Values for the area of this parallelogram [Slovenia 2010]
Source:
November 16, 2010
geometryparallelogramtrapezoidsymmetrycombinatorics proposedcombinatorics
Problem Statement
Let be an equilateral triangle with the side of units. Amir divides this triangle into smaller equilateral triangles with the sides of unit. Reza then picks of the vertices of these smaller triangles. The vertices lie inside the triangle and form a parallelogram with sides parallel to the sides of the triangle There are exactly smaller triangles that have at least one point in common with the sides of this parallelogram. Find all possible values for the area of this parallelogram.[asy]
unitsize(150);
defaultpen(linewidth(0.7));
int n = 20; /* # of vertical lines, including BC */
pair A = (0,0), B = dir(-30), C = dir(30);
draw(A--B--C--cycle,linewidth(1)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0));
label("",A,W); label("",C,NE); label("",B,SE);
for(int i = 1; i < n; ++i) {
draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n);
draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n);
draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n);
}[/asy][Thanks azjps for drawing the diagram.][hide="Note"]Note: Vid changed to Amir, and Eva change to Reza!