MathDB

1

Part of 2022 Saudi Arabia IMO TST

Problems(2)

exists index i, such prime p|a_i, a_{n+1}=a_n^2 + n a_n .

Source: 2022 Saudi Arabia IMO TST 1.1

11/1/2022
Let (an)(a_n) be the integer sequence which is defined by a1=1a_1= 1 and an+1=an2+n⋅an  ,  ∀n≥1. a_{n+1}=a_n^2 + n \cdot a_n \,\, , \,\, \forall n \ge 1. Let SS be the set of all primes pp such that there exists an index ii such that p∣aip|a_i. Prove that the set SS is an infinite set and it is not equal to the set of all primes.
number theorydividesnumber theory with sequencesrecurrence relation
2022 / 2023 plates around a round table, 2 player game

Source: 2022 Saudi Arabia IMO TST 4.1

11/1/2022
There are a) 20222022, b) 20232023 plates placed around a round table and on each of them there is one coin. Alice and Bob are playing a game that proceeds in rounds indefinitely as follows. In each round, Alice first chooses a plate on which there is at least one coin. Then Bob moves one coin from this plate to one of the two adjacent plates, chosen by him. Determine whether it is possible for Bob to select his moves so that, no matter how Alice selects her moves, there are never more than two coins on any plate.
combinatoricsgamegame strategy