8

Part of 2007 All-Russian Olympiad

Problems(2)

decimal fractions of 1/k!

Source: All-Russian 2007

Dima has written number

1/80!,\,1/81!,\,\dots,1/99!

20

infinite pieces of papers as decimal fractions (the following is written on the last piece: \frac {1}{99!} \equal{} 0{,}{00\dots 00}10715\dots, 155 0-s before 1). Sasha wants to cut a fragment of

N

consecutive digits from one of pieces without the comma. For which maximal

N

he may do it so that Dima may not guess, from which piece Sasha has cut his fragment? A. Golovanov

number theory unsolvednumber theory

each cyle contains edges of both colors

Source: All-Russian 2007

Given an undirected graph with

N

vertices. For any set of

k

vertices, where

1\le k\le N

, there are at most

2k-2

edges, which join vertices of this set. Prove that the edges may be coloured in two colours so that each cycle contains edges of both colours. (Graph may contain multiple edges). I. Bogdanov, G. Chelnokov

inductionsymmetrycombinatorics unsolvedcombinatorics