2023 IOQM/Problem 11
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
