Problem:
In a given plane, points and are units apart. How many points are there in the plane such that the perimeter of is units and the area of is square units?
Answer Choices:
A.
B.
C.
D.
E. infinitely many
Solution:
Notice that whatever point we pick for will be the base of the triangle. Without loss of generality, let points and be and , since for any other combination of points, we can just rotate the plane to make them and under a new coordinate system. When we pick point , we have to make sure that its -coordinate is , because that's the only way the area of the triangle can be .
Now when the perimeter is minimized, by symmetry, we put in the middle, at (5,20). We can easily see that and will both be . The perimeter of this minimal triangle is , which is larger than . Since the minimum perimeter is greater than , there is no triangle that satisfies the condition, giving us .
OR
Without loss of generality, let be a horizontal segment of length . Now realize that has to lie on one of the lines parallel to and vertically units away from it. But is already , and this doesn't form a triangle. Otherwise, without loss of generality, . Dropping altitude , we have a right triangle with hypotenuse and leg , which is clearly impossible, again giving the answer as .
The problems on this page are the property of the MAA's American Mathematics Competitions