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National and Regional Contests
Turkey Contests
National Olympiad First Round
1999 National Olympiad First Round
5
5
Part of
1999 National Olympiad First Round
Problems
(1)
Turkish NMO First Round - 1999 P-05 (Geometry)
Source:
7/3/2012
Let
A
B
C
ABC
A
BC
be an isosceles triangle with \left|AB\right| \equal{} \left|AC\right| \equal{} 10 and \left|BC\right| \equal{} 12.
P
P
P
and
R
R
R
are points on
[
B
C
]
\left[BC\right]
[
BC
]
such that \left|BP\right| \equal{} \left|RC\right| \equal{} 3.
S
S
S
and
T
T
T
are midpoints of
[
A
B
]
\left[AB\right]
[
A
B
]
and
[
A
C
]
\left[AC\right]
[
A
C
]
, respectively. If
M
M
M
and
N
N
N
are the foot of perpendiculars from
S
S
S
and
R
R
R
to
P
T
PT
PT
, then find
∣
M
N
∣
\left|MN\right|
∣
MN
∣
.
(A)
\ \frac {9\sqrt {13} }{26} \qquad
(B)
\ \frac {12 \minus{} 2\sqrt {13} }{13} \qquad
(C)
\ \frac {5\sqrt {13} \plus{} 20}{13} \qquad
(D)
\ 15\sqrt {3} \qquad
(E)
\ \frac {10\sqrt {13} }{13}
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