IMO Shortlist 1997 problem 3
Kvaliteta:
Avg: 0,0Težina:
Avg: 0,0 For each finite set
of nonzero vectors in the plane we define
to be the length of the vector that is the sum of all vectors in
Given a finite set
of nonzero vectors in the plane, a subset
of
is said to be maximal if
is greater than or equal to
for each nonempty subset
of
(a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively.
(b) Show that, for any set
consisting of
vectors the number of maximal subsets is less than or equal to
![U](/media/m/d/f/a/dfa3ccb1bb2d14869d77a98d0d2baf97.png)
![l(U)](/media/m/f/6/6/f66ddf5c685905000840a2b56c1ad077.png)
![U.](/media/m/a/9/8/a9879cdc160d05c3471b108fd675e2fb.png)
![V](/media/m/5/d/1/5d1544cc9c474ed7006c60d2c6dfebf6.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![V](/media/m/5/d/1/5d1544cc9c474ed7006c60d2c6dfebf6.png)
![l(B)](/media/m/2/d/e/2de673e536bf42633860d269e341d1a1.png)
![l(A)](/media/m/b/f/a/bfa352482f896ab12e0011155a0b18ac.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![V.](/media/m/c/e/6/ce643105afd61b7e2d481b5fadc3234f.png)
(a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively.
(b) Show that, for any set
![V](/media/m/5/d/1/5d1544cc9c474ed7006c60d2c6dfebf6.png)
![n \geq 1](/media/m/a/9/8/a982fcac3e2c9e0d94e965d6efb5a582.png)
![2n.](/media/m/f/9/f/f9f2aed0d189e77bc754847b3a52d6b4.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1997