Mathematics Journals

Crux Mathematicorum
an international problem-solving journal published by the Canadian Mathematical Society

A number of my proposals for Crux have been published. Here they are:

2111. [1996: 35]

Does there exist a function  f: N --> N satisfying the 3 conditions:
(i) f(1996) = 1;
(ii) for all primes p, every prime occurs in the sequence f(p), f(2p), f(3p),...,f(kp),... infinitely often; and
(iii) f(f(n)) = 1 for all natural numbers n?

2119. [1996: 76]

(a) Show that for any positive integer m > 2, there is a permutation of m 1's, m 2's and m 3's such that

(i) no block of consecutive terms of the permutation (other than the entire permutation) contains equal numbers of 1's, 2's and 3's; and

(ii) there is no block of m consecutive terms of the permutation which are all equal.

(b) For m = 3, how many such permutations are there?

Find all positive integers n > 1 with the following property: there exists a cyclic permutation of (1, 1, 2, 2, ..., n, n) satisfying:
(i) no two adjacent terms of the permutation (including the last and first terms) are equal, and
(ii) no block of n consecutive terms consists of n distinct integers.

A number of my solutions for Crux problems have also been selected for publication. They include:

1. Problem 2004: an algebra problem proposed by Waldemar Pompe from Poland.
2. Problem 2010: a geometry problem proposed by Marcin E. Kuczma from Poland.
3. Problem 2027: a geometry problem proposed by D. J. Smeenk from the Netherlands.
4. Problem 2028: a number theory problem also proposed by Marcin E. Kuczma from Poland.

My solution to Problem 1487 has been selected to appear in the December 1996 issue of the journal. That's my third contribution to the journal, so I have still got a long way to go!