Difference between revisions of "2021 WSMO Team Round/Problem 10"
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The minimum possible value of<cmath>\sqrt{m^2+n^2}+\sqrt{3m^2+3n^2-6m+12n+15}</cmath>can be expressed as <math>a.</math> Find <math>a^2.</math> | The minimum possible value of<cmath>\sqrt{m^2+n^2}+\sqrt{3m^2+3n^2-6m+12n+15}</cmath>can be expressed as <math>a.</math> Find <math>a^2.</math> | ||
| − | ''Proposed by | + | ''Proposed by pinkpig'' |
==Solution== | ==Solution== | ||
Revision as of 19:27, 23 March 2023
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:
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
