Balkan 00/3 solution

17th Balkan 2000

Problem 3

How many 1 by 10√2 rectangles can be cut from a 50 x 90 rectangle using cuts parallel to its edges.

Solution

Answer: 315.

We can place 3 tiles long-ways across the width and 6 tiles long-ways across the length of the rectangle. So place 50 x 6 = 300 tiles parallel to the long side to fill a 50 x 90√2 rectangle. Then place 5 rows of 3 tiles the other way. That shows that 315 is possible. The problem is to prove we cannot do better.

The basic idea is to draw a set of equally-spaced diagonal lines (at 45 deg to the sides of the rectangle) across the rectangle. If the lines are a distance 10 apart, so that they cut the sides at a spacing of 10√2, then either one or two lines will intersect each tile. Moreover the intersection of a tile with the lines will be one or two line segments of total length √2. The idea is to show that the total length of the parallel lines (inside the rectangle) is less than 316√2.

This is only true for some positions of the lines. For example, if we take the rectangle to have diagonal (0, 0) to (50, 90), then taking a line through the origin (to the point (50,50) ) does not work. We may, however, take lines as follows:


(0, 90 - 10√2) to (10√2,90) length 20

(0, 90 - 20√2) to (20√2, 90) length 40

(0, 90 - 30√2) to (30√2, 90) length 60

(0, 90 - 40√2) to (50, 140 - 40√2) length 50√2

(0, 90 - 50√2) to (50,140 - 50√2) length 50√2

(0, 90 - 60√2) to (50, 140 - 60√2) length 50√2

(70√2 - 90, 0) to (50, 140 - 70√2) length 140√2 - 140

(80√2 - 90, 0) to (50, 140 - 80√2) length 140√2 - 160

(90√2 - 90, 0) to (50, 140 - 90√2) length 140√2 - 180.



Total length 570√2 - 360.

Now we have that 570√2 - 360 < 316√2 which proves the result.

17th Balkan 2000