Let
![n_1, n_2](/media/m/b/9/d/b9ded74c4cdc4352c308e9188ff99458.png)
be positive integers. Consider in a plane
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
two disjoint sets of points
![M_1](/media/m/9/7/2/9728b6a1cf3e905234b6989eae0cc038.png)
and
![M_2](/media/m/4/7/1/4718b55cbd1692b474c94c4104ddb007.png)
consisting of
![2n_1](/media/m/2/6/b/26bc264bb3b29105f01c9288c0fd3a66.png)
and
![2n_2](/media/m/3/1/8/3183846c0187e36d7ee93a3556bddcbc.png)
points, respectively, and such that no three points of the union
![M_1 \cup M_2](/media/m/1/9/4/19430cca4946cf9ca8b22b40aa7648f3.png)
are collinear. Prove that there exists a straightline
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
with the following property: Each of the two half-planes determined by
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
on
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
(
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
not being included in either) contains exactly half of the points of
![M_1](/media/m/9/7/2/9728b6a1cf3e905234b6989eae0cc038.png)
and exactly half of the points of
%V0
Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$