IMO Shortlist 1968 problem 17
Dodao/la:
arhiva2. travnja 2012. Given a point
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
and lengths
![x, y, z](/media/m/e/1/6/e160f3439547ca8c1afcc35a1c26f080.png)
, prove that there exists an equilateral triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
for which
![OA = x, OB = y, OC = z](/media/m/2/a/0/2a06203bc2a0c9fef33f7468d193828e.png)
, if and only if
![x+y \geq z, y+z \geq x, z+x \geq y](/media/m/a/a/3/aa3ebc2ddbc85e116c0740eaf275726e.png)
(the points
![O,A,B,C](/media/m/e/e/0/ee038039bd619442ef8876b7d60069aa.png)
are coplanar).
%V0
Given a point $O$ and lengths $x, y, z$, prove that there exists an equilateral triangle $ABC$ for which $OA = x, OB = y, OC = z$, if and only if $x+y \geq z, y+z \geq x, z+x \geq y$ (the points $O,A,B,C$ are coplanar).
Izvor: Međunarodna matematička olimpijada, shortlist 1968