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Lusophon Mathematical Olympiad
2022 Lusophon Mathematical Olympiad
5
5
Part of
2022 Lusophon Mathematical Olympiad
Problems
(1)
Weird construction
Source: Lusophon Mathematical Olympiad 2022 Problem 5
11/3/2022
Tow circumferences of radius
R
1
R_1
R
1
and
R
2
R_2
R
2
are tangent externally between each other. Besides that, they are both tangent to a semicircle with radius of 1, as shown in the figure. (Diagram is in the attachment)a) If
A
1
A_1
A
1
and
A
2
A_2
A
2
are the tangency points of the two circumferences with the diameter of the semicircle, find the length of
A
1
A
2
‾
\overline{A_1 A_2}
A
1
A
2
.b) Prove that
R
1
+
R
2
=
2
R
1
R
2
(
2
−
R
1
R
2
)
R_{1}+R_{2}=2\sqrt{R_{1}R_{2}}(\sqrt{2}-\sqrt{R_{1}R_{2}})
R
1
+
R
2
=
2
R
1
R
2
(
2
−
R
1
R
2
)
.
geometry