Found problems: 19
2021 Olimphíada, 3
Let $n$ be a positive integer. In the $\mathit{philand}$ language, words are all finite sequences formed by the letters "$P$", "$H$" and "$I$". Philipe, who speaks only the $\mathit{philand}$ language, writes the word $PHIPHI\ldots PHI$ on a piece of paper, where $PHI$ is repeated $n$ times. He can do the following operations:
• Erase two identical letters and write in their place two different letters from the original and from each other;
(Ex: $PP\rightarrow HI$)
• Erase two distinct letters and rewrite them changing the order in which they appear;
(Ex: $PI\rightarrow IP$)
• Erase two distinct letters and write the letter distinct from the two he erased.
(Ex: $PH\rightarrow I$)
Find the largest integer $C$ such that any Philandese word of up to $C$ letters can be written by Philip through the above operations.
Note: Operations are taken on adjacent letters.
2023 Centroamerican and Caribbean Math Olympiad, 2
Octavio writes an integer $n \geq 1$ on a blackboard and then he starts a process in which, at each step he erases the integer $k$ written on the blackboard and replaces it with one of the following numbers:
$$3k-1, \quad 2k+1, \quad \frac{k}{2}.$$
provided that the result is an integer.
Show that for any integer $n \geq 1$, Octavio can write on the blackboard the number $3^{2023}$ after a finite number of steps.
2009 Belarus Team Selection Test, 1
On R a binary algebraic operation ''*'' is defined which satisfies the following two conditions:
i) for all $a,b \in R$, there exists a unique $x \in R$ such that $x *a=b$ (write $x=b/a$)
ii) $(a*b)*c= (a*c)* (b*c)$ for all $a,b,c \in R$
a) Is this operation necesarily commutative (i.e. $a*b=b*a$ for all $a,b \in R$) ?
b) Prove that $(a/b)/c = (a/c) / (b/c)$ and $(a/b)*c = (a*c) / (b*c)$ for all $a,b,c \in R$.
A. Mirotin
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.
2021 Olimphíada, 5
Let $p$ be an odd prime. The numbers $1, 2, \ldots, d$ are written on a blackboard, where $d \geq p-1$ is a positive integer. A valid operation is to delete two numbers $x$ and $y$ and write $x + y - c \cdot xy$ in their place, where $c$ is a positive integer. One moment there is only one number $A$ left on the board. Show that if there is an order of operations such that $p$ divides $A$, then $p | d$ or $p | d + 1$.
1993 Tournament Of Towns, (364) 3
An operation denoted by $*$ defines, for each pair of numbers $(x, y)$, a number $x*y$ so that for all $x, y$ and $z$ the identities
$$x*x = 0 \,\,\,\,\, (1)$$
and
$$x*(*z) = (x* y)+ z \,\,\,\,\, (2)$$
hold ($+$ denoting ordinary addition of numbers). Find $1993* 1932$.
(G Galperin)
2019 Tuymaada Olympiad, 4
A calculator can square a number or add $1$ to it. It cannot add $1$ two times in a row. By several operations it transformed a number $x$ into a number $S > x^n + 1$ ($x, n,S$ are positive integers). Prove that $S > x^n + x - 1$.
2024 Kyiv City MO Round 1, Problem 4
For real numbers $a_1, a_2, \ldots, a_{200}$, we consider the value $S = a_1a_2 + a_2a_3 + \ldots + a_{199}a_{200} + a_{200}a_1$. In one operation, you can change the sign of any number (that is, change $a_i$ to $-a_i$), and then calculate the value of $S$ for the new numbers again. What is the smallest number of operations needed to always be able to make $S$ nonnegative?
[i]Proposed by Oleksii Masalitin[/i]
2017 Junior Balkan Team Selection Tests - Moldova, Problem 8
The bottom line of a $2\times 13$ rectangle is filled with $13$ tokens marked with the numbers $1, 2, ..., 13$ and located in that order. An operation is a move of a token from its cell into a free adjacent cell (two cells are called adjacent if they have a common side). What is the minimum number of operations needed to rearrange the chips in reverse order in the bottom line of the rectangle?
