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...") |
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*<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 11:20, 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 , .
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