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Contests
National and Regional Contests
Canada Contests
Canadian Students Math Olympiad
2011 Canadian Students Math Olympiad
2011 Canadian Students Math Olympiad
Part of
Canadian Students Math Olympiad
Subcontests
(4)
2
1
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Canadian Students Math Olympiad 2011 Problem 2
For a fixed positive integer
k
k
k
, prove that there exist infinitely many primes
p
p
p
such that there is an integer
w
w
w
, where
w
2
−
1
w^2-1
w
2
−
1
is not divisible by
p
p
p
, and the order of
w
w
w
in modulus
p
p
p
is the same as the order of
w
w
w
in modulus
p
k
p^k
p
k
.Author: James Rickards
3
1
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Canadian Students Math Olympiad 2011 Problem 3
Find the largest
C
∈
R
C \in \mathbb{R}
C
∈
R
such that
x
+
z
(
x
−
z
)
2
+
x
+
w
(
x
−
w
)
2
+
y
+
z
(
y
−
z
)
2
+
y
+
w
(
y
−
w
)
2
+
∑
c
y
c
1
x
≥
C
x
+
y
+
z
+
w
\frac{x+z}{(x-z)^2} +\frac{x+w}{(x-w)^2} +\frac{y+z}{(y-z)^2}+\frac{y+w}{(y-w)^2} + \sum_{cyc} \frac{1}{x} \ge \frac{C}{x+y+z+w}
(
x
−
z
)
2
x
+
z
+
(
x
−
w
)
2
x
+
w
+
(
y
−
z
)
2
y
+
z
+
(
y
−
w
)
2
y
+
w
+
cyc
∑
x
1
≥
x
+
y
+
z
+
w
C
where
x
,
y
,
z
,
w
∈
R
+
x,y,z,w \in \mathbb{R^+}
x
,
y
,
z
,
w
∈
R
+
.Author: Hunter Spink
4
1
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Canadian Students Math Olympiad 2011 Problem 4
Circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
have centers
O
1
O_1
O
1
and
O
2
O_2
O
2
and intersect at
P
P
P
and
Q
Q
Q
. A line through
P
P
P
intersects
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
at
A
A
A
and
B
B
B
, respectively, such that
A
B
AB
A
B
is not perpendicular to
P
Q
PQ
PQ
. Let
X
X
X
be the point on
P
Q
PQ
PQ
such that
X
A
=
X
B
XA=XB
X
A
=
XB
and let
Y
Y
Y
be the point within
A
O
1
O
2
B
AO_1 O_2 B
A
O
1
O
2
B
such that
A
Y
O
1
AYO_1
A
Y
O
1
and
B
Y
O
2
BYO_2
B
Y
O
2
are similar. Prove that
2
∠
O
1
A
Y
=
∠
A
X
B
2\angle{O_1 AY}=\angle{AXB}
2∠
O
1
A
Y
=
∠
A
XB
.Author: Matthew Brennan
1
1
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Canadian Students Math Olympiad 2011 Problem 1
In triangle
A
B
C
ABC
A
BC
,
∠
B
A
C
=
6
0
∘
\angle{BAC}=60^\circ
∠
B
A
C
=
6
0
∘
and the incircle of
A
B
C
ABC
A
BC
touches
A
B
AB
A
B
and
A
C
AC
A
C
at
P
P
P
and
Q
Q
Q
, respectively. Lines
P
C
PC
PC
and
Q
B
QB
QB
intersect at
G
G
G
. Let
R
R
R
be the circumradius of
B
G
C
BGC
BGC
. Find the minimum value of
R
/
B
C
R/BC
R
/
BC
.Author: Alex Song