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Poland Contests
Poland - Second Round
1955 Poland - Second Round
6
6
Part of
1955 Poland - Second Round
Problems
(1)
1/ OM + 1/ ON + 1/OP is fixed, trihedral angle related
Source: Polish MO second round 1955 p6
8/29/2024
Inside the trihedral angle
O
A
B
C
OABC
O
A
BC
, whose plane angles
A
O
B
AOB
A
OB
,
B
O
C
BOC
BOC
,
C
O
A
COA
CO
A
are equal, a point
S
S
S
is chosen equidistant from the faces of this angle. Through point
S
S
S
a plane is drawn that intersects the edges
O
A
OA
O
A
,
O
B
OB
OB
,
O
C
OC
OC
at points
M
M
M
,
N
N
N
,
P
P
P
, respectively. Prove that the sum
1
O
M
+
1
O
N
+
1
O
P
\frac{1}{OM} + \frac{1}{ON} + \frac{1}{OP}
OM
1
+
ON
1
+
OP
1
has a constant value, i.e. independent of the position of the plane
M
N
P
MNP
MNP
.
3D geometry
geometry
fixed