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Contests
National and Regional Contests
Canada Contests
Canadian Mathematical Olympiad Qualification Repechage
2012 Canadian Mathematical Olympiad Qualification Repechage
2012 Canadian Mathematical Olympiad Qualification Repechage
Part of
Canadian Mathematical Olympiad Qualification Repechage
Subcontests
(8)
8
1
Hide problems
Show that <I1MI2=<O1MO2
Suppose circles
W
1
\mathit{W}_1
W
1
and
W
2
\mathit{W}2
W
2
, with centres
O
1
\mathit{O}_1
O
1
and
O
2
\mathit{O}_2
O
2
respectively, intersect at points
M
\mathit{M}
M
and
N
\mathit{N}
N
. Let the tangent on
W
2
\mathit{W}_2
W
2
at point
N
\mathit{N}
N
intersect
W
1
\mathit{W}_1
W
1
for the second time at
B
1
\mathit{B}_1
B
1
. Similarly, let the tangent on
W
1
\mathit{W}_1
W
1
at point
N
\mathit{N}
N
intersect
W
2
\mathit{W}_2
W
2
for the second time at
B
2
\mathit{B}_2
B
2
. Let
A
1
\mathit{A}_1
A
1
be a point on
W
1
\mathit{W}_1
W
1
which is on arc
B
1
N
\mathit{B}_1\mathit{N}
B
1
N
not containing
M
\mathit{M}
M
and suppose line
A
1
N
\mathit{A}_1\mathit{N}
A
1
N
intersects
W
2
\mathit{W}_2
W
2
at point
A
2
\mathit{A}_2
A
2
. Denote the incentres of triangles
B
1
A
1
N
\mathit{B}_1\mathit{A}_1\mathit{N}
B
1
A
1
N
and
B
2
A
2
N
\mathit{B}_2\mathit{A}_2\mathit{N}
B
2
A
2
N
by
I
1
\mathit{I}_1
I
1
and
I
2
\mathit{I}_2
I
2
, respectively.*[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(10.1cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -0.9748626324969808, xmax = 13.38440254515721, ymin = 0.5680051903627492, ymax = 10.99430986899034; /* image dimensions */pair O_2 = (7.682929606970993,6.084708172218866), O_1 = (2.180000000000002,6.760000000000007), M = (4.560858774883258,8.585242858926296), B_2 = (10.07334553576748,9.291873850408265), A_2 = (11.49301008867042,4.866805580476367), B_1 = (2.113311869970955,9.759258690628950), A_1 = (0.2203184186713625,4.488514120712773); /* draw figures */ draw(circle(O_2, 4.000000000000000)); draw(circle(O_1, 3.000000000000000)); draw((4.048892687647541,4.413249028538064)--B_2); draw(B_2--A_2); draw(A_2--(4.048892687647541,4.413249028538064)); draw((4.048892687647541,4.413249028538064)--B_1); draw(B_1--A_1); draw(A_1--(4.048892687647541,4.413249028538064)); /* dots and labels */ dot(O_2,dotstyle); label("
O
2
O_2
O
2
", (7.788512439159622,6.243082420501817), NE * labelscalefactor); dot(O_1,dotstyle); label("
O
1
O_1
O
1
", (2.298205165350667,6.929370829727937), NE * labelscalefactor); dot(M,dotstyle); label("
M
M
M
", (4.383466101076183,8.935444641311980), NE * labelscalefactor); dot((4.048892687647541,4.413249028538064),dotstyle); label("
N
N
N
", (3.855551940133015,3.761885864068922), NE * labelscalefactor); dot(B_2,dotstyle); label("
B
2
B_2
B
2
", (10.19052187145104,9.463358802255147), NE * labelscalefactor); dot(A_2,dotstyle); label("
A
2
A_2
A
2
", (11.80066006232771,4.659339937672310), NE * labelscalefactor); dot(B_1,dotstyle); label("
B
1
B_1
B
1
", (1.981456668784765,10.09685579538695), NE * labelscalefactor); dot(A_1,dotstyle); label("
A
1
A_1
A
1
", (0.08096568938935705,3.973051528446190), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]Show that
∠
I
1
M
I
2
=
∠
O
1
M
O
2
.
\angle\mathit{I}_1\mathit{MI}_2=\angle\mathit{O}_1\mathit{MO}_2.
∠
I
1
MI
2
=
∠
O
1
MO
2
.
