Apparently, I bit off more than I could chew when tackling this question. I don’t have formal mathematical solution to this problem, but I will show you what I came up with.

Question: Determine the maximum value of , where m and n are integers in the range 1,2,…,1981 satisfying .

Analysis: In an attempt to maximize , we should first investigate what values of m and n satisfy the constraints which can be reduced to . If we look for solutions to this equation for small values of m and n, you will find the following sequence:

The solution appears to be Fibonacci numbers. This would mean that if (m, n) are solutions then (n, m+n) would also be a solution. If we put (n, m+n) into the constraints, we can verify that all Fibonacci numbers are a solution.

This shows that the following sequence is a solution to the given constraints:

The part that I wasn’t able to prove is that the Fibonacci numbers are the ONLY solution to the constraints. However, if they were, the function would be maximized when .

In order to prove that this was the correct solution, I wrote the following perl script:

$max_val = 0;
$max_m = 0;
$max_n = 0;
 
for ($m = 1; $m < 1982; $m++) {
   for ($n = 1; $n < 1982; $n++) {
      $val = $n*$n - $m*$n - $m*$m;
      $val = $val*$val;
 
      if($val == 1){
         $val = $m*$m + $n*$n;
 
         if($val > $max_val){
            $max_val = $val;
            $max_m = $m;
            $max_n = $n;
         }
      }
   }
}
 
print "Max Value: ".$max_val." m: ". $max_m." n: ".$max_n;

This script confirmed that the maximum value is when

$ perl find.pl
Max Value: 3524578 m: 987 n: 1597

This week has been really busy, so I haven’t put a lot of thought into my weekly question. I was reading about problems dealing with rotating frames of references and gravitational fields in my Classical Mechanics Textbook, but I couldn’t find any interesting problems. I decided to follow Rod from Reasonable Deviations and post today’s Putnam problem from the Harvard Mathematics Website.

Question: Determine the maximum value of , where m and n are integers in the range 1,2,…,1981 satisfying

I found an interesting problem on the blog Reasonable Deviations.

Question:
Given a plane , a point in the plane and a point not in the plane, find all points in such that the ratio is a maximum.

Solution:
First, create a plane that intersects the points Q, P, and R. In addition to these 3-points, I’m going to create a 4th point point called R’ which is perpendicular to the plane.

We can simplify the formula by using the following relations: . The important thing to note is that x is the only variable. The other variables are constant.

The formula can be reduced to:

The maximum of f(x) will occur when its derivative is 0. Therefore,

Therefore, the point R will a distance x away from R’ defined by the following equation: