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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1996 Canada National Olympiad
1996 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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Max and min of a function involving [x]
Let
r
1
r_1
r
1
,
r
2
r_2
r
2
,
…
\ldots
…
,
r
m
r_m
r
m
be a given set of
m
m
m
positive rational numbers such that
∑
k
=
1
m
r
k
=
1
\sum_{k=1}^m r_k = 1
∑
k
=
1
m
r
k
=
1
. Define the function
f
f
f
by
f
(
n
)
=
n
−
∑
k
=
1
m
[
r
k
n
]
f(n)= n-\sum_{k=1}^m \: [r_k n]
f
(
n
)
=
n
−
∑
k
=
1
m
[
r
k
n
]
for each positive integer
n
n
n
. Determine the minimum and maximum values of
f
(
n
)
f(n)
f
(
n
)
. Here
[
x
]
{\ [ x ]}
[
x
]
denotes the greatest integer less than or equal to
x
x
x
.
4
1
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Find the angle of an isosceles triangle with BC = BD + AD
Let triangle
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
. Suppose that the angle bisector of its angle
∠
B
\angle B
∠
B
meets the side
A
C
AC
A
C
at a point
D
D
D
and that
B
C
=
B
D
+
A
D
BC = BD+AD
BC
=
B
D
+
A
D
. Determine
∠
A
\angle A
∠
A
.
3
1
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Sequence and divisibility
We denote an arbitrary permutation of the integers
1
1
1
,
2
2
2
,
…
\ldots
…
,
n
n
n
by
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\ldots
…
,
a
n
a_n
a
n
. Let
f
(
n
)
f(n)
f
(
n
)
denote the number of these permutations such that: (1)
a
1
=
1
a_1 = 1
a
1
=
1
; (2):
∣
a
i
−
a
i
+
1
∣
≤
2
|a_i - a_{i+1}| \leq 2
∣
a
i
−
a
i
+
1
∣
≤
2
,
i
=
1
,
…
,
n
−
1
i = 1, \ldots, n - 1
i
=
1
,
…
,
n
−
1
. Determine whether
f
(
1996
)
f(1996)
f
(
1996
)
is divisible by 3.
2
1
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System of equations
Find all real solutions to the following system of equations. Carefully justify your answer.
{
4
x
2
1
+
4
x
2
=
y
4
y
2
1
+
4
y
2
=
z
4
z
2
1
+
4
z
2
=
x
\left\{ \begin{array}{c} \displaystyle\frac{4x^2}{1+4x^2} = y \\ \\ \displaystyle\frac{4y^2}{1+4y^2} = z \\ \\ \displaystyle\frac{4z^2}{1+4z^2} = x \end{array} \right.
⎩
⎨
⎧
1
+
4
x
2
4
x
2
=
y
1
+
4
y
2
4
y
2
=
z
1
+
4
z
2
4
z
2
=
x
1
1
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Roots of third degree equation
If
α
\alpha
α
,
β
\beta
β
, and
γ
\gamma
γ
are the roots of
x
3
−
x
−
1
=
0
x^3 - x - 1 = 0
x
3
−
x
−
1
=
0
, compute
1
+
α
1
−
α
+
1
+
β
1
−
β
+
1
+
γ
1
−
γ
\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}
1
−
α
1
+
α
+
1
−
β
1
+
β
+
1
−
γ
1
+
γ
.