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Contests
National and Regional Contests
Canada Contests
Canadian Mathematical Olympiad Qualification Repechage
2019 Canadian Mathematical Olympiad Qualification
2019 Canadian Mathematical Olympiad Qualification
Part of
Canadian Mathematical Olympiad Qualification Repechage
Subcontests
(8)
8
1
Hide problems
for every prime p there is an integer k for which p divides k and k is a peak
For
t
≥
2
t \ge 2
t
≥
2
, defi ne
S
(
t
)
S(t)
S
(
t
)
as the number of times
t
t
t
divides into
t
!
t!
t
!
. We say that a positive integer
t
t
t
is a peak if
S
(
t
)
>
S
(
u
)
S(t) > S(u)
S
(
t
)
>
S
(
u
)
for all values of
u
<
t
u < t
u
<
t
. Prove or disprove the following statement: For every prime
p
p
p
, there is an integer
k
k
k
for which
p
p
p
divides
k
k
k
and
k
k
k
is a peak.
7
1
Hide problems
ways of n passengers in a line, waiting to board a plane with n seats
There are
n
n
n
passengers in a line, waiting to board a plane with
n
n
n
seats. For
1
≤
k
≤
n
1 \le k \le n
1
≤
k
≤
n
, the
k
t
h
k^{th}
k
t
h
passenger in line has a ticket for the
k
t
h
k^{th}
k
t
h
seat. However, the rst passenger ignores his ticket, and decides to sit in a seat at random. Thereafter, each passenger sits as follows: If his/her assigned is empty, then he/she sits in it. Otherwise, he/she sits in an empty seat at random. How many different ways can all
n
n
n
passengers be seated?
6
1
Hide problems
4 lines concurrent means 5 lines concur, pentagon and perpendiculars
Pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is given in the plane. Let the perpendicular from
A
A
A
to line
C
D
CD
C
D
be
F
F
F
, the perpendicular from
B
B
B
to
D
E
DE
D
E
be
G
G
G
, from
C
C
C
to
E
A
EA
E
A
be
H
H
H
, from
D
D
D
to
A
B
AB
A
B
be
I
I
I
,and from
E
E
E
to
B
C
BC
BC
be
J
J
J
. Given that lines
A
F
,
B
G
,
C
H
AF,BG,CH
A
F
,
BG
,
C
H
, and
D
I
DI
D
I
concur, show that they also concur with line
E
J
EJ
E
J
.
5
1
Hide problems
winning strategy on a game in m x n array , winning condition |x|\ge N
Let
(
m
,
n
,
N
)
(m,n,N)
(
m
,
n
,
N
)
be a triple of positive integers. Bruce and Duncan play a game on an m\times n array, where the entries are all initially zeroes. The game has the following rules.
∙
\bullet
∙
The players alternate turns, with Bruce going first.
∙
\bullet
∙
On Bruce's turn, he picks a row and either adds
1
1
1
to all of the entries in the row or subtracts
1
1
1
from all the entries in the row.
∙
\bullet
∙
On Duncan's turn, he picks a column and either adds
1
1
1
to all of the entries in the column or subtracts
1
1
1
from all of the entries in the column.
∙
\bullet
∙
Bruce wins if at some point there is an entry
x
x
x
with
∣
x
∣
≥
N
|x|\ge N
∣
x
∣
≥
N
. Find all triples
(
m
,
n
,
N
)
(m, n,N)
(
m
,
n
,
N
)
such that no matter how Duncan plays, Bruce has a winning strategy.
4
1
Hide problems
product of 2 different elements in same subset is never a perfect square
Let
n
n
n
be a positive integer. For a positive integer
m
m
m
, we partition the set
{
1
,
2
,
3
,
.
.
.
,
m
}
\{1, 2, 3,...,m\}
{
1
,
2
,
3
,
...
,
m
}
into
n
n
n
subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of
n
n
n
, fi nd the largest positive integer
m
m
m
for which such a partition exists.
3
1
Hide problems
f(x) = x^3 + 3x^2 - 1 with roots a,b,c find a^3 + b^3 + c^3, a^2b + b^2c + c^2a
Let
f
(
x
)
=
x
3
+
3
x
2
−
1
f(x) = x^3 + 3x^2 - 1
f
(
x
)
=
x
3
+
3
x
2
−
1
have roots
a
,
b
,
c
a,b,c
a
,
b
,
c
. (a) Find the value of
a
3
+
b
3
+
c
3
a^3 + b^3 + c^3
a
3
+
b
3
+
c
3
(b) Find all possible values of
a
2
b
+
b
2
c
+
c
2
a
a^2b + b^2c + c^2a
a
2
b
+
b
2
c
+
c
2
a
2
1
Hide problems
Rosemonde is stacking spheres to make 2 types of pyramids
Rosemonde is stacking spheres to make pyramids. She constructs two types of pyramids
S
n
S_n
S
n
and
T
n
T_n
T
n
. The pyramid
S
n
S_n
S
n
has
n
n
n
layers, where the top layer is a single sphere and the
i
t
h
i^{th}
i
t
h
layer is an
i
×
i\times
i
×
i square grid of spheres for each
2
≤
i
≤
n
2 \le i \le n
2
≤
i
≤
n
. Similarly, the pyramid
T
n
T_n
T
n
has
n
n
n
layers where the top layer is a single sphere and the
i
t
h
i^{th}
i
t
h
layer is
i
(
i
+
1
)
2
\frac{i(i+1)}{2}
2
i
(
i
+
1
)
spheres arranged into an equilateral triangle for each
2
≤
i
≤
n
2 \le i \le n
2
≤
i
≤
n
.
1
1
Hide problems
1/f(n)+1/f(m)=4/(f(n) + f(m)), injective f => m=n
A function
f
f
f
is called injective if when
f
(
n
)
=
f
(
m
)
f(n) = f(m)
f
(
n
)
=
f
(
m
)
, then
n
=
m
n = m
n
=
m
. Suppose that
f
f
f
is injective and
1
f
(
n
)
+
1
f
(
m
)
=
4
f
(
n
)
+
f
(
m
)
\frac{1}{f(n)}+\frac{1}{f(m)}=\frac{4}{f(n) + f(m)}
f
(
n
)
1
+
f
(
m
)
1
=
f
(
n
)
+
f
(
m
)
4
. Prove
m
=
n
m = n
m
=
n