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Bulgaria National Olympiad
1987 Bulgaria National Olympiad
Problem 2
Problem 2
Part of
1987 Bulgaria National Olympiad
Problems
(1)
double rotation, rational ratio of angles
Source: Bulgaria 1987 P2
6/15/2021
Let there be given a polygon
P
P
P
which is mapped onto itself by two rotations:
ρ
1
\rho_1
ρ
1
with center
O
1
O_1
O
1
and angle
ω
1
\omega_1
ω
1
, and
ρ
2
\rho_2
ρ
2
with center
O
2
O_2
O
2
and angle
ω
2
(
0
<
ω
i
<
2
π
)
\omega_2~(0<\omega_i<2\pi)
ω
2
(
0
<
ω
i
<
2
π
)
. Show that the ratio
ω
1
ω
2
\frac{\omega_1}{\omega_2}
ω
2
ω
1
is rational.
geometry
geometric transformation
rotation
ratio