MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1972 Canada National Olympiad
1972 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(10)
10
1
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Geometric progression
What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?
9
1
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4 lines in the plane
Four distinct lines
L
1
,
L
2
,
L
3
,
L
4
L_1,L_2,L_3,L_4
L
1
,
L
2
,
L
3
,
L
4
are given in the plane:
L
1
L_1
L
1
and
L
2
L_2
L
2
are respectively parallel to
L
3
L_3
L
3
and
L
4
L_4
L
4
. Find the locus of a point moving so that the sum of its perpendicular distances from the four lines is constant.
8
1
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Election campaign
During a certain election campaign,
p
p
p
different kinds of promises are made by the different political parties (
p
>
0
p>0
p
>
0
). While several political parties may make the same promise, any two parties have at least one promise in common; no two parties have exactly the same set of promises. Prove that there are no more than
2
p
−
1
2^{p-1}
2
p
−
1
parties.
7
1
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\sqrt{2}
a) Prove that the values of
x
x
x
for which
x
=
(
x
2
+
1
)
/
198
x=(x^2+1)/198
x
=
(
x
2
+
1
)
/198
lie between
1
/
198
1/198
1/198
and
197.99494949
⋯
197.99494949\cdots
197.99494949
⋯
. b) Use the result of problem a) to prove that
2
<
1.41
421356
‾
\sqrt{2}<1.41\overline{421356}
2
<
1.41
421356
. c) Is it true that
2
<
1.41421356
\sqrt{2}<1.41421356
2
<
1.41421356
?
6
1
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Prove that there exist integers m and n...
Let
a
a
a
and
b
b
b
be distinct real numbers. Prove that there exist integers
m
m
m
and
n
n
n
such that
a
m
+
b
n
<
0
am+bn<0
am
+
bn
<
0
,
b
m
+
a
n
>
0
bm+an>0
bm
+
an
>
0
.
5
1
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No solution in positive integers
Prove that the equation
x
3
+
1
1
3
=
y
3
x^3+11^3=y^3
x
3
+
1
1
3
=
y
3
has no solution in positive integers
x
x
x
and
y
y
y
.
4
1
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Construct a quadrilateral given...
Describe a construction of quadrilateral
A
B
C
D
ABCD
A
BC
D
given: (i) the lengths of all four sides; (ii) that
A
B
AB
A
B
and
C
D
CD
C
D
are parallel; (iii) that
B
C
BC
BC
and
D
A
DA
D
A
do not intersect.
3
1
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Composite in all bases
a) Prove that
10201
10201
10201
is composite in all bases greater than 2. b) Prove that
10101
10101
10101
is composite in all bases.
2
1
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Sum of all products of pairs
Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
be non-negative real numbers. Define
M
M
M
to be the sum of all products of pairs
a
i
a
j
a_ia_j
a
i
a
j
(
i
<
j
)
(i<j)
(
i
<
j
)
,
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
i
.
e
.
<
/
s
p
a
n
>
<span class='latex-italic'>i.e.</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
i
.
e
.
<
/
s
p
an
>
,
M
=
a
1
(
a
2
+
a
3
+
⋯
+
a
n
)
+
a
2
(
a
3
+
a
4
+
⋯
+
a
n
)
+
⋯
+
a
n
−
1
a
n
.
M = a_1(a_2+a_3+\cdots+a_n)+a_2(a_3+a_4+\cdots+a_n)+\cdots+a_{n-1}a_n.
M
=
a
1
(
a
2
+
a
3
+
⋯
+
a
n
)
+
a
2
(
a
3
+
a
4
+
⋯
+
a
n
)
+
⋯
+
a
n
−
1
a
n
.
Prove that the square of at least one of the numbers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
does not exceed
2
M
/
n
(
n
−
1
)
2M/n(n-1)
2
M
/
n
(
n
−
1
)
.
1
1
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Three unit circles
Given three distinct unit circles, each of which is tangent to the other two, find the radii of the circles which are tangent to all three circles.