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# Problem

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Le Bron has three pins, labeled A,  B, and C, respectively, located at the origin of the coordinate plane. In a move, Le Bron may move a pin to an adjacent lattice point at distance 1 away. What is the least number of moves that Le Bron can make in order for triangle ABC to have area 2021?

Jan 13, 2024

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To find the minimum number of moves for Le Bron to create a triangle with area 2021, we need to consider the possible configurations of points A, B, and C.

Since the points are initially at the origin, Le Bron can move each pin to any lattice point with integer coordinates. Let's consider the possible side lengths of the triangle. The distance between any two lattice points is an integer.

If Le Bron moves one pin to a point with coordinates (a,b), another to (c,d), and the third stays at the origin, the side lengths of the triangle will be a,b,c,d.

The area of the triangle with sides a,b,c can be calculated using the formula:

Area=21​∣ac−bd∣

To achieve an area of 2021, we need ∣ac−bd∣=4042 (as 2×2021=40422×2021=4042). Now, we need to find two pairs (a,c) and (b,d) whose product difference is 4042.

Let's factorize 4042:

4042=2×2021=2×7×17×17

Now, we need to find two pairs of factors whose product difference is 4042. One possible configuration could be:

(a,c)=(2,2021)

(b,d)=(7,17)

So, Le Bron can achieve a triangle with area 2021 by moving one pin to (2, 2021), another to (7, 17), and leaving the third at the origin.

The minimum number of moves required is the sum of the absolute values of the coordinates, which is 2+2021+7+17=2047.

Therefore, the least number of moves Le Bron can make for triangle ABC to have an area of 2021 is 2047 moves.

Jan 13, 2024