MathDB

5

Part of 2010 Slovenia National Olympiad

Problems(4)

Find all possible values for the area [Slovenia 2010]

Source:

11/14/2010
Let ABCDABCD be a square with the side of 2020 units. Amir divides this square into 400400 unit squares. Reza then picks 44 of the vertices of these unit squares. These vertices lie inside the square ABCDABCD and define a rectangle with the sides parallel to the sides of the square ABCD.ABCD. There are exactly 2424 unit squares which have at least one point in common with the sides of this rectangle. Find all possible values for the area of a rectangle with these properties.
[hide="Note"]Note: Vid changed to Amir, and Eva change to Reza!
geometryrectanglecombinatorics proposedcombinatorics
Values for the area of this parallelogram [Slovenia 2010]

Source:

11/16/2010
Let ABCABC be an equilateral triangle with the side of 2020 units. Amir divides this triangle into 400400 smaller equilateral triangles with the sides of 11 unit. Reza then picks 44 of the vertices of these smaller triangles. The vertices lie inside the triangle ABCABC and form a parallelogram with sides parallel to the sides of the triangle ABC.ABC. There are exactly 4646 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("AA",A,W); label("CC",C,NE); label("BB",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!
geometryparallelogramtrapezoidsymmetrycombinatorics proposedcombinatorics
Pirates and the chest [Slovenia 2010]

Source:

11/16/2010
Ten pirates find a chest filled with golden and silver coins. There are twice as many silver coins in the chest as there are golden. They divide the golden coins in such a way that the difference of the numbers of coins given to any two of the pirates is not divisible by 10.10. Prove that they cannot divide the silver coins in the same way.
combinatorics proposedcombinatorics
Find n for which there exists a polygon [Slovenia 2010]

Source:

11/16/2010
For what positive integers n≥3n \geq 3 does there exist a polygon with nn vertices (not necessarily convex) with property that each of its sides is parallel to another one of its sides?
geometry unsolvedgeometry