MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2018 Canada National Olympiad
2018 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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Always divisible by 2018 at some point [CMO 2018 - P5]
Let
k
k
k
be a given even positive integer. Sarah first picks a positive integer
N
N
N
greater than
1
1
1
and proceeds to alter it as follows: every minute, she chooses a prime divisor
p
p
p
of the current value of
N
N
N
, and multiplies the current
N
N
N
by
p
k
−
p
−
1
p^k -p^{-1}
p
k
−
p
−
1
to produce the next value of
N
N
N
. Prove that there are infinitely many even positive integers
k
k
k
such that, no matter what choices Sarah makes, her number
N
N
N
will at some point be divisible by
2018
2018
2018
.
4
1
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p(1) + p(2) + p(3) + ... + p(n) = p(n)q(n) [CMO 2018 - P4]
Find all polynomials
p
(
x
)
p(x)
p
(
x
)
with real coefficients that have the following property: there exists a polynomial
q
(
x
)
q(x)
q
(
x
)
with real coefficients such that
p
(
1
)
+
p
(
2
)
+
p
(
3
)
+
⋯
+
p
(
n
)
=
p
(
n
)
q
(
n
)
p(1) + p(2) + p(3) +\dots + p(n) = p(n)q(n)
p
(
1
)
+
p
(
2
)
+
p
(
3
)
+
⋯
+
p
(
n
)
=
p
(
n
)
q
(
n
)
for all positive integers
n
n
n
.
3
1
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Prime-related integers [CMO 2018 - P3]
Two positive integers
a
a
a
and
b
b
b
are prime-related if
a
=
p
b
a = pb
a
=
p
b
or
b
=
p
a
b = pa
b
=
p
a
for some prime
p
p
p
. Find all positive integers
n
n
n
, such that
n
n
n
has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related. Note that
1
1
1
and
n
n
n
are included as divisors.
2
1
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Five points [CMO 2018 - P2]
Let five points on a circle be labelled
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
, and
E
E
E
in clockwise order. Assume
A
E
=
D
E
AE = DE
A
E
=
D
E
and let
P
P
P
be the intersection of
A
C
AC
A
C
and
B
D
BD
B
D
. Let
Q
Q
Q
be the point on the line through
A
A
A
and
B
B
B
such that
A
A
A
is between
B
B
B
and
Q
Q
Q
and
A
Q
=
D
P
AQ = DP
A
Q
=
D
P
Similarly, let
R
R
R
be the point on the line through
C
C
C
and
D
D
D
such that
D
D
D
is between
C
C
C
and
R
R
R
and
D
R
=
A
P
DR = AP
D
R
=
A
P
. Prove that
P
E
PE
PE
is perpendicular to
Q
R
QR
QR
.
1
1
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Collapsible arrangements [CMO 2018 - P1]
Consider an arrangement of tokens in the plane, not necessarily at distinct points. We are allowed to apply a sequence of moves of the following kind: select a pair of tokens at points
A
A
A
and
B
B
B
and move both of them to the midpoint of
A
A
A
and
B
B
B
. We say that an arrangement of
n
n
n
tokens is collapsible if it is possible to end up with all
n
n
n
tokens at the same point after a finite number of moves. Prove that every arrangement of
n
n
n
tokens is collapsible if and only if
n
n
n
is a power of
2
2
2
.