MathDB

3

Part of 2021 Final Mathematical Cup

Problems(2)

S(1)S(3)...S(2n-1) >=S(2)S(4)...S(2n) , sum of digits inequality

Source: 2021 3nd Final Mathematical Cup Junior Division P3 FMC

10/13/2021
For every positive integer nn, s(n)s(n) denotes the sum of the digits in the decimal representation of nn. Prove that for every integer n≥5n \ge 5, we have S(1)S(3)...S(2n−1)≥S(2)S(4)...S(2n)S(1)S(3)...S(2n-1) \ge S(2)S(4)...S(2n)
inequalitiesnumber theorysum of digitsalgebra
max X_1^{X_2^{...^{X_k}}} for n = X_1X_2...X_k

Source: 2021 3nd Final Mathematical Cup Senior Division P3 FMC

10/30/2022
For a positive integer nn we define f(n)=max⁡X1X2...Xkf (n) = \max X_1^{X_2^{...^{X_k}}} where the maximum is taken over all possible decompositions of natural numbers n=X1X2...Xkn = X_1X_2...X_k. Determine f(n)f(n).
algebranumber theory