The numbers $2,4,\ldots,2^{100}$ are written on a board. At a move, one may erase the numbers $a,b$ from the board and replace them with $ab/(a+b).$ Prove that the last numer on the board will be greater than 1. [i]From the folklore[/i]

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 $p | d + 1$.

Silverio is very happy for the 25th year of the PMO. In his jubilation, he ends up writing a finite sequence of As and Gs on a nearby blackboard. He then performs the following operation: if he finds at least one occurrence of the string "AG", he chooses one at random and replaces it with "GAAA". He performs this operation repeatedly until there is no more "AG" string on the blackboard. Show that for any initial sequence of As and Gs, Silverio will eventually be unable to continue doing the operation.

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.

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.

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 $a*b \in C, b*c \in A, c*a \in B$

Let $S$ be the set of [b]lattice[/b] points whose both coordinates are positive integers no larger than $2022$. i.e., $S=\{(x, y) \mid x, y\in \mathbb{N}, \, 1\leq x, y\leq 2022\}$. We put a card with one gold side and one black side on each point in $S$. We call a rectangle [i]"good"[/i] if: (i) All of its sides are parallel to the axes and have positive integer coordinates no larger than $2022$. (ii) The cards on its top-left and bottom-right corners are showing gold, and the cards on its top-right and bottom-left corners are showing black. Each [i]"move"[/i] consists of choosing a good rectangle and flipping all cards simultaneously on its four corners. Find the maximum possible number of moves one can perform, or show that one can perform infinitely many moves. [i]Proposed by CSJL[/i]

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.

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 $k \leq 4284$.

On the table there are $20$ coins of weights $1,2,3,\ldots,15,37,38,39,40$ and $41$ grams. They all look alike but their colours are all distinct. Now Miss Adams knows the weight and colour of each coin, but Mr. Bean knows only the weights of the coins. There is also a balance on the table, and each comparison of weights of two groups of coins is called an operation. Miss Adams wants to tell Mr. Bean which coin is the $1$ gram coin by performing some operations. What is the minimum number of operations she needs to perform?

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

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 $k$.

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?

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.

We have a set of 2n positive integers whose sum is a multiple of n. One operation consists of choosing n of them and adding the same positive integer to all of them. Show that, starting from the initial 2n numbers, we can get all are equal, performing a maximum of 2n - 1 operations.

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]

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)

Let $M$ be a set of $13$ positive integers with the property that $\forall \ m\in M, \ 100\leq m\leq 999$. Prove that there exists a subset $S\subset M$ and a combination of arithmetic operations (addition, subtraction, multiplication, division – without using parentheses) between the elements of $S$, such that the value of the resulting expression is a rational number in the interval $(3,4)$.

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 $S > x^n + x - 1$.

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]

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.

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.

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?

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 $k$.

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 $k$.