MathDB

1

Part of 1988 Vietnam National Olympiad

Problems(2)

Birds and cages

Source: Vietnam NMO 1988 Problem 1

2/2/2009
There are 1988 1988 birds in 994 994 cages, two in each cage. Every day we change the arrangement of the birds so that no cage contains the same two birds as ever before. What is the greatest possible number of days we can keep doing so?
combinatorics unsolvedcombinatorics
Prove that the sequence has a finite limit

Source: Vietnam NMO 1988 Problem 4

2/2/2009
A bounded sequence (xn)n≥1 (x_n)_{n\ge 1} of real numbers satisfies x_n \plus{} x_{n \plus{} 1} \ge 2x_{n \plus{} 2} for all n≥1 n \ge 1. Prove that this sequence has a finite limit.
inductionalgebra unsolvedalgebra