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> | ||
+ | Now we try the possible combinations of <math>n</math> and <math>m</math>: | ||
+ | <math>\begin{cases} | ||
+ | m=0\text{; }n=0\text{; }P=(0^2+0^2)/2=0; & \text{NO}\\ | ||
+ | m=0\text{; }n=1\text{; }P=(0^2+1^2)/2=2; & \text{YES}\\ | ||
+ | m=0\text{; }n=2\text{; }P=(0^2+2^2)/2=8; & \text{YES}\\ | ||
+ | m=0\text{; }n=3\text{; }P=(0^2+3^2)/2=18; & \text{YES}\\ | ||
+ | m=0\text{; }n=4\text{; }P=(0^2+4^2)/2=32; & \text{YES}\\ | ||
+ | m=0\text{; }n=5\text{; }P=(0^2+5^2)/2=50; & \text{YES}\\ | ||
+ | m=0\text{; }n=6\text{; }P=(0^2+6^2)/2=72; & \text{YES}\\ | ||
+ | m=0\text{; }n=7\text{; }P=(0^2+7^2)/2=98; & \text{YES}\\ | ||
+ | m=1\text{; }n=1\text{; }P=(1^2+1^2)/2=4; & \text{YES}\\ | ||
+ | m=1\text{; }n=2\text{; }P=(1^2+2^2)/2=10; & \text{YES}\\ | ||
+ | m=1\text{; }n=3\text{; }P=(1^2+3^2)/2=20; & \text{YES}\\ | ||
+ | m=1\text{; }n=4\text{; }P=(1^2+4^2)/2=34; & \text{YES}\\ | ||
+ | m=1\text{; }n=5\text{; }P=(1^2+5^2)/2=52; & \text{YES}\\ | ||
+ | m=1\text{; }n=6\text{; }P=(1^2+6^2)/2=74; & \text{YES}\\ | ||
+ | m=1\text{; }n=7\text{; }P=(1^2+7^2)/2=100; & \text{YES}\\ | ||
+ | m=2\text{; }n=2\text{; }P=(2^2+2^2)/2=16; & \text{YES}\\ | ||
+ | m=2\text{; }n=3\text{; }P=(2^2+3^2)/2=26; & \text{YES}\\ | ||
+ | m=2\text{; }n=4\text{; }P=(2^2+4^2)/2=40; & \text{YES}\\ | ||
+ | m=2\text{; }n=5\text{; }P=(2^2+5^2)/2=58; & \text{YES}\\ | ||
+ | m=2\text{; }n=6\text{; }P=(2^2+6^2)/2=80; & \text{YES}\\ | ||
+ | m=2\text{; }n=7\text{; }P=(2^2+7^2)/2=106; & \text{NO}\\ | ||
+ | m=3\text{; }n=3\text{; }P=(3^2+3^2)/2=36; & \text{YES}\\ | ||
+ | m=3\text{; }n=4\text{; }P=(3^2+4^2)/2=50; & \text{YES}\\ | ||
+ | m=3\text{; }n=5\text{; }P=(3^2+5^2)/2=68; & \text{YES}\\ | ||
+ | m=3\text{; }n=6\text{; }P=(3^2+6^2)/2=90; & \text{YES}\\ | ||
+ | m=3\text{; }n=7\text{; }P=(3^2+7^2)/2=116; & \text{NO}\\ | ||
+ | m=4\text{; }n=4\text{; }P=(4^2+4^2)/2=64; & \text{YES}\\ | ||
+ | m=4\text{; }n=5\text{; }P=(4^2+5^2)/2=82; & \text{YES}\\ | ||
+ | m=4\text{; }n=6\text{; }P=(4^2+6^2)/2=104; & \text{NO}\\ | ||
+ | m=4\text{; }n=7\text{; }P=(4^2+7^2)/2=130; & \text{NO}\\ | ||
+ | m=5\text{; }n=5\text{; }P=(5^2+5^2)/2=100; & \text{YES}\\ | ||
+ | m=5\text{; }n=6\text{; }P=(5^2+6^2)/2=122; & \text{NO}\\ | ||
+ | m=5\text{; }n=7\text{; }P=(5^2+7^2)/2=148; & \text{NO}\\ | ||
+ | m=6\text{; }n=6\text{; }P=(6^2+6^2)/2=144; & \text{NO}\\ | ||
+ | m=6\text{; }n=7\text{; }P=(6^2+7^2)/2=170; & \text{NO}\\ | ||
+ | m=7\text{; }n=7\text{; }P=(7^2+7^2)/2=196; & \text{NO} | ||
+ | \end{cases}</math> | ||
* 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 20:21, 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.