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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1994 Canada National Olympiad
4
4
Part of
1994 Canada National Olympiad
Problems
(1)
Determine cosine of angle
Source: Canadian Mathematical Olympiad - 1994 - Problem 4.
5/13/2011
Let
A
B
AB
A
B
be a diameter of a circle
Ω
\Omega
Ω
and
P
P
P
be any point not on the line through
A
B
AB
A
B
. Suppose that the line through
P
A
PA
P
A
cuts
Ω
\Omega
Ω
again at
U
U
U
, and the line through
P
B
PB
PB
cuts
Ω
\Omega
Ω
at
V
V
V
. Note that in case of tangency,
U
U
U
may coincide with
A
A
A
or
V
V
V
might coincide with
B
B
B
. Also, if
P
P
P
is on
Ω
\Omega
Ω
then
P
=
U
=
V
P=U=V
P
=
U
=
V
. Suppose that
∣
P
U
∣
=
s
∣
P
A
∣
|PU|=s|PA|
∣
P
U
∣
=
s
∣
P
A
∣
and
∣
P
V
∣
=
t
∣
P
B
∣
|PV|=t|PB|
∣
P
V
∣
=
t
∣
PB
∣
for some
0
≤
s
,
t
∈
R
0\le s,t\in \mathbb{R}
0
≤
s
,
t
∈
R
. Determine
cos
∠
A
P
B
\cos \angle APB
cos
∠
A
PB
in terms of
s
,
t
s,t
s
,
t
.