MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Competitions 143
Figure 5.21: International Mathematical Olympiad (Shortlist) 1996a
Problem 52 (International Mathematical Olympiad (Shortlist) 1996a).
Let ABC be equilateral and let P be a point in its interior. Let the lines AP,
BP, CP meet the sides BC, CA, AB in the points A1, B1, C1, respectively
(Figure 5.21). Prove that
[306, p. 13]
A1B1 · B1C1 · C1A1 ≥ A1B · B1C · C1A.
Figure 5.22: International Mathematical Olympiad (Shortlist) 1996b
Problem 53 (International Mathematical Olympiad (Shortlist) 1996b).
Let ABC be an acute-angled triangle with circumcenter O and circumradius
R. Let AO meet the circle BOC again in A ′ , let BO meet the circle COA
again in B ′ , and let CO meet the circle AOB again in C ′ (Figure 5.22). Prove
that
OA ′ · OB ′ · OC ′ ≥ 8R 3 ,
with equality if and only if ∆ABC is equilateral. [306, p. 13]