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Swedish Mathematical Competition
1997 Swedish Mathematical Competition
1
1
Part of
1997 Swedish Mathematical Competition
Problems
(1)
1/(a^2+1/2 )< b/a < 1/a^2 , geo inequality related to a circle
Source: 1997 Swedish Mathematical Competition p1
4/2/2021
Let
A
C
AC
A
C
be a diameter of a circle and
A
B
AB
A
B
be tangent to the circle. The segment
B
C
BC
BC
intersects the circle again at
D
D
D
. Show that if
A
C
=
1
AC = 1
A
C
=
1
,
A
B
=
a
AB = a
A
B
=
a
, and
C
D
=
b
CD = b
C
D
=
b
, then
1
a
2
+
1
2
<
b
a
<
1
a
2
\frac{1}{a^2+ \frac12 }< \frac{b}{a}< \frac{1}{a^2}
a
2
+
2
1
1
<
a
b
<
a
2
1
Geometric Inequalities
geometry
inequalities