MathDB

11.4

Part of VMEO III 2006

Problems(2)

marbles on lattice points on infinite grid

Source: 2006 VMEO III Juniors 11.4 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463155_vmeo_iii

9/11/2021
On an infi nite grid, a square with four vertices lie at (m,n)(m, n), (m−1,n)(m-1, n), (m,n−1)(m,n-1), (m−1,n−1)(m-1, n-1) is denoted as cell (m,n)(m,n) (m,n∈Z)(m, n \in Z). Some marbles are dropped on some cell. Each cell may have more than one marble or have no marble at all. Consider a "move" can be conducted in one of two following ways: i) Remove one marble from cell (m,n)(m,n) (if there is marble at that cell), then add one marble to each of cell (m−1,n−2)(m - 1, n- 2) and cell (m−2,n−1)(m -2, n - 1). ii) Remove two marbles from cell (m,n)(m,n) (if there is marble at that cell), then add one marble to each of cell (m+1,n−2)(m +1, n - 2) and cell (m−2,n+1)(m - 2, n +1). Assume that initially, there are nn marbles at the cell (1,n),(2,n−1),...,(n,1)(1,n), (2,n - 1),..., (n, 1) (each cell contains one marble). Can we conduct an finite amount of moves such that both cells (n+1,n)(n + 1, n) and (n,n+1)(n, n + 1) have marbles?
combinatoricsgame strategygame
sequence with coprime terms wanted

Source: 2006 VMEO III Seniors 11.4 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463155_vmeo_iii

9/17/2021
Given an integer a>1a>1. Let p1<p2<...<pkp_1 < p_2 <...< p_k be all prime divisors of aa. For each positive integer nn we define:
C0(n)=a2n,C1(n)=a2np12,....,Ck(n)=a2npk2C_0(n) = a^{2n}, C_1(n) =\frac{a^{2n}}{p^2_1}, .... , C_k(n) =\frac{a^{2_n}}{p^2_k} A=a2+1A = a^2 + 1 T(n)=AC0(n)−1T(n) = A^{C_0(n)} - 1 M(n)=LCM(a2n+2,AC1(n)−1,...,ACk(n)−1)M(n) = LCM(a^{2n+2}, A^{C_1(n)} - 1, ..., A^{C_k(n)} - 1) An=T(n)M(n)A_n =\frac{T(n)}{M(n)}
Prove that the sequence A1,A2,...A_1, A_2, ... satisfies the properties: (i) Every number in the sequence is an integer greater than 11 and has only prime divisors of the form am+1am + 1. (ii) Any two different numbers in the sequence are coprime.
number theorycoprimeLCM