IMO Shortlist 1972 problem 4
Kvaliteta:
Avg: 0,0Težina:
Avg: 0,0 Let
be positive integers. Consider in a plane
two disjoint sets of points
and
consisting of
and
points, respectively, and such that no three points of the union
are collinear. Prove that there exists a straightline
with the following property: Each of the two half-planes determined by
on
(
not being included in either) contains exactly half of the points of
and exactly half of the points of
![n_1, n_2](/media/m/b/9/d/b9ded74c4cdc4352c308e9188ff99458.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![M_1](/media/m/9/7/2/9728b6a1cf3e905234b6989eae0cc038.png)
![M_2](/media/m/4/7/1/4718b55cbd1692b474c94c4104ddb007.png)
![2n_1](/media/m/2/6/b/26bc264bb3b29105f01c9288c0fd3a66.png)
![2n_2](/media/m/3/1/8/3183846c0187e36d7ee93a3556bddcbc.png)
![M_1 \cup M_2](/media/m/1/9/4/19430cca4946cf9ca8b22b40aa7648f3.png)
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
![M_1](/media/m/9/7/2/9728b6a1cf3e905234b6989eae0cc038.png)
![M_2.](/media/m/9/9/4/9941c9a2eb4c6e796d477ce090523c70.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1972