MathDB

2

Part of 1999 Bundeswettbewerb Mathematik

Problems(2)

a_1 = b_1 = 1 and a_{n+1} = a_n +b_n, b_{n+1} = a_nb_n, coprime terms

Source: 1999 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p2

1/27/2020
The sequences (an)(a_n) and (bn)(b_n) are defined by a1=b1=1a_1 = b_1 = 1 and an+1=an+bn,bn+1=anbna_{n+1} = a_n +b_n, b_{n+1} = a_nb_n for n=1,2,...n = 1,2,... Show that every two distinct terms of the sequence (an)(a_n) are coprime
number theorynumber theory with sequencesSequencesSequencecoprime
Q(3^k) \ge Q(3^{k+1} for infinite n, whereas Q(n) is sum of decimal digits of n

Source: 1999 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p2

1/27/2020
For every natural number nn, let Q(n)Q(n) denote the sum of the decimal digits of nn. Prove that there are infinitely many positive integers kk with Q(3k)≥Q(3k+1)Q(3^k) \ge Q(3^{k+1}).
number theorysum of digitsDigitsinequalities