MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

140 Mathematical Competitions
Problem 39 (Russian Mathematical Olympiad 1998). A family S of
equilateral triangles in the plane is given, all translates of each other, and any
two having nonempty intersection. Prove that there exist three points such that
every member of S contains one of the points. [7, p. 136]
Problem 40 (Bulgarian Mathematical Olympiad 1999). Each interior
point of an equilateral triangle of side 1 lies in one of six congruent circles of
radius r. Prove that r ≥ √ 3.
[8, p. 33]
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Problem 41 (French Mathematical Olympiad 1999). For which acuteangled
triangle is the ratio of the shortest side to the inradius maximal? (Answer:
The maximum ratio of 2 √ 3 is attained with an equilateral triangle.) [8,
p. 57]
Problem 42 (Romanian Mathematical Olympiad 1999). Let SABC be
a right pyramid with equilateral base ABC, let O be the center of ABC, and
let M be the midpoint of BC. If AM = 2SO and N is a point on edge SA
such that SA = 25SN, prove that planes ABP and SBC are perpendicular,
where P is the intersection of lines SO and MN. [8, p. 119]
Problem 43 (Romanian IMO Selection Test 1999). Let ABC be an
acute triangle with angle bisectors BL and CM. Prove that ∠A = 60 ◦ if and
only if there exists a point K on BC (K �= B, C) such that triangle KLM is
equilateral. [8, p. 127]
Problem 44 (Russian Mathematical Olympiad 1999). An equilateral
triangle of side length n is drawn with sides along a triangular grid of side
length 1. What is the maximum number of grid segments on or inside the
triangle that can be marked so that no three marked segments form a triangle?
(Answer: n(n + 1).) [8, p. 156]
Problem 45 (Belarusan Mathematical Olympiad 2000). In an equilateral
triangle of n(n+1)
pennies, with n pennies along each side of the triangle,
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all but one penny shows heads. A “move” consists of choosing two adjacent
pennies with centers A and B and flipping every penny on line AB. Determine
all initial arrangements - the value of n and the position of the coin initially
showing tails - from which one can make all the coins show tails after finitely
many moves. (Answer: For any value of n, the desired initial arrangements
are those in which the coin showing tails is in a corner.) [9, p. 1]
Problem 46 (Romanian Mathematical Olympiad 2000). Let P1P2 · · ·Pn
be a convex polygon in the plane. Assume that, for any pair of vertices Pi, Pj,
there exists a vertex V of the polygon such that ∠PiV Pj = π/3. Show that
n = 3, i.e. show that the polygon is an equilateral triangle. [9, p. 96]