MathDB

3

Part of 1999 Tuymaada Olympiad

Problems(2)

Tuymaada 1999, Q3

Source: Tuymaada 1999, Q3

7/31/2012
What maximum number of elements can be selected from the set {1,2,3,…,100}\{1, 2, 3, \dots, 100\} so that no sum of any three selected numbers is equal to a selected number?
Proposed by A. Golovanov
combinatorics unsolvedcombinatorics
Tuymaada 1999, Q7

Source: Tuymaada 1999, Q7

7/31/2012
A sequence of integers a0, a1,…an…a_0,\ a_1,\dots a_n \dots is defined by the following rules: a0=0, a1=1, an+1>ana_0=0,\ a_1=1,\ a_{n+1} > a_n for each n∈Nn\in \mathbb{N}, and an+1a_{n+1} is the minimum number such that no three numbers among a0, a1,…an+1a_0,\ a_1,\dots a_{n+1} form an arithmetical progression. Prove that a2n=3na_{2^n}=3^n for each n∈N.n \in \mathbb{N}.
algebra unsolvedalgebra