Math 289 โ Winter 2007 โ Problem Set 11
Due April 4, 2007
1.(15) A positive integer is wavy if for each of its three consecutive digits (in the
decimal representation) holds
(aโb)(bโc)<0.
Prove that there are more than 25,000 different 10-digit wavy numbers such that
each of them starts with a non-zero digit and it contains all the digits 1,2,...,9,0.
2.(15) Find all ordered triples (x, y, z) of mutually distinct real numbers which
satisfy the following set equality:
{x, y, z}=๎xโy
yโz,yโz
zโx,zโx
xโy๎.
3.(15) The set Mhas the following properties
โขMcontains all integers between 1 and 2007 (including the end points).
โขIf nโMthen Mcontains all the terms of an infinite arithmetic sequence with
the first term nand the difference n+ 1.
Determine whether Mmust necessary contain all integers bigger than a certain
number m.
4.(20) Let Nbe the set of all positive integers and let f:NโNbe a function
satisfying
f(xf(y)) = yf (x) for all x, y โN.
Find the smallest possible value of f(2007).