2021 WSMO Team Round/Problem 10
Revision as of 20:28, 23 March 2023 by Megaboy6679 (talk | contribs)
Problem
The minimum possible value of![\[\sqrt{m^2+n^2}+\sqrt{3m^2+3n^2-6m+12n+15}\]](http://latex.artofproblemsolving.com/6/8/d/68d0e30357a8b66f429013f413c224dca472dc7c.png)
can be expressed as 
Find 
Proposed by pinkpig
Solution
Notice that we can complete the square inside the second square root:
![\[\sqrt{3m^2+3n^2-6m+12n+15} =\sqrt{3(m^2+n^2-2m+4n+5)} =\sqrt{3(m^2-2m+1+n^2+4n+4)} =\sqrt{3((m-1)^2+(n+2)^2)}\]](http://latex.artofproblemsolving.com/b/3/c/b3c7bbf690f08f4c7cee0de7a5e93260f3575d87.png)
Notice that we can find the minimum by setting this to 
, which occurs when 
and 
. This gives us the minimum of 
. (If we set the other square root to , we get a minimum of 
which is larger than 
.) Therefore 
.
~programmeruser
