MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Competitions 139
Figure 5.16: All-Union Russian Olympiad 1980
Problem 34 (All-Union Russian Olympiad 1980). A line parallel to
the side AC of equilateral triangle ABC intersects AB at M and BC at P,
thus making ∆BMP equilateral as well (Figure 5.16). Let D be the center of
∆BMP and E be the midpoint of AP. Show that ∆CDE is a 30 ◦ − 60 ◦ − 90 ◦
triangle. [182, p. 125]
Problem 35 (Bulgarian Mathematical Olympiad 1998a). On the sides
of a non-obtuse triangle ABC are constructed externally a square, a regular ngon
and a regular m-gon (m, n > 5) whose centers form an equilateral triangle.
Prove that m = n = 6, and find the angles of triangle ABC. (Answer: The
angles are 90 ◦ , 45 ◦ , 45 ◦ .) [7, p. 9]
Problem 36 (Bulgarian Mathematical Olympiad 1998b). Let ABC be
an equilateral triangle and n > 1 be a positive integer. Denote by S the set
of n − 1 lines which are parallel to AB and divide triangle ABC into n parts
of equal area, and by S ′ the set of n − 1 lines which are parallel to AB and
divide triangle ABC into n parts of equal perimeter. Prove that S and S ′ do
not share a common element. [7, p. 18]
Problem 37 (Irish Mathematical Olympiad 1998). Show that the area
of an equilateral triangle containing in its interior a point P whose distances
from the vertices are 3, 4, and 5 is equal to 9 + 25√ 3
4 . [7, p. 74]
Problem 38 (Korean Mathematical Olympiad 1998). Let D, E, F be
points on the sides BC, CA, AB, respectively, of triangle ABC. Let P, Q, R
be the second intersections of AD, BE, CF, respectively, with the circumcircle
of ABC. Show that AD BE CF + + ≥ 9, with equality if and only if ABC is
PD QE RF
equilateral. [7, p. 84]