MathDB

2

Part of 2015 Poland - Second Round

Problems(2)

Inequality

Source: Polish National Olympiad 2015 2nd round, 2nd problem

3/3/2015
Let AA be an integer and A>1A>1. Let a1=AAa_{1}=A^{A}, an+1=Aana_{n+1}=A^{a_{n}} and b1=AA+1b_{1}=A^{A+1}, bn+1=2bnb_{n+1}=2^{b_{n}}, n=1,2,3,...n=1, 2, 3, .... Prove that an<bna_{n}<b_{n} for each nn.
inequalities
no of sequences with terms in {0,1,2,3} so that n=a_0+2a_1+2^2a_2+...+2^na_n

Source: Polish National Olympiad 2015 2nd round, p5

8/28/2019
Let nn be a positive integer. Determine the number of sequences a0,a1,…,ana_0, a_1, \ldots, a_n with terms in the set {0,1,2,3}\{0,1,2,3\} such that n=a0+2a1+22a2+…+2nan.n=a_0+2a_1+2^2a_2+\ldots+2^na_n.
algebranumber theorycombinatorics