Difference between revisions of "2023 IOQM/Problem 11"
(Created page with "==Problem== A positive integer <math>m</math> haas the property that <math>m^2</math> is expressible in the form <math>4n^2-5n+16</math>, where n is an integer. Find the maxim...") |
(→Solution) |
||
| Line 7: | Line 7: | ||
*<math>\textbf{Case III}</math>: <math>(4m-8n+5)=7</math> and <math>(4m+8n-5)=33</math> <math>\Rightarrow</math> <math>m=5</math> and <math>n=-1</math>, <math>|m-n|=6</math>. | *<math>\textbf{Case III}</math>: <math>(4m-8n+5)=7</math> and <math>(4m+8n-5)=33</math> <math>\Rightarrow</math> <math>m=5</math> and <math>n=-1</math>, <math>|m-n|=6</math>. | ||
*<math>\textbf{Case IV}</math>: <math>(4m-8n+5)=11</math> and <math>(4m+8n-5)=21</math> <math>\Rightarrow</math> <math>m=4</math> and <math>n=0</math>, <math>|m-n|=4</math>. | *<math>\textbf{Case IV}</math>: <math>(4m-8n+5)=11</math> and <math>(4m+8n-5)=21</math> <math>\Rightarrow</math> <math>m=4</math> and <math>n=0</math>, <math>|m-n|=4</math>. | ||
| + | Other cases in which the value of <math>(4m-8n+5)</math> and <math>(4m+8n-5)</math> interchange, the values of m and n will not change in those cases. | ||
Thus the maximum value of <math>|m-n|=\boxed{14}</math>. | Thus the maximum value of <math>|m-n|=\boxed{14}</math>. | ||
~Lakshya Pamecha and Parveen Sir | ~Lakshya Pamecha and Parveen Sir | ||
Revision as of 12:20, 2 May 2024
Problem
A positive integer 
haas the property that 
is expressible in the form 
, where n is an integer. Find the maximum value of 
.
Solution
. Now we try to complete the square, multiplying by 
and 
won't complete the square but on multiplication with 
we get 
or 
.
: 
and 
and 
, 
.
: 
and 
and 
,.
: 
and 
and 
, 
.
: 
and 
and 
, 
.
: 
and 
and 
, 
.
: 
and 
and 
,
: 
and 
and 
, 
.
: 
and 
and 
, 
.Other cases in which the value of 
and 
interchange, the values of m and n will not change in those cases.
Thus the maximum value of 
.
~Lakshya Pamecha and Parveen Sir
