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National and Regional Contests
Turkey Contests
National Olympiad First Round
2002 National Olympiad First Round
9
9
Part of
2002 National Olympiad First Round
Problems
(1)
P09 [Geometry] - Turkish NMO 1st Round - 2002
Source:
8/8/2014
Let
A
B
C
ABC
A
BC
be triangle such that
∣
A
B
∣
=
5
|AB| = 5
∣
A
B
∣
=
5
,
∣
B
C
∣
=
9
|BC| = 9
∣
BC
∣
=
9
and
∣
A
C
∣
=
8
|AC| = 8
∣
A
C
∣
=
8
. The angle bisector of
B
C
A
^
\widehat{BCA}
BC
A
meets
B
A
BA
B
A
at
X
X
X
and the angle bisector of
C
A
B
^
\widehat{CAB}
C
A
B
meets
B
C
BC
BC
at
Y
Y
Y
. Let
Z
Z
Z
be the intersection of lines
X
Y
XY
X
Y
and
A
C
AC
A
C
. What is
∣
A
Z
∣
|AZ|
∣
A
Z
∣
?
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104
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145
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89
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9
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10
<span class='latex-bold'>a)</span>\ \sqrt{104} \qquad<span class='latex-bold'>b)</span>\ \sqrt{145} \qquad<span class='latex-bold'>c)</span>\ \sqrt{89} \qquad<span class='latex-bold'>d)</span>\ 9 \qquad<span class='latex-bold'>e)</span>\ 10
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145
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89
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9
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10
geometry
angle bisector