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IMO Shortlist
1968 IMO Shortlist
17
17
Part of
1968 IMO Shortlist
Problems
(1)
prove that there exists an equilateral triangle ABC iff
Source:
9/24/2010
Given a point
O
O
O
and lengths
x
,
y
,
z
x, y, z
x
,
y
,
z
, prove that there exists an equilateral triangle
A
B
C
ABC
A
BC
for which
O
A
=
x
,
O
B
=
y
,
O
C
=
z
OA = x, OB = y, OC = z
O
A
=
x
,
OB
=
y
,
OC
=
z
, if and only if
x
+
y
≥
z
,
y
+
z
≥
x
,
z
+
x
≥
y
x+y \geq z, y+z \geq x, z+x \geq y
x
+
y
≥
z
,
y
+
z
≥
x
,
z
+
x
≥
y
(the points
O
,
A
,
B
,
C
O,A,B,C
O
,
A
,
B
,
C
are coplanar).
geometry
Triangle
triangle inequality
IMO Shortlist