MathDB

7

Part of 2023 Caucasus Mathematical Olympiad

Problems(2)

Sasha has $10$ cards with numbers $1, 2, 4, 8,\ldots, 512$

Source: Caucasus MO 2023

7/16/2023
Sasha has 1010 cards with numbers 1,2,4,8,…,5121, 2, 4, 8,\ldots, 512. He writes the number 00 on the board and invites Dima to play a game. Dima tells the integer 0<p<10,p0 < p < 10, p can vary from round to round. Sasha chooses pp cards before which he puts a “++” sign, and before the other cards he puts a “−-" sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha’s moves.
combinatorics
Find all $n$ such that after $n-1$ moves it is possible to obtain $0$.

Source: Caucasus MO 2023

7/16/2023
Numbers 1,2,…,n1, 2,\ldots, n are written on the board. By one move, we replace some two numbers a,b a, b with the number a2−ba^2-b{}. Find all nn{} such that after n−1n-1 moves it is possible to obtain 00.