IMO Shortlist 2008 problem C3
Kvaliteta:
Avg: 0,0Težina:
Avg: 7,0 In the coordinate plane consider the set
of all points with integer coordinates. For a positive integer
, two distinct points
,
will be called
-friends if there is a point
such that the area of the triangle
is equal to
. A set
will be called
-clique if every two points in
are
-friends. Find the least positive integer
for which there exits a
-clique with more than 200 elements.
Proposed by Jorge Tipe, Peru
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
![B\in S](/media/m/c/9/b/c9b989e70116c21d70d6f364f3ab5c46.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![C\in S](/media/m/d/f/e/dfe4404f3c4ea234623ee22672b69697.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![T\subset S](/media/m/e/8/1/e810227b48c4446c9913194f844a6787.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)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
Proposed by Jorge Tipe, Peru
Izvor: Međunarodna matematička olimpijada, shortlist 2008