MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2005 Canada National Olympiad
2005 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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powerful numbers
Let's say that an ordered triple of positive integers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
is
n
n
n
-powerful if
a
≤
b
≤
c
,
gcd
(
a
,
b
,
c
)
=
1
a\le b\le c,\gcd (a,b,c)=1
a
≤
b
≤
c
,
g
cd
(
a
,
b
,
c
)
=
1
and
a
n
+
b
n
+
c
n
a^n+b^n+c^n
a
n
+
b
n
+
c
n
is divisible by
a
+
b
+
c
a+b+c
a
+
b
+
c
. For example,
(
1
,
2
,
2
)
(1,2,2)
(
1
,
2
,
2
)
is
5
5
5
-powerful.
a
)
a)
a
)
Determine all ordered triples (if any) which are
n
n
n
-powerful for all
n
≥
1
n\ge 1
n
≥
1
.
b
)
b)
b
)
Determine all ordered triples (if any) which are
2004
2004
2004
-powerful and
2005
2005
2005
-powerful, but not
2007
2007
2007
-powerful.
4
1
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triangle
Let
A
B
C
ABC
A
BC
be a triangle with circumradius
R
R
R
, perimeter
P
P
P
and area
K
K
K
. Determine the maximum value of:
K
P
R
3
\frac{KP}{R^3}
R
3
K
P
.
3
1
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circle
Let
S
S
S
be a set of
n
≥
3
n\ge 3
n
≥
3
points in the interior of a circle.
a
)
a)
a
)
Show that there are three distinct points
a
,
b
,
c
∈
S
a,b,c\in S
a
,
b
,
c
∈
S
and three distinct points
A
,
B
,
C
A,B,C
A
,
B
,
C
on the circle such that
a
a
a
is (strictly) closer to
A
A
A
than any other point in
S
S
S
,
b
b
b
is closer to
B
B
B
than any other point in
S
S
S
and
c
c
c
is closer to
C
C
C
than any other point in
S
S
S
.
b
)
b)
b
)
Show that for no value of
n
n
n
can four such points in
S
S
S
(and corresponding points on the circle) be guaranteed.
2
1
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Pythagorean triples
Let
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
be a Pythagorean triple, i.e. a triplet of positive integers with a^2\plus{}b^2\equal{}c^2.
a
)
a)
a
)
Prove that \left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2>8.
b
)
b)
b
)
Prove that there are no integer
n
n
n
and Pythagorean triple
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
satisfying \left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2\equal{}n.
1
1
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equilateral triangle
An equilateral triangle of side length
n
n
n
is divided into unit triangles. Let
f
(
n
)
f(n)
f
(
n
)
be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in a path share a common edge and the path never travels up (from a lower row to a higher row) or revisits a triangle. An example is shown on the picture for n \equal{} 5. Determine the value of
f
(
2005
)
f(2005)
f
(
2005
)
.