Difference between revisions of "1991 OIM Problems/Problem 5"
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<math>x=\frac{3y \pm \sqrt{2P-y^2}}{2}</math> | <math>x=\frac{3y \pm \sqrt{2P-y^2}}{2}</math> | ||
− | Let <math>K</math> be an integer and <math>K^2=2P-y^2</math>. Therefore, <math>P=\frac{K^2+y^2}{2}</math> Since <math>1 \le P \le 100</math>, then <math>0 \le K,y \le 14</math> because <math>15^2/2>100</math> | + | Let <math>K</math> be an integer and <math>K^2=2P-y^2</math>. Therefore, <math>P=\frac{K^2+y^2}{2}</math> Since <math>1 \le P \le 100</math>, then <math>0 \le K \le 14</math>, <math>-14 \le y \le 14</math> because <math>15^2/2>100</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 | ||
* 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 20:05, 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
- 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.