In this regular feature, you are invited to send us problems for publication - but with each problem you must send your solution, and, if the problem is not your own invention, a reference to its source. A subscription to Zimaths for 2003 (or equivalent backcopies) will be awarded for every Reader whose Problem(s) we publish in 2002 The best solutions we receive from other readers will be published under their name. The following Readers' Problems remain unsolved: 4.3-1, 5.1-1, 5.3-4, 5.3-5.

6.2-1 From Munyaradzi Mukachana of Victoria High School
FOR FORMS 3 & 4 ONLY:
Let f(x)=x3-18x2+72x.
(i) Verify that the roots of f(x)=0 form an arithmetic sequence.
(ii) If k is a constant and the roots of f(x)-k=0 form a geometric sequence, what is the value of k?
[Source: Mathematical Digest July 1996]
6.2-2 From Tawanda E Mungure of Victoria High School
Show that given any 5 numbers it is possible to choose 3 whose sum is divisible by 3. Show also that given any 17 numbers it is possible to choose 5 whose sum is divisible by 5. Now generalise to sets of numbers whose sum is to be divisible by 7, etc.

6.2-3 From Tawanda E Mungure of Victoria High School
Inside a cube of side 15 units there are 11 000 given points. Prove that there is a sphere of unit radius (that is, radius 1) within which there are at least 6 of the given points.