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Moscow Mathematical Olympiad
1997 Moscow Mathematical Olympiad
3
1997 MMO Grade 9 #3
1997 MMO Grade 9 #3
Source:
October 3, 2016
Grade 9
1997
Problem Statement
Convex octagon
A
C
1
B
A
1
C
B
1
AC_1BA_1CB_1
A
C
1
B
A
1
C
B
1
satisfies:
A
B
1
=
A
C
1
AB_1=AC_1
A
B
1
=
A
C
1
,
B
C
1
=
B
A
1
BC_1=BA_1
B
C
1
=
B
A
1
,
C
A
1
=
C
B
1
CA_1=CB_1
C
A
1
=
C
B
1
and
∠
A
+
∠
B
+
∠
C
=
∠
A
1
+
∠
B
1
+
∠
C
1
\angle{A}+\angle{B}+\angle{C}=\angle{A_1}+\angle{B_1}+\angle{C_1}
∠
A
+
∠
B
+
∠
C
=
∠
A
1
+
∠
B
1
+
∠
C
1
. Prove that the area of
△
A
B
C
\triangle{ABC}
△
A
BC
is equal to half the area of the octagon.
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