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Contests
International Contests
IMO Longlists
1978 IMO Longlists
8
8
Part of
1978 IMO Longlists
Problems
(1)
Proving inequalities of areas given inequality of sides
Source:
10/30/2010
For two given triangles
A
1
A
2
A
3
A_1A_2A_3
A
1
A
2
A
3
and
B
1
B
2
B
3
B_1B_2B_3
B
1
B
2
B
3
with areas
Δ
A
\Delta_A
Δ
A
and
Δ
B
\Delta_B
Δ
B
, respectively,
A
i
A
k
≥
B
i
B
k
,
i
,
k
=
1
,
2
,
3
A_iA_k \ge B_iB_k, i, k = 1, 2, 3
A
i
A
k
≥
B
i
B
k
,
i
,
k
=
1
,
2
,
3
. Prove that
Δ
A
≥
Δ
B
\Delta_A \ge \Delta_B
Δ
A
≥
Δ
B
if the triangle
A
1
A
2
A
3
A_1A_2A_3
A
1
A
2
A
3
is not obtuse-angled.
inequalities
geometry
geometry unsolved