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Moscow Mathematical Olympiad
1997 Moscow Mathematical Olympiad
3
1997 MMO Grade 8 #3
1997 MMO Grade 8 #3
Source:
October 3, 2016
Grade 8
1997
Problem Statement
Inside acute
∠
X
O
Y
,
\angle{XOY},
∠
XO
Y
,
points
M
M
M
and
N
N
N
are taken so that
∠
X
O
N
=
∠
Y
O
M
\angle{XON}=\angle{YOM}
∠
XON
=
∠
Y
OM
. Point
Q
Q
Q
is taken on segment
O
X
OX
OX
such that
∠
N
Q
O
=
∠
M
Q
X
.
\angle{NQO}=\angle{MQX}.
∠
NQO
=
∠
MQX
.
Point
P
P
P
is taken such that
∠
N
P
O
=
∠
M
P
Y
.
\angle{NPO}=\angle{MPY}.
∠
NPO
=
∠
MP
Y
.
Prove the lengths of the broken lines
M
P
N
MPN
MPN
and
M
Q
N
MQN
MQN
are equal.
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