MathDB

7

Part of 2008 Saint Petersburg Mathematical Olympiad

Problems(2)

Sequence

Source: St Petersburg 2008 10th grade #7

7/29/2011
In a sequence, x1=12x_1=\frac{1}{2} and xn+1=1−x1x2x3...xnx_{n+1}=1-x_1x_2x_3...x_n for n≥1n\ge 1. Prove that 0.99<x100<0.9910.99<x_{100}<0.991.
Fresh translation. This problem may be similar to one of the 9th grade problems.
inequalitiesinequalities proposed
Roads in country

Source: St Petersburg Olympiad 2008, Grade 11, P7

8/30/2017
There are 1000010000 cities in country, and roads between some cities. Every city has <100<100 roads. Every cycle route with odd number of road consists of ≥101\geq 101 roads. Prove that we can divide all cities in 100100 groups with 100100 cities, such that every road leads from one group to other.
combinatoricsgraph theory