Difference between revisions of "1991 OIM Problems/Problem 5"
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Since <math>(-y)^2=y^2</math> we can look at the combinations of <math>y</math> with <math>K</math> for non-negative values. So, we can use: <math>0 \le y \le 14</math> to find the values of <math>P</math> | Since <math>(-y)^2=y^2</math> we can look at the combinations of <math>y</math> with <math>K</math> for non-negative values. So, we can use: <math>0 \le y \le 14</math> to find the values of <math>P</math> | ||
− | Since | + | Since <math>x=\frac{3y \pm K}{2}</math>, <math>P=\frac{K^2+y^2}{2}</math>, then to get integers <math>x</math> and <math>P</math>, both expressions <math>K^2+y^2</math> and <math>3y \pm K</math> need to be even. This happens when either <math>K</math> and <math>y</math> are both odd, or both even. |
* 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. | * 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. |
Revision as of 21:08, 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.
- 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.