2023 Olimphíada, 3
Let $n$ be a positive integer. On a blackboard is a circle, and around it $n$ non-negative integers are written. Raphinha plays a game in which an operation consists of erasing a number $a$ neighboring $b$ and $c$, with $b \geq c$, and writing in its place $b + c$ if $b + c \leq 5a/4$ and $b - c$ otherwise.
Your goal is to make all the numbers on the board equal $0$. Find all $n$ such that Raphinha always manages to reach her goal.
2001 Saint Petersburg Mathematical Olympiad, 10.2
The computer "Intel stump-V" can do only one operation with a number: add 1 to it, then rearrange all the zeros in the decimal representation to the end and rearrenge the left digits in any order. (For example from 1004 you could get 1500 or 5100). The number $12345$ was written on the computer and after performing 400 operations, the number 100000 appeared on the screen. How many times has a number with the last digit 0 appeared on the screen?
2023 Belarusian National Olympiad, 11.1
On a set $G$ we are given an operation $*: G \times G \to G$, that for every pair $(x,y)$ of elements of $G$ gives back $x*y \in G$, and for every elements $x,y,z \in G$ the equation $(x*y)*z=x*(y*z)$ holds. $G$ is partitioned into three non-empty sets $A,B$ and $C$.
Can it be that for every three elements $a \in A, b \in B, c \in C$ we have $a*b \in C, b*c \in A, c*a \in B$
2014 Canada National Olympiad, 5
Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.
Kvant 2019, M2584
We have 2019 boxes. Initially, they are all empty. At one operation, we can add exactly 100 stones to some 100 boxes and exactly one stone in each of several other (perhaps none) boxes. What is the smallest possible number of moves after which all boxes will have the same (positive) number of stones.
[i]Proposed by P. Kozhevnikov[/i]
2022 Korea -Final Round, P2
There are $n$ boxes $A_1, ..., A_n$ with non-negative number of pebbles inside it(so it can be empty). Let $a_n$ be the number of pebbles in the box $A_n$. There are total $3n$ pebbles in the boxes. From now on, Alice plays the following operation.
In each operation, Alice choose one of these boxes which is non-empty. Then she divide this pebbles into $n$ group such that difference of number of pebbles in any two group is at most 1, and put these $n$ group of pebbles into $n$ boxes one by one. This continues until only one box has all the pebbles, and the rest of them are empty. And when it's over, define $Length$ as the total number of operations done by Alice.
Let $f(a_1, ..., a_n)$ be the smallest value of $Length$ among all the possible operations on $(a_1, ..., a_n)$. Find the maximum possible value of $f(a_1, ..., a_n)$ among all the ordered pair $(a_1, ..., a_n)$, and find all the ordered pair $(a_1, ..., a_n)$ that equality holds.
2014 Postal Coaching, 5
Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.
1990 Nordic, 4
It is possible to perform three operations $f, g$, and $h$ for positive integers: $f(n) = 10n, g(n) = 10n + 4$, and $h(2n) = n$; in other words, one may write $0$ or $4$ in the end of the number and one may divide an even number by $2$. Prove: every positive integer can be constructed starting from $4$ and performing a finite number of the operations $f,
g,$ and $h$ in some order.
2020 Brazil Team Selection Test, 4
A quadruple of integers $(a, b, c, d)$ is said good if $ad-bc=2020$. Two good quadruplets are said to be dissimilar if it is not possible to obtain one from the other using a finite number of applications of the following operations:
$$(a,b,c,d) \rightarrow (-c,-d,a,b)$$
$$(a,b,c,d) \rightarrow (a,b,c+a,d+b)$$
$$(a,b,c,d) \rightarrow (a,b,c-a,d-b)$$
Let $A$ be a set of $k$ good quadruples, two by two dissimilar. Show that $k \leq 4284$.
Russian TST 2015, P3
Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.