MathDB

1

Part of 2004 IMO Shortlist

Problems(2)

problem involving the tau function, Australia TST 1

Source: IMO Shortlist 2004, number theory problem 1

4/20/2005
Let τ(n)\tau(n) denote the number of positive divisors of the positive integer nn. Prove that there exist infinitely many positive integers aa such that the equation τ(an)=n \tau(an)=n does not have a positive integer solution nn.
number theoryDivisorsequationIMO Shortlist
10001 students at an university

Source: IMO ShortList 2004, combinatorics problem 1

6/15/2005
There are 1000110001 students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of kk societies. Suppose that the following conditions hold:
i.) Each pair of students are in exactly one club.
ii.) For each student and each society, the student is in exactly one club of the society.
iii.) Each club has an odd number of students. In addition, a club with 2m+1{2m+1} students (mm is a positive integer) is in exactly mm societies.
Find all possible values of kk.
Proposed by Guihua Gong, Puerto Rico
combinatoricsIMO Shortlistdouble counting