MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2017 Canada National Olympiad
2017 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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CMO 2017 P5
There are
100
100
100
circles of radius one in the plane. A triangle formed by the centres of any three given circles has area at most
2017
2017
2017
. Prove that there is a line intersecting at least three of the circles.
4
1
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CMO 2017 P4
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. Points
P
P
P
and
Q
Q
Q
lie inside
A
B
C
D
ABCD
A
BC
D
such that
△
A
B
P
\bigtriangleup ABP
△
A
BP
and
△
B
C
Q
\bigtriangleup{BCQ}
△
BCQ
are equilateral. Prove that the intersection of the line through
P
P
P
perpendicular to
P
D
PD
P
D
and the line through
Q
Q
Q
perpendicular to
D
Q
DQ
D
Q
lies on the altitude from
B
B
B
in
△
A
B
C
\bigtriangleup{ABC}
△
A
BC
.
3
1
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CMO 2017 P3
Define
S
n
S_n
S
n
as the set
1
,
2
,
⋯
,
n
{1,2,\cdots,n}
1
,
2
,
⋯
,
n
. A non-empty subset
T
n
T_n
T
n
of
S
n
S_n
S
n
is called
b
a
l
a
n
c
e
d
balanced
ba
l
an
ce
d
if the average of the elements of
T
n
T_n
T
n
is equal to the median of
T
n
T_n
T
n
. Prove that, for all
n
n
n
, the number of balanced subsets
T
n
T_n
T
n
is odd.
2
1
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CMO 2017 P2
Define a function
f
(
n
)
f(n)
f
(
n
)
from the positive integers to the positive integers such that
f
(
f
(
n
)
)
f(f(n))
f
(
f
(
n
))
is the number of positive integer divisors of
n
n
n
. Prove that if
p
p
p
is a prime, then
f
(
p
)
f(p)
f
(
p
)
is prime.
1
1
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CMO 2017 P1
For pairwise distinct nonnegative reals
a
,
b
,
c
a,b,c
a
,
b
,
c
, prove that
a
2
(
b
−
c
)
2
+
b
2
(
c
−
a
)
2
+
c
2
(
b
−
a
)
2
>
2
\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(b-a)^2}>2
(
b
−
c
)
2
a
2
+
(
c
−
a
)
2
b
2
+
(
b
−
a
)
2
c
2
>
2
.