President's Problems

We continue a new feature called the President's Problems, with new problems posed by Karl Zilm. If you wish, you may send proposed solutions to Karl at Lewis & Clark. His address is in the mailing list earlier in this issue. Some of these exercises are very easy if you use a Computer Algebra System such as DERIVE. Of course, a graphics calculator or a TI-92 will help as well. (Extra Credit: submit solutions by e-mail using an attached file).

These problems all emphasize proofs of varying levels of complexity.

1. Prove: the sum of the cubes of the first "n" positive integers is a perfect square.

2. Prove: if the product of any four consecutive positive integers is increased by one, then the result is a perfect square.

3. A palindrome is a word, phrase or number which reads the same forward and backward. For example, "noon" is a word palindrome, "toot, Otto, toot" and "If I had a Hi-fi" are palindromic phrases, and 1,234,321 is a palindromic number. Prove that every palindromic number with an even number of digits is divisible by eleven.