MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Competitions 137
Figure 5.14: USAMO 2007
Problem 27 (USAMO 2007). Let ABC be an acute triangle with ω, Ω,
and R being its incircle, circumcircle, and circumradius, respectively (Figure
5.14). Circle ωA is tangent internally to Ω at A and tangent externally to ω.
Circle ΩA is tangent internally to Ω at A and tangent internally to ω. Let PA
and QA denote the centers of ωA and ΩA, respectively. Define points PB, QB,
PC, QC analogously. Prove that
8PAQA · PBQB · PCQC ≤ R 3 ,
with equality if an only if triangle ABC is equilateral. [104, p. 28].
Problem 28 (IMO 1961). Let a, b, c be the sides of a triangle, and T its
area. Prove: a 2 + b 2 + c 2 ≥ 4 √ 3T and that equality holds if and only if the
triangle is equilateral. [155, p. 3]
Problem 29 (IMO 1983). Let ABC be an equilateral triangle and E the set
of all points contained in the three segments AB, BC and CA (including A, B
and C). Show that, for every partition of E into two disjoint subsets, at least
one of the two subsets contains the vertices of a right-angled triangle. [195, p.
6]