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International Contests
IMO Longlists
1967 IMO Longlists
1967 IMO Longlists
Part of
IMO Longlists
Subcontests
(2)
4
1
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Orthogonal Medians
Suppose, medians
m
a
m_a
m
a
and
m
b
m_b
m
b
of a triangle are orthogonal. Prove that: (a) The medians of the triangle correspond to the sides of a right-angled triangle. (b) If
a
,
b
,
c
a,b,c
a
,
b
,
c
are the side-lengths of the triangle, then, the following inequality holds:
5
(
a
2
+
b
2
−
c
2
)
≥
8
a
b
5(a^2+b^2-c^2)\geq 8ab
5
(
a
2
+
b
2
−
c
2
)
≥
8
ab
39
1
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sines fraction and cosines fraction
Show that the triangle whose angles satisfy the equality
s
i
n
2
(
A
)
+
s
i
n
2
(
B
)
+
s
i
n
2
(
C
)
c
o
s
2
(
A
)
+
c
o
s
2
(
B
)
+
c
o
s
2
(
C
)
=
2
\frac{sin^2(A) + sin^2(B) + sin^2(C)}{cos^2(A) + cos^2(B) + cos^2(C)} = 2
co
s
2
(
A
)
+
co
s
2
(
B
)
+
co
s
2
(
C
)
s
i
n
2
(
A
)
+
s
i
n
2
(
B
)
+
s
i
n
2
(
C
)
=
2
is a rectangular triangle.