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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1994 Canada National Olympiad
1994 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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Prove angles are equal
Let
A
B
C
ABC
A
BC
be an acute triangle. Let
A
D
AD
A
D
be the altitude on
B
C
BC
BC
, and let
H
H
H
be any interior point on
A
D
AD
A
D
. Lines
B
H
,
C
H
BH,CH
B
H
,
C
H
, when extended, intersect
A
C
,
A
B
AC,AB
A
C
,
A
B
at
E
,
F
E,F
E
,
F
respectively. Prove that
∠
E
D
H
=
∠
F
D
H
\angle EDH=\angle FDH
∠
E
DH
=
∠
F
DH
.
4
1
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Determine cosine of angle
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
.
3
1
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25 men around a table
25
25
25
men sit around a circular table. Every hour there is a vote, and each must respond yes or no. Each man behaves as follows: on the
n
th
n^{\text{th}}
n
th
, vote if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the
(
n
+
1
)
th
(n+1)^{\text{th}}
(
n
+
1
)
th
vote as on the
n
th
n^{\text{th}}
n
th
vote; but if his response is different from that of both his neighbours on the
n
th
n^{\text{th}}
n
th
vote, then his response on the
(
n
+
1
)
th
(n+1)^{\text{th}}
(
n
+
1
)
th
vote will be different from his response on the
n
th
n^{\text{th}}
n
th
vote. Prove that, however everybody responded on the first vote, there will be a time after which nobody's response will ever change.
2
1
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Decompose powers of (sqrt{2}-1)
Prove that
(
2
−
1
)
n
(\sqrt{2}-1)^n
(
2
−
1
)
n
∀
n
∈
Z
+
\forall n\in \mathbb{Z}^{+}
∀
n
∈
Z
+
can be represented as
m
−
m
−
1
\sqrt{m}-\sqrt{m-1}
m
−
m
−
1
for some
m
∈
Z
+
m\in \mathbb{Z}^{+}
m
∈
Z
+
.
1
1
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Evaluate Sum - Parity, Factorial, Polynomial
Evaluate
∑
n
=
1
1994
(
(
−
1
)
n
⋅
(
n
2
+
n
+
1
n
!
)
)
\sum_{n=1}^{1994}{\left((-1)^{n}\cdot\left(\frac{n^2 + n + 1}{n!}\right)\right)}
∑
n
=
1
1994
(
(
−
1
)
n
⋅
(
n
!
n
2
+
n
+
1
)
)
.