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Saudi Arabia Contests
Saudi Arabia IMO TST
2010 Saudi Arabia IMO TST
2
AM x AN / AB x AC + BM x BN/ BA x BC + CM x CN / CA x CB = 1
AM x AN / AB x AC + BM x BN/ BA x BC + CM x CN / CA x CB = 1
Source: 2010 Saudi Arabia IMO TST III p2
December 27, 2021
ratio
geometry
equal angles
Problem Statement
Points
M
M
M
and
N
N
N
are considered in the interior of triangle
A
B
C
ABC
A
BC
such that
∠
M
A
B
=
∠
N
A
C
\angle MAB = \angle NAC
∠
M
A
B
=
∠
N
A
C
and
∠
M
B
A
=
∠
N
B
C
\angle MBA = \angle NBC
∠
MB
A
=
∠
NBC
. Prove that
A
M
⋅
A
N
A
B
⋅
A
C
+
B
M
⋅
B
N
B
A
⋅
B
C
+
C
M
⋅
C
N
C
A
⋅
C
B
=
1
\frac{AM \cdot AN}{AB \cdot AC}+ \frac{BM\cdot BN}{BA \cdot BC}+ \frac{CM \cdot CN }{CA \cdot CB}=1
A
B
⋅
A
C
A
M
⋅
A
N
+
B
A
⋅
BC
BM
⋅
BN
+
C
A
⋅
CB
CM
⋅
CN
=
1
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