IMO Shortlist 1967 problem 4
Dodao/la:
arhiva2. travnja 2012. Let
![k_1](/media/m/3/5/6/35656cbf3adb55dddd30996fc068363b.png)
and
![k_2](/media/m/6/a/b/6abbe24dbf6713b55498fe55ab050d06.png)
be two circles with centers
![O_1](/media/m/7/2/b/72b270d556043f6f393afbf50620eb57.png)
and
![O_2](/media/m/f/2/d/f2de7ab4fb5625160a4d2f2ac2dd707d.png)
and equal radius
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
such that
![O_1O_2 = r](/media/m/c/0/0/c00edda21c821b2426727f837e82ca1c.png)
. Let
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
be two points lying on the circle
![k_1](/media/m/3/5/6/35656cbf3adb55dddd30996fc068363b.png)
and being symmetric to each other with respect to the line
![O_1O_2](/media/m/d/f/d/dfdefa801f9dce3b3b6138913c5ea15c.png)
. Let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
be an arbitrary point on
![k_2](/media/m/6/a/b/6abbe24dbf6713b55498fe55ab050d06.png)
. Prove that
%V0
Let $k_1$ and $k_2$ be two circles with centers $O_1$ and $O_2$ and equal radius $r$ such that $O_1O_2 = r$. Let $A$ and $B$ be two points lying on the circle $k_1$ and being symmetric to each other with respect to the line $O_1O_2$. Let $P$ be an arbitrary point on $k_2$. Prove that
$$PA^2 + PB^2 \geq 2r^2.$$
Izvor: Međunarodna matematička olimpijada, shortlist 1967