2

Part of 2023 Francophone Mathematical Olympiad

Problems(2)

whatever strategy is the list of numbers on the blackboard remains the same

Source: P2 Francophone Math Olympiad Junior 2023

On her blackboard, Alice has written

n

integers strictly greater than

1

. Then, she can, as often as she likes, erase two numbers

a

b

a \neq b

, and replace them with

q

q^2

q

is the product of the prime factors of

ab

(each prime factor is counted only once). For instance, if Alice erases the numbers

4

6

, the prime factors of

ab = 2^3 \times 3

2

3

, and Alice writes

q = 6

q^2 =36

. Prove that, after some time, and whatever Alice's strategy is, the list of numbers written on the blackboard will never change anymore.
Note: The order of the numbers of the list is not important.

combinatoricsgamegame strategy

Scrooge McDuck owns k gold coins

Source: P2 Francophone Math Olympiad Senior 2023

k

be a positive integer. Scrooge McDuck owns

k

gold coins. He also owns infinitely many boxes

B_1, B_2, B_3, \ldots

B_1

contains one coin, and the

k-1

other coins are on McDuck's table, outside of every box. Then, Scrooge McDuck allows himself to do the following kind of operations, as many times as he likes: - if two consecutive boxes

B_i

B_{i+1}

both contain a coin, McDuck can remove the coin contained in box

B_{i+1}

and put it on his table; - if a box

B_i

contains a coin, the box

B_{i+1}

is empty, and McDuck still has at least one coin on his table, he can take such a coin and put it in box

B_{i+1}

. As a function of

k

, which are the integers

n

for which Scrooge McDuck can put a coin in box

B_n