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National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1967 Spain Mathematical Olympiad
5
5
Part of
1967 Spain Mathematical Olympiad
Problems
(1)
path by equal angles across a semicircle , billiard type
Source: Spanish Mathematical Olympiad 1967 P5
12/4/2022
Let
γ
\gamma
γ
be a semicircle with diameter
A
B
AB
A
B
. A creek is built with origin in
A
A
A
, which has its vertices alternately in the diameter
A
B
AB
A
B
and in the semicircle
γ
\gamma
γ
, so that its sides make equal angles
α
\alpha
α
with the diameter (but alternately in either direction). It is requested: a) Values of the angle
α
\alpha
α
for the ravine to pass through the other end
B
B
B
of the diameter. b) The total length of the ravine, in the case that it ends in
B
B
B
, as a function of the length
d
d
d
of the diameter and of the angle
α
\alpha
α
. https://cdn.artofproblemsolving.com/attachments/3/c/54c71cdf1bf8fbb3bdfa38f4b14a1dc961c5fe.png
geometry
equal angles