*Given a triangle ABC, the incentre of the triangle is defined to be the intersection of the angle bisectors of A, B, and C. To avoid cluttering, the incentre is omitted in the provided diagram. Note also that the diagram serves only as an aid and is not necessarily drawn to scale.
7
1
Hide problems
Triplet of three players is cyclic
Six tennis players gather to play in a tournament where each pair of persons play one game, with one person declared the winner and the other person the loser. A triplet of three players
{
A
,
B
,
C
}
\{\mathit{A}, \mathit{B}, \mathit{C}\}
{
A
,
B
,
C
}
is said to be cyclic if
A
\mathit{A}
A
wins against
B
\mathit{B}
B
,
B
\mathit{B}
B
wins against
C
\mathit{C}
C
and
C
\mathit{C}
C
wins against
A
\mathit{A}
A
. [*] (a) After the tournament, the six people are to be separated in two rooms such that none of the two rooms contains a cyclic triplet. Prove that this is always possible.[*] (b) Suppose there are instead seven people in the tournament. Is it always possible that the seven people can be separated in two rooms such that none of the two rooms contains a cyclic triplet?
6
1
Hide problems
Determine whether there exist two real numbers a and b
Determine whether there exist two real numbers
a
a
a
and
b
b
b
such that both
(
x
−
a
)
3
+
(
x
−
b
)
2
+
x
(x-a)^3+ (x-b)^2+x
(
x
−
a
)
3
+
(
x
−
b
)
2
+
x
and
(
x
−
b
)
3
+
(
x
−
a
)
2
+
x
(x-b)^3 + (x-a)^2 +x
(
x
−
b
)
3
+
(
x
−
a
)
2
+
x
contain only real roots.
5
1
Hide problems
Terms in a sequence divisible by 3^2011
Given a positive integer
n
n
n
, let
d
(
n
)
d(n)
d
(
n
)
be the largest positive divisor of
n
n
n
less than
n
n
n
. For example,
d
(
8
)
=
4
d(8) = 4
d
(
8
)
=
4
and
d
(
13
)
=
1
d(13) = 1
d
(
13
)
=
1
. A sequence of positive integers
a
1
,
a
2
,
…
a_1, a_2,\dots
a
1
,
a
2
,
…
satisfies
a
i
+
1
=
a
i
+
d
(
a
i
)
,
a_{i+1} = a_i +d(a_i),
a
i
+
1
=
a
i
+
d
(
a
i
)
,
for all positive integers
i
i
i
. Prove that regardless of the choice of
a
1
a_1
a
1
, there are infinitely many terms in the sequence divisible by
3
2011
3^{2011}
3
2011
.
4
1
Hide problems
Six sets of all points P inside triangle ABC
Let
A
B
C
ABC
A
BC
be a triangle such that
∠
B
A
C
=
9
0
∘
\angle BAC = 90^\circ
∠
B
A
C
=
9
0
∘
and
A
B
<
A
C
AB < AC
A
B
<
A
C
. We divide the interior of the triangle into the following six regions: \begin{align*} S_1=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA
49
:
1
49 : 1
49
:
1
. Determine the ratio
A
C
:
A
B
AC : AB
A
C
:
A
B
.
3
1
Hide problems
Fantastic Triplets
We say that
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
form a fantastic triplet if
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive integers,
a
,
b
,
c
a,b,c
a
,
b
,
c
form a geometric sequence, and
a
,
b
+
1
,
c
a,b+1,c
a
,
b
+
1
,
c
form an arithmetic sequence. For example,
(
2
,
4
,
8
)
(2,4,8)
(
2
,
4
,
8
)
and
(
8
,
12
,
18
)
(8,12,18)
(
8
,
12
,
18
)
are fantastic triplets. Prove that there exist infinitely many fantastic triplets.
2
1
Hide problems
d(n)+d(n+1)=5
Given a positive integer
m
m
m
, let
d
(
m
)
d(m)
d
(
m
)
be the number of positive divisors of
m
m
m
. Determine all positive integers
n
n
n
such that
d
(
n
)
+
d
(
n
+
1
)
=
5
d(n) +d(n+ 1) = 5
d
(
n
)
+
d
(
n
+
1
)
=
5
.
1
1
Hide problems
Problems on 45 seat row of a movie theater
The front row of a movie theatre contains
45
45
45
seats. [*] (a) If
42
42
42
people are sitting in the front row, prove that there are
10
10
10
consecutive seats that are all occupied.[*] (b) Show that this conclusion doesn’t necessarily hold if only
41
41
41
people are sitting in the front row.