ​​for the vertex point of the parabola. p3. It is known that $T.ABC$ is a regular triangular pyramid with side lengths of $2$ cm. The points $P, Q, R$, and $S$ are the centroids of triangles $ABC$, $TAB$, $TBC$ and $TCA$, respectively . Determine the volume of the triangular pyramid $P.QRS$ . p4. At an event invited $13$ special guests consisting of $ 8$ people men and $5$ women. Especially for all those special guests provided $13$ seats in a special row. If it is not expected two women sitting next to each other, determine the number of sitting positions possible for all those special guests. p5. A table of size $n$ rows and $n$ columns will be filled with numbers $ 1$ or $-1$ so that the product of all the numbers in each row and the product of all the numbers in each column is $-1$. How many different ways to fill the table?

Vijay has a stash of different size stones: in particular, he has $2021$ types of stones, with sizes from $0$ through $2020$, and he has $2r+1$ stones of size $r$. Vijay starts randomly (and without replacement) taking out stones from his stash and laying them out in a line. Vijay notices that the first stone of size $2020$ comes before the first stone of size $2019$, the first stone of size $2019$ is before the first stone of size $2018$, and so on. What is the probability of this happening? Express your answer in terms of only basic arithmetic operations (division, exponentiation, etc.) and the factorial function. [i]Proposed by Misha Ivkov[/i]

$ A$ is a set of points on the plane, $ L$ is a line on the same plane. If $ L$ passes through one of the points in $ A$, then we call that $ L$ passes through $ A$. (1) Prove that we can divide all the rational points into $ 100$ pairwisely non-intersecting point sets with infinity elements. If for any line on the plane, there are two rational points on it, then it passes through all the $ 100$ sets. (2) Find the biggest integer $ r$, so that if we divide all the rational points on the plane into $ 100$ pairwisely non-intersecting point sets with infinity elements with any method, then there is at least one line that passes through $ r$ sets of the $ 100$ point sets.

The squares in a $3\times 7$ grid are colored either blue or yellow. Consider all $m\times n$ rectangles in this grid, where $m \in \{2,3\}$, $n \in \{2,...,7\}$. Prove that at least one of these rectangles has all four corner squares the same color.

In La-La-Land there are 5784 cities. Alpaca chooses for each pair of cities to either build a road or a river between them, and additionally she places a fish in each city to defend it. Subsequently Bear chooses a city to start his trip. At first, he chooses whether to take his trip in a car or in a boat. A boat can sail through rivers but not drive on roads, and a car can drive on roads but not sail through rivers. When Bear enters a city he takes the fish defending it, and consequently the city collapses and he can't return to it again. What is the maximum number of fish Bear can guarantee himself, regardless of the construction of the paths? Remarks: Bear takes a fish also from the city he begins his trip from (and the city collapses). All roads and rivers are two-way.

Find the largest positive integer $n$ such that no two adjacent digits are the same, and for any two distinct digits $0 \leq a,b \leq 9 $, you can't get the string $abab$ just by removing digits from $n$.

$ABCD$ is a square side $10$. There are four points $P_1, P_2, P_3, P_4$ inside the square. Show that we can always construct line segments parallel to the sides of the square of total length $25$ or less, so that each $P_i$ is linked by the segments to both of the sides $AB$ and $CD$. Show that for some points $P_i$ it is not possible with a total length less than $25$.

A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.

The director of a Zoo has bought eight elephants numbered by $1, 2, \ldots , 8$. He has forgotten their masses but he remembers that each elephant starting with the third one has the mass equal to the sum of the masses of two preceding ones. Suddenly the director hears a rumor that one of the elephants has lost his mass. How can the director perform two weightings on balancing scales without weights to either find this elephant or make sure that this was just a rumor? (It is known that no elephant gained mass and no more than one elephant lost mass.) [i]Alexandr Gribalko[/i]

Let be given integers k and n such that 1<=k<=n. Find the number of ordered k-tuples (a_1,a_2,...,a_n), where a_1, a_2, ..., a_k are different and in the set {1,2,...,n}, satisfying 1) There exist s, t such that 1<=s<t<=k and a_s>a_t. 2) There exists s such that 1<=s<=k and a_s is not congruent to s mod 2. P.S. This is the only problem from VMO 1996 I cannot find a solution or I cannot solve. But I'm not good at all in combinatoric...

Define a colouring in tha plane the following way: - we pick a positive integer $m$; - let $K_{1}$, $K_{2}$, ..., $K_{m}$ be different circles with nonzero radii such that $K_{i}\subset K_{j}$ or $K_{j}\subset K_{i}$ if $i \neq j$; - the points in the plane that lie outside an arbitrary circle (one that is amongst the circles we pick) are coloured differently than the points that lie inside the circle. There are $2019$ points in the plane such that any $3$ of them are not collinear. Determine the maximum number of colours which we can use to colour the given points.

Consider a regular polygon $A_1A_2\ldots A_{6n+3}$. The vertices $A_{2n+1}, A_{4n+2}, A_{6n+3}$ are called [i]holes[/i]. Initially there are three pebbles in some vertices of the polygon, which are also vertices of equilateral triangle. Players $A$ and $B$ take moves in turn. In each move, starting from $A$, the player chooses pebble and puts it to the next vertex clockwise (for example, $A_2\rightarrow A_3$, $A_{6n+3}\rightarrow A_1$). Player $A$ wins if at least two pebbles lie in holes after someone's move. Does player $A$ always have winning strategy? [i]Proposed by Bohdan Rublov [/i]

Determine whether there exists an arithimethical progression consisting of 40 terms and each of whose terms can be written in the form $ 2^m \plus{} 3^n$ or not. where $ m,n$ are nonnegative integers.

Find the number of representations of the number $180$ in the form $180 =x+y+z$, where $x, y, z$ are positive integers that are proportional with some three consecutive positive integers

There are $n{}$ passengers in the queue to board a $n{}$-seat plane. The first one in the queue is an absent-minded old lady who, after boarding the plane, sits down at a randomly selected place. Each subsequent passenger sits in his seat if it is free, and in a random seat otherwise. How many passengers will be out of their seats on average? [i]Proposed by A. Zaslavsky[/i]