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2015 Bangladesh Mathematical Olympiad
7
Bangladesh Mathematical Olympiad 2015, Secondary, P7
Bangladesh Mathematical Olympiad 2015, Secondary, P7
Source:
March 31, 2015
national olympiad
geometry
algebra
area
Cevas Theorem
Problem Statement
In triangle
△
A
B
C
\triangle ABC
△
A
BC
, the points
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
are on sides
B
C
,
A
C
,
A
B
BC, AC, AB
BC
,
A
C
,
A
B
respectively. Also,
A
A
′
,
B
B
′
,
C
C
′
AA', BB', CC'
A
A
′
,
B
B
′
,
C
C
′
intersect at the point
O
O
O
(they are concurrent at
O
O
O
). Also,
A
O
O
A
′
+
B
O
O
B
′
+
C
O
O
C
′
=
92
\frac {AO}{OA'}+\frac {BO}{OB'}+\frac {CO}{OC'} = 92
O
A
′
A
O
+
O
B
′
BO
+
O
C
′
CO
=
92
. Find the value of
A
O
O
A
′
×
B
O
O
B
′
×
C
O
O
C
′
\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}
O
A
′
A
O
×
O
B
′
BO
×
O
C
′
CO
.
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