MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1987 Canada National Olympiad
1987 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
Hide problems
Show that [√(4n + 2)] = [√(4n + 3)] = [√n + √(n + 1)]
For every positive integer
n
n
n
show that
[
4
n
+
1
]
=
[
4
n
+
2
]
=
[
4
n
+
3
]
=
[
n
+
n
+
1
]
[\sqrt{4n + 1}] = [\sqrt{4n + 2}] = [\sqrt{4n + 3}] = [\sqrt{n} + \sqrt{n + 1}]
[
4
n
+
1
]
=
[
4
n
+
2
]
=
[
4
n
+
3
]
=
[
n
+
n
+
1
]
where
[
x
]
[x]
[
x
]
is the greatest integer less than or equal to
x
x
x
(for example
[
2.3
]
=
2
[2.3] = 2
[
2.3
]
=
2
,
[
π
]
=
3
[\pi] = 3
[
π
]
=
3
,
[
5
]
=
5
[5] = 5
[
5
]
=
5
).
4
1
Hide problems
n stationary people
On a large, flat field
n
n
n
people are positioned so that for each person the distances to all the other people are different. Each person holds a water pistol and at a given signal fires and hits the person who is closest. When
n
n
n
is odd show that there is at least one person left dry. Is this always true when
n
n
n
is even?
3
1
Hide problems
Find the angles of the parallelogram [Canada 1987 - P3]
Suppose
A
B
C
D
ABCD
A
BC
D
is a parallelogram and
E
E
E
is a point between
B
B
B
and
C
C
C
on the line
B
C
BC
BC
. If the triangles
D
E
C
DEC
D
EC
,
B
E
D
BED
BE
D
and
B
A
D
BAD
B
A
D
are isosceles what are the possible values for the angle
D
A
B
DAB
D
A
B
?
2
1
Hide problems
All ways of representing 1987 in another base [Canada 1987]
The number 1987 can be written as a three digit number
x
y
z
xyz
x
yz
in some base
b
b
b
. If
x
+
y
+
z
=
1
+
9
+
8
+
7
x + y + z = 1 + 9 + 8 + 7
x
+
y
+
z
=
1
+
9
+
8
+
7
, determine all possible values of
x
x
x
,
y
y
y
,
z
z
z
,
b
b
b
.
1
1
Hide problems
On the equation n! = a^2 + b^2
Find all solutions of
a
2
+
b
2
=
n
!
a^2 + b^2 = n!
a
2
+
b
2
=
n
!
for positive integers
a
a
a
,
b
b
b
,
n
n
n
with
a
≤
b
a \le b
a
≤
b
and
n
<
14
n < 14
n
<
14
.