MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1997 Canada National Olympiad
1997 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
Hide problems
Evaluate Sum
Write the sum
∑
i
=
0
n
(
−
1
)
i
⋅
(
n
i
)
i
3
+
9
i
2
+
26
i
+
24
\sum_{i=0}^{n}{\frac{(-1)^i\cdot\binom{n}{i}}{i^3 +9i^2 +26i +24}}
∑
i
=
0
n
i
3
+
9
i
2
+
26
i
+
24
(
−
1
)
i
⋅
(
i
n
)
as the ratio of two explicitly defined polynomials with integer coefficients.
4
1
Hide problems
Point inside parallelogram
The point
O
O
O
is situated inside the parallelogram
A
B
C
D
ABCD
A
BC
D
such that
∠
A
O
B
+
∠
C
O
D
=
18
0
∘
\angle AOB+\angle COD=180^{\circ}
∠
A
OB
+
∠
CO
D
=
18
0
∘
. Prove that
∠
O
B
C
=
∠
O
D
C
\angle OBC=\angle ODC
∠
OBC
=
∠
O
D
C
.
3
1
Hide problems
Product is between two numbers
Prove that
1
1999
<
∏
i
=
1
999
2
i
−
1
2
i
<
1
44
\frac{1}{1999}< \prod_{i=1}^{999}{\frac{2i-1}{2i}}<\frac{1}{44}
1999
1
<
∏
i
=
1
999
2
i
2
i
−
1
<
44
1
.
2
1
Hide problems
Closed interval
The closed interval
A
=
[
0
,
50
]
A = [0, 50]
A
=
[
0
,
50
]
is the union of a finite number of closed intervals, each of length
1
1
1
. Prove that some of the intervals can be removed so that those remaining are mutually disjoint and have total length greater than
25
25
25
. Note: For reals
a
≤
b
a\le b
a
≤
b
, the closed interval
[
a
,
b
]
:
=
{
x
∈
R
:
a
≤
x
≤
b
}
[a, b] := \{x\in \mathbb{R}:a\le x\le b\}
[
a
,
b
]
:=
{
x
∈
R
:
a
≤
x
≤
b
}
has length
b
−
a
b-a
b
−
a
; disjoint intervals have empty intersection.
1
1
Hide problems
Determine all pairs
Determine the number of pairs of positive integers
x
,
y
x,y
x
,
y
such that
x
≤
y
x\le y
x
≤
y
,
gcd
(
x
,
y
)
=
5
!
\gcd (x,y)=5!
g
cd
(
x
,
y
)
=
5
!
and
lcm
(
x
,
y
)
=
50
!
\text{lcm}(x,y)=50!
lcm
(
x
,
y
)
=
50
!
.