MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

144 Mathematical Competitions
Figure 5.23: Nordic Mathematics Competition 1994
Problem 54 (Nordic Mathematics Competition 1994). Let O be a point
in the interior of an equilateral triangle ABC with side length a. The lines
AO, BO and CO intersect the sides of the triangle at the points A1, B1 and
C1, respectively (Figure 5.23). Prove that
[306, p. 15]
|OA1| + |OB1| + |OC1| < a.
Problem 55 (Latvian Mathematical Olympiad 1997). An equilateral
triangle of side 1 is dissected into n triangles. Prove that the sum of squares
of all sides of all triangles is at least 3 and that there is equality if and only if
the triangle can be dissected into n equilateral triangles. [306, p. 29]
Figure 5.24: Irish Mathematical Olympiad 1997
Problem 56 (Irish Mathematical Olympiad 1997). Let ABC be an equilateral
triangle. For a point M inside ABC, let D, E, F be the feet of the
perpendiculars from M onto BC, CA, AB, respectively (Figure 5.24). Show
that the locus of all such points M for which ∠FDE is a right angle is the arc
of the circle interior to ∆ABC subtending 150 ◦ on the line segment BC. [306,
p. 31]