IMO Shortlist 2006 problem C6
Kvaliteta:
Avg: 0,0Težina:
Avg: 8,0 A holey triangle is an upward equilateral triangle of side length
with
upward unit triangular holes cut out. A diamond is a
unit rhombus.
Prove that a holey triangle
can be tiled with diamonds if and only if the following condition holds: Every upward equilateral triangle of side length
in
contains at most
holes, for
.
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![60^\circ-120^\circ](/media/m/3/4/2/342eaa2270843840efe07ac86653fb2c.png)
Prove that a holey triangle
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![1\leq k\leq n](/media/m/d/8/f/d8f38d64dff0e16509f9f41559db7f4c.png)
Izvor: Međunarodna matematička olimpijada, shortlist 2006