Olympiad problems

2021 Hong Kong TST, 3

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?

2017 Peru Iberoamerican Team Selection Test, P4

Tags: combinatorics , Operations

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.

2023 Philippine MO, 5

Tags: combinatorics , Strings , Operations , PMO

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.

2022 Iran-Taiwan Friendly Math Competition, 5

Tags: Taiwan , Iran , combinatorics , Operations

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]

Kvant 2023, M2758

Tags: combinatorics , Operations

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]

2023 Pan-American Girls’ Mathematical Olympiad, 6

Tags: grid , Operations

Let $n \geq 2$ be an integer. Lucia chooses $n$ real numbers $x_1,x_2,\ldots,x_n$ such that $\left| x_i-x_j \right|\geq 1$ for all $i\neq j$. Then, in each cell of an $n \times n$ grid, she writes one of these numbers, in such a way that no number is repeated in the same row or column. Finally, for each cell, she calculates the absolute value of the difference between the number in the cell and the number in the first cell of its same row. Determine the smallest value that the sum of the $n^2$ numbers that Lucia calculated can take.

2021 Junior Balkan Team Selection Tests - Romania, P4

Tags: combinatorics , Sets , Operations , romania , Romanian TST

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 $(3,4)$.