Found problems: 12
2014 Bosnia and Herzegovina Junior BMO TST, 4
It is given $5$ numbers $1$, $3$, $5$, $7$, $9$. We get the new $5$ numbers such that we take arbitrary $4$ numbers(out of current $5$ numbers) $a$, $b$, $c$ and $d$ and replace them with $\frac{a+b+c-d}{2}$, $\frac{a+b-c+d}{2}$, $\frac{a-b+c+d}{2}$ and $\frac{-a+b+c+d}{2}$. Can we, with repeated iterations, get numbers:
$a)$ $0$, $2$, $4$, $6$ and $8$
$b)$ $3$, $4$, $5$, $6$ and $7$
2000 Saint Petersburg Mathematical Olympiad, 10.5
Cells of a $2000\times2000$ board are colored according to the following rules:
1)At any moment a cell can be colored, if none of its neighbors are colored
2)At any moment a $1\times2$ rectangle can be colored, if exactly two of its neighbors are colored.
3)At any moment a $2\times2$ squared can be colored, if 8 of its neighbors are colored
(Two cells are considered to be neighboring, if they share a common side). Can the entire $2000\times2000$ board be colored?
[I]Proposed by K. Kohas[/i]
2021 Science ON grade VI, 4
The numbers $\frac 32$, $\frac 43$ and $\frac 65$ are intially written on the blackboard. A move consists of erasing one of the numbers from the blackboard, call it $a$, and replacing it with $bc-b-c+2$, where $b,c$ are the other two numbers currently written on the blackboard. Is it possible that $\frac{1000}{999}$ would eventually appear on the blackboard? What about $\frac{113}{108}$?
[i] (Andrei Bâra)[/i]
2021 Science ON all problems, 4
The numbers $\frac 32$, $\frac 43$ and $\frac 65$ are intially written on the blackboard. A move consists of erasing one of the numbers from the blackboard, call it $a$, and replacing it with $bc-b-c+2$, where $b,c$ are the other two numbers currently written on the blackboard. Is it possible that $\frac{1000}{999}$ would eventually appear on the blackboard? What about $\frac{113}{108}$?
[i] (Andrei Bâra)[/i]
2015 Olympic Revenge, 2
Given $v = (a,b,c,d) \in \mathbb{N}^4$, let $\Delta^{1} (v) = (|a-b|,|b-c|,|c-d|,|d-a|)$ and $\Delta^{k} (v) = \Delta(\Delta^{k-1} (v))$ for $k > 1$. Define $f(v) = \min\{k \in \mathbb{N} : \Delta^k (v) = (0,0,0,0)\}$ and $\max(v) = \max\{a,b,c,d\}.$ Show that $f(v) < 1000\log \max(v)$ for all sufficiently large $v$ and $f(v) > 0.001 \log \max (v)$ for infinitely many $v$.
2017-IMOC, C2
On a large chessboard, there are $4$ puddings that form a square with size $1$. A pudding $A$ could jump over a pudding $B$, or equivalently, $A$ moves to the symmetric point with respect to $B$. Is it possible that after finite times of jumping, the puddings form a square with size $2$?
2020 Cono Sur Olympiad, 5
There is a pile with $15$ coins on a table. At each step, Pedro choses one of the piles in the table with $a>1$ coins and divides it in two piles with $b\geq1$ and $c\geq1$ coins and writes in the board the product $abc$. He continues until there are $15$ piles with $1$ coin each. Determine all possible values that the final sum of the numbers in the board can have.
2019 Tournament Of Towns, 2
Consider 2n+1 coins lying in a circle. At the beginning, all the coins are heads up. Moving clockwise, 2n+1 flips are performed: one coin is flipped, the next coin is skipped, the next coin is flipped, the next two coins are skipped, the next coin is flipped,the next three coins are skipped and so on, until finally 2n coins are skipped and the next coin is flipped.Prove that at the end of this procedure,exactly one coin is heads down.
2021 Cyprus JBMO TST, 3
George plays the following game: At every step he can replace a triple of integers $(x,y,z)$ which is written on the blackboard, with any of the following triples:
(i) $(x,z,y)$
(ii) $(-x,y,z)$
(iii) $(x+y,y,2x+y+z)$
(iv) $(x-y,y,y+z-2x)$
Initially, the triple $(1,1,1)$ is written on the blackboard. Determine whether George can, with a sequence of allowed steps, end up at the triple $(2021,2019,2023)$, fully justifying your answer.
2019 SIMO, Q1
[i]George the grasshopper[/i] lives of the real line, starting at $0$ . He is given the following sequence of numbers: $2, 3, 4, 8, 9, ... ,$ which are all the numbers of the form $2^k$ or $3^l$, $k, l \in \mathbb{N}$, arranged in increasing order. Starting from $2$, for each number $x$ in the sequence in order, he (currently at $a$) must choose to jump to either $a+x$ or $a-x$. Show that [i]George the grasshopper[/i] can jump in a way that he reaches every integer on the real line.
2025 Bulgarian Winter Tournament, 11.3
We have \( n \) chips that are initially placed on the number line at position 0. On each move, we select a position \( x \in \mathbb{Z} \) where there are at least two chips; we take two of these chips, then place one at \( x-1 \) and the other at \( x+1 \).
a) Prove that after a finite number of moves, regardless of how the moves are chosen, we will reach a final position where no two chips occupy the same number on the number line.
b) For every possible final position, let \( \Delta \) represent the difference between the numbers where the rightmost and the leftmost chips are located. Find all possible values of \( \Delta \) in terms of \( n \).
2000 Saint Petersburg Mathematical Olympiad, 9.5
The numbers $1,2,\dots,2000$ are written on the board. Two players are playing a game with alternating moves. A move consists of erasing two number $a,b$ and writing $a^b$. After some time only one number is left. The first player wins, if the numbers last digit is $2$, $7$ or $8$. If not, the second player wins. Who has a winning strategy?
[I]Proposed by V. Frank[/i]