Difference between revisions of "1991 OIM Problems/Problem 5"
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Let <math>K=2n</math>, <math>y=2m</math> where integers <math>n</math> and <math>m</math> with <math>0 \le n \le 7</math> and <math>0 \le m \le 7</math> | Let <math>K=2n</math>, <math>y=2m</math> where integers <math>n</math> and <math>m</math> with <math>0 \le n \le 7</math> and <math>0 \le m \le 7</math> | ||
| + | |||
| + | <math>P=\frac{K^2+y^2}{2}=\frac{4n^2+4m^2}{2}=2(n^2+m^2)</math> | ||
Now we try the possible combinations of <math>n</math> and <math>m</math>: | Now we try the possible combinations of <math>n</math> and <math>m</math>: | ||
Revision as of 21:23, 22 December 2023
Problem
Let 
. We will say that an integer 
is a value of 
if there exist integers 
and 
such that 
.
i. Determine how many elements of {1, 2, 3, ... ,100} are values of .
ii. Prove that the product of values of is a value of .
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
Part i.
Let 
, 
, be integers
, then solving for using the quadratic equation we have:
Let 
be an integer and 
. Therefore, 
Since 
, then 
, 
because 
Since 
we can look at the combinations of with for non-negative values. So, we can use: 
to find the values of
Since 
, , then to get integers and , both expressions 
and 
need to be even. This happens when either and are both odd, or both even. Thus we will try both cases:
Case 1: Both and are even.
Let 
, 
where integers 
and 
with 
and 
Now we try the possible combinations of and :
- Note. I actually competed at this event in Argentina when I was in High School representing Puerto Rico. I have no idea what I did on this one nor how many points they gave me. Probably close to zero on this one.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
