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Contests
National and Regional Contests
Canada Contests
Canadian Mathematical Olympiad Qualification Repechage
2010 Canadian Mathematical Olympiad Qualification Repechage
2010 Canadian Mathematical Olympiad Qualification Repechage
Part of
Canadian Mathematical Olympiad Qualification Repechage
Subcontests
(8)
8
1
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3 Layered Parallolegrams
Consider three parallelograms
P
1
,
P
2
,
P
3
P_1,~P_2,~ P_3
P
1
,
P
2
,
P
3
. Parallelogram
P
3
P_3
P
3
is inside parallelogram
P
2
P_2
P
2
, and the vertices of
P
3
P_3
P
3
are on the edges of
P
2
P_2
P
2
. Parallelogram
P
2
P_2
P
2
is inside parallelogram
P
1
P_1
P
1
, and the vertices of
P
2
P_2
P
2
are on the edges of
P
1
P_1
P
1
. The sides of
P
3
P_3
P
3
are parallel to the sides of
P
1
P_1
P
1
. Prove that one side of
P
3
P_3
P
3
has length at least half the length of the parallel side of
P
1
P_1
P
1
.
7
1
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Function g for triples of real numbers
If
(
a
,
b
,
c
)
(a,~b,~c)
(
a
,
b
,
c
)
is a triple of real numbers, de fine [*]
g
(
a
,
b
,
c
)
=
(
a
+
b
,
b
+
c
,
a
+
c
)
g(a,~b,~c)=(a+b,~b+c,~a+c)
g
(
a
,
b
,
c
)
=
(
a
+
b
,
b
+
c
,
a
+
c
)
, and [*]
g
n
(
a
,
b
,
c
)
=
g
(
g
n
−
1
(
a
,
b
,
c
)
)
g^n(a,~b,~c)=g(g^{n-1}(a,~b,~c))
g
n
(
a
,
b
,
c
)
=
g
(
g
n
−
1
(
a
,
b
,
c
))
for
n
≥
2
n\ge 2
n
≥
2
Suppose that there exists a positive integer
n
n
n
so that
g
n
(
a
,
b
,
c
)
=
(
a
,
b
,
c
)
g^n(a,~b,~c)=(a,~b,~c)
g
n
(
a
,
b
,
c
)
=
(
a
,
b
,
c
)
for some
(
a
,
b
,
c
)
≠
(
0
,
0
,
0
)
(a,~b,~c)\neq (0,~0,~0)
(
a
,
b
,
c
)
=
(
0
,
0
,
0
)
. Prove that
g
6
(
a
,
b
,
c
)
=
(
a
,
b
,
c
)
g^6(a,~b,~c)=(a,~b,~c)
g
6
(
a
,
b
,
c
)
=
(
a
,
b
,
c
)
6
1
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8/15 of the area with magazines
There are
15
15
15
magazines on a table, and they cover the surface of the table entirely. Prove that one can always take away
7
7
7
magazines in such a way that the remaining ones cover at least
8
15
\dfrac{8}{15}
15
8
of the area of the table surface
5
1
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Pythagorean Triangles and the Fibonacci Sequence
The Fibonacci sequence is de ned by
f
1
=
f
2
=
1
f_1=f_2=1
f
1
=
f
2
=
1
and
f
n
=
f
n
−
1
+
f
n
−
2
f_n=f_{n-1}+f_{n-2}
f
n
=
f
n
−
1
+
f
n
−
2
for
n
≥
3
n\ge 3
n
≥
3
. A Pythagorean triangle is a right-angled triangle with integer side lengths. Prove that
f
2
k
+
1
f_{2k+1}
f
2
k
+
1
is the hypotenuse of a Pythagorean triangle for every positive integer
k
k
k
with
k
≥
2
k\ge 2
k
≥
2
4
1
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m^3-3m^2+2m is divisible by both 79 and 83
Determine the smallest positive integer
m
m
m
with the property that
m
3
−
3
m
2
+
2
m
m^3-3m^2+2m
m
3
−
3
m
2
+
2
m
is divisible by both
79
79
79
and
83
83
83
.
3
1
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Prove that there is no real number x
Prove that there is no real number
x
x
x
satisfying both equations \begin{align*}2^x+1=2\sin x \\ 2^x-1=2\cos x.\end{align*}
2
1
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Two tangents AT and BT touch a circle at A and B
Two tangents
A
T
AT
A
T
and
B
T
BT
BT
touch a circle at
A
A
A
and
B
B
B
, respectively, and meet perpendicularly at
T
T
T
.
Q
Q
Q
is on
A
T
AT
A
T
,
S
S
S
is on
B
T
BT
BT
, and
R
R
R
is on the circle, so that
Q
R
S
T
QRST
QRST
is a rectangle with
Q
T
=
8
QT = 8
QT
=
8
and
S
T
=
9
ST = 9
ST
=
9
. Determine the radius of the circle.
1
1
Hide problems
Logarithmic Proof
Suppose that
a
a
a
,
b
b
b
and
x
x
x
are positive real numbers. Prove that
log
a
b
x
=
log
a
x
log
b
x
log
a
x
+
log
b
x
\log_{ab} x =\dfrac{\log_a x\log_b x}{\log_ax+\log_bx}
lo
g
ab
x
=
lo
g
a
x
+
lo
g
b
x
lo
g
a
x
lo
g
b
x
.