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3
SMT 2021 Geometry #3
SMT 2021 Geometry #3
Source:
August 9, 2023
geometry
Problem Statement
If
r
r
r
is a rational number, let
f
(
r
)
=
(
1
−
r
2
1
+
r
2
,
2
r
1
+
r
2
)
f(r) = \left( \frac{1-r^2}{1+r^2} , \frac{2r}{1+r^2} \right)
f
(
r
)
=
(
1
+
r
2
1
−
r
2
,
1
+
r
2
2
r
)
. Then the images of
f
f
f
forms a curve in the
x
y
xy
x
y
plane. If
f
(
1
/
3
)
=
p
1
f(1/3) = p_1
f
(
1/3
)
=
p
1
and
f
(
2
)
=
p
2
f(2) = p_2
f
(
2
)
=
p
2
, what is the distance along the curve between
p
1
p_1
p
1
and
p
2
p_2
p
2
?
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