MathDB

Problem 4

Part of 1989 French Mathematical Olympiad

Problems(1)

recurrence operation, exponential

Source: France 1989 P4

5/18/2021
For natural numbers x1,…,xkx_1,\ldots,x_k, let [xk,…,x1][x_k,\ldots,x_1] be defined recurrently as follows: [x2,x1]=x2x1[x_2,x_1]=x_2^{x_1} and, for k≥3k\ge3, [xk,xk−1,…,x1]=xk[xk−1,…,x1][x_k,x_{k-1},\ldots,x_1]=x_k^{[x_{k-1},\ldots,x_1]}.
(a) Let 3≤a1≤a2≤…≤an3\le a_1\le a_2\le\ldots\le a_nbe integers. For a permutation σ\sigma of the set {1,2,…,n}\{1,2,\ldots,n\}, we set P(σ)=[aσ(n),aσ(n−1),…,aσ(1)]P(\sigma)=[a_{\sigma(n)},a_{\sigma(n-1)},\ldots,a_{\sigma(1)}]. Find the permutations σ\sigma for which P(σ)P(\sigma) is minimal or maximal. (b) Find all integers a,b,c,da,b,c,d, greater than or equal to 22, for which [178,9]≤[a,b,c,d]≤[198,9][178,9]\le[a,b,c,d]\le[198,9].
recurrence relationalgebra