2021 USAJMO Problems/Problem 4
Carina has three pins, labeled , and , respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance away. What is the least number of moves that Carina can make in order for triangle to have area 2021?
(A lattice point is a point in the coordinate plane where and are both integers, not necessarily positive.)
The answer is , achievable by . We now show the bound.
We first do the following optimizations:
-if you have a point goes both left and right, we may obviously delete both of these moves and decrease the number of moves by .
-if all of lie on one side of the plane, for example , we shift them all down, decreasing the number of moves by , until one of the points is on for the first time.
Now we may assume that , , where . Note we may still shift all down by if , decreasing the number of moves by , until one of is on for the first time. So we may assume one of and is , by symmetry. In particular, by shoelace the answer to 2021 JMO Problem 4 is the minimum of the answers to the following problems:
Case 1 (where ) if , find the minimum possible value of .
Case 2 (else) , find the minimum possible value of .
Note that so if is fixed then is maximized exactly when is minimized. In particular, if then as desired.
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