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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2009 Canada National Olympiad
2009 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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Set of points on the plane
A set of points is marked on the plane, with the property that any three marked points can be covered with a disk of radius
1
1
1
. Prove that the set of all marked points can be covered with a disk of radius
1
1
1
.
4
1
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Find all ordered pairs
Find all ordered pairs of integers
(
a
,
b
)
(a,b)
(
a
,
b
)
such that
3
a
+
7
b
3^a + 7^b
3
a
+
7
b
is a perfect square.
3
1
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Determine the range
Define
f
(
x
,
y
,
z
)
=
(
x
y
+
y
z
+
z
x
)
(
x
+
y
+
z
)
(
x
+
y
)
(
y
+
z
)
(
z
+
x
)
f(x,y,z)=\frac{(xy+yz+zx)(x+y+z)}{(x+y)(y+z)(z+x)}
f
(
x
,
y
,
z
)
=
(
x
+
y
)
(
y
+
z
)
(
z
+
x
)
(
x
y
+
yz
+
z
x
)
(
x
+
y
+
z
)
. Determine the set of real numbers
r
r
r
for which there exists a triplet of positive real numbers satisfying
f
(
x
,
y
,
z
)
=
r
f(x,y,z)=r
f
(
x
,
y
,
z
)
=
r
.
2
1
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Circles cut from cardboard
Two circles of different radii are cut out of cardboard. Each circle is subdivided into
200
200
200
equal sectors. On each circle
100
100
100
sectors are painted white and the other
100
100
100
are painted black. The smaller circle is then placed on top of the larger circle, so that their centers coincide. Show that one can rotate the small circle so that the sectors on the two circles line up and at least
100
100
100
sectors on the small circle lie over sectors of the same color on the big circle.
1
1
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"Stranded" squares closed formula
Given an
m
×
n
m\times n
m
×
n
grid with unit squares coloured either black or white, a black