1991 OIM Problems/Problem 5
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
- 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.