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 $a<k<b$.

$2000$ consecutive integers (not necessarily positive) are written on the board. A student takes several turns. On each turn, he partitions the $2000$ integers into $1000$ pairs, and substitutes each pair by the difference arid the sum of that pair (note that the difference does not need to be positive as the student may choose to subtract the greater number from the smaller one; in addition, all the operations are carried simultaneously). Prove that the student will never again write $2000$ consecutive integers on the board.

Find all natural numbers $n$ for which there exists set $S$ consisting of $n$ points in the plane, satisfying the condition: For each point $A \in S$ there exist at least three points say $X, Y, Z$ from $S$ such that the segments $AX, AY$ and$ AZ$ have length $1$ (it means that $AX = AY = AZ = 1$).

A ball is placed on a plane and a point on the ball is marked. Our goal is to roll the ball on a polygon in the plane in a way that it comes back to where it started and the marked point comes to the top of it. Note that We are not allowed to rotate without moving, but only rolling. Prove that it is possible. Time allowed for this problem was 90 minutes.

Someone draws at least three lines on paper. Each cuts the other lines two by two. No three lines pass through one point. He chooses a line and counts the intersection points on either side of the line. The numbers of intersections turn out to be the same. He chooses another line. Now the intersections number on one side appears to be six times as large as that on the other side. What is the minimum number of lines where this is possible? [hide=original wording of second sentence]De lijnen snijden elkaar twee aan twee.[/hide]

Olya and Tolya are playing a game on $[0,1]$ segment. In the beginning it is white. In the first round Tolya chooses a number $0 \leq l \leq 1$, and then Olya chooses a subsegment of $[0,1]$ of length $l$ and recolors every its point to the opposite color(white to black, black to white). In the next round players change roles, etc. The game lasts $2024$ rounds. Let $L$ be the sum of length of white segments after the end of the game. If $L > \frac{1}{2}$ Olya wins, otherwise Tolya wins. Which player has a strategy to guarantee his win? [i]A. Naradzetski[/i]

Let $ k$ and $ s$ be positive integers. For sets of real numbers $ \{\alpha_1, \alpha_2, \ldots , \alpha_s\}$ and $ \{\beta_1, \beta_2, \ldots, \beta_s\}$ that satisfy \[ \sum^s_{i\equal{}1} \alpha^j_i \equal{} \sum^s_{i\equal{}1} \beta^j_i \quad \forall j \equal{} \{1,2 \ldots, k\}\] we write \[ \{\alpha_1, \alpha_2, \ldots , \alpha_s\} \overset{k}{\equal{}} \{\beta_1, \beta_2, \ldots , \beta_s\}.\] Prove that if \[ \{\alpha_1, \alpha_2, \ldots , \alpha_s\} \overset{k}{\equal{}} \{\beta_1, \beta_2, \ldots , \beta_s\}\] and $ s \leq k,$ then there exists a permutation $ \pi$ of $ \{1, 2, \ldots , s\}$ such that \[ \beta_i \equal{} \alpha_{\pi(i)} \quad \forall i \equal{} 1,2, \ldots, s.\]

I have an $n \times n$ sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let $b(n)$ be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants $c$ and $d$ such that \[ \dfrac{1}{7} n^2 - cn \leq b(n) \leq \dfrac{1}{5} n^2 + dn \] for all $n > 0$.

A $3 \times 3 \times 4$ cuboid is constructed out of 36 white-coloured unit cubes. Then, all six of the cuboid's sides are coloured red. After that, the cuboid is dismantled into its constituent unit cubes. Then, randomly, all said unit cubes are constructed into the cuboid of its original size (and position). a) How many ways are there to position eight of its corner cubes so that the apparent sides of eight corner cubes are still red? (Cube rotations are still considered distinct configurations, and the position of the cuboid remains unchanged.) b) Determine the probability that after the reconstruction, all of its apparent sides are still red-coloured. (The cuboid is still upright, with the same dimensions as the original cuboid, without rotation.) [hide=Notes]The problem might have multiple interpretations. We agreed that this problem's wording was a bit ambiguous. Here's the original Indonesian version: Suatu balok berukuran $3 \times 3 \times 4$ tersusun dari 36 kubus satuan berwarna putih. Kemudian keenam permukaan balok diwarnai merah. Setelah itu, balok yang tersusun dari kubus-kubus satuan tersebut dibongkar. Kemudian, secara acak, semua kubus satuan disusun lagi menjadi balok seperti balok semula. a) Ada berapa cara menempatkan kedelapan kubus satuan yang berasal dari pojok sehingga kedelapan kubus di pojok yang tampak tetap berwarna merah? (Rotasi kubus dianggap konfigurasi yang berbeda, namun posisi balok tidak diubah.) b) Tentukan probabilitas balok yang tersusun lagi semua permukaannya berwarna merah. (Balok tegak tetap tegak dan balok tetap dalam suatu posisi.) [/hide]

Let $n\ge 3$ be an integer. Let $s(n)$ be the number of (ordered) pairs $(a;b)$ consisting of positive integers $a,b$ from the set $\{1,2,\dots ,n\}$ which satisfy $\gcd (a,b,n)=1$. Prove that $s(n)$ is divisible by $4$ for all $n\ge 3$. [i] (Vlad Robu) [/i]

Show that given any 9 points inside a square of side 1 we can always find 3 which form a triangle with area less than $\frac 18$. [i]Bulgaria[/i]

The price of $175$ Humpties is more than the price of $125$ Dumpties but less than that of $126$ Dumpties. Prove that you cannot buy three Humpties and one Dumpty for (a) $80$ cents. (b) $1$ dollar. (S Fomin, Leningrad) PS. (a) for Juniors , (a),(b) for Seniors

A straight road consists of green and red segments in alternating colours, the first and last segment being green. Suppose that the lengths of all segments are more than a centimeter and less than a meter, and that the length of each subsequent segment is larger than the previous one. A grasshopper wants to jump forward along the road along these segments, stepping on each green segment at least once an without stepping on any red segment (or the border between neighboring segments). Prove that the grasshopper can do this in such a way that among the lengths of his jumps no more than $8$ different values occur. [i]Proposed by T. Korotchenko[/i]

In a company of 7 sousliks each souslik has 4 friends. Is it always possible to find in this company two non-intersecting groups of 3 sousliks each such that in both groups all sousliks are friends?

In an $n$ by $n$ grid, each cell is filled with an integer between $1$ and $6$. The outmost cells all contain the number $1$, and any two cells that share a vertex has difference not equal to $3$. For any vertex $P$ inside the grid (not including the boundary), there are $4$ cells that have $P$ has a vertex. If these four cells have exactly three distinct numbers $i$, $j$, $k$ (two cells have the same number), and the two cells with the same number have a common side, we call $P$ an $ijk$-type vertex. Let there be $A_{ijk}$ vertices that are $ijk$-type. Prove that $A_{123}\equiv A_{246} \pmod 2$.

$2025$ light bulbs are operated by some switches. Each switch works on a subset of the light bulbs. When we use a switch, all the light bulbs in the subset change their state: bulbs that were on turn off, and bulbs that were off turn on. We know that every light bulb is operated by at least one of the switches. Initially, all lamps were off. Find the biggest number $k$ for which we can surely turn on at least $k$ light bulbs. [i]Based on an OKTV problem[/i]

Given are $8$ weights weighing $1, 2, . . . , 8$ grams, but it is not known which one how much does it weigh. Baron Munchausen claims that he remembers which of the weights weighs how much, and to prove that he is right he is ready to conduct one weighing, as a result of which the weight of at least one of the weights will be unambiguously established. Is he cheating?

Jody has $6$ distinguishable balls and $6$ distinguishable sticks, all of the same length. How many ways are there to use the sticks to connect the balls so that two disjoint non-interlocking triangles are formed? Consider rotations and reflections of the same arrangement to be indistinguishable.

Let $n \ge 2$ be a positive integer. Each square of an $n\times n$ board is coloured red or blue. We put dominoes on the board, each covering two squares of the board. A domino is called [i]even [/i] if it lies on two red or two blue squares and [i]colourful [/i] if it lies on a red and a blue square. Find the largest positive integer $k$ having the following property: regardless of how the red/blue-colouring of the board is done, it is always possible to put $k$ non-overlapping dominoes on the board that are either all [i]even [/i] or all [i]colourful[/i].

(a) In how many ways can $1003$ distinct integers be chosen from the set $\{1, 2, ... , 2003\}$ so that no two of the chosen integers differ by $10?$ (b) Show that there are $(3(5151) + 7(1700)) 101^7$ ways to choose $1002$ distinct integers from the set $\{1, 2, ... , 2003\}$ so that no two of the chosen integers differ by $10.$

In a community of more than six people each member exchanges letters with exactly three other members of the community. Show that the community can be partitioned into two nonempty groups so that each member exchanges letters with at least two members of the group he belongs to.

One country has $n$ cities and every two of them are linked by a railroad. A railway worker should travel by train exactly once through the entire railroad system (reaching each city exactly once). If it is impossible for worker to travel by train between two cities, he can travel by plane. What is the minimal number of flights that the worker will have to use?

Each of two houses $A$ and $B$ is divided into two flats. Several cats and dogs live there. It is known that the fraction of cats in the first flat of $A$ (the ratio between the number of cats and the total number of animals in the flat) is greater than the fraction of cats in the first flat of $B$, and the fraction of cats in the second flat of $A$ is greater than the fraction of cats in the second flat of $B$. Is it true that the fraction of cats in house $A$ is greater than the fraction of cats in house $B$? (AK Kovaldji)

Two identical blue blocks, two identical red blocks, two identical green blocks, and two identical purple blocks are placed next to each other in a row. Find the number of distinct arrangements of these blocks where no blue block is placed next to a red block, and no green block is placed next to a purple block.

Each point in the plane with integer coordinates is colored red or blue such that the following two properties hold. For any two red points, the line segment joining them does not contain any blue points. For any two blue points that are distance $2$ apart, the midpoint of the line segment joining them is blue. Prove that if three red points are the vertices of a triangle, then the interior of the triangle does not contain any blue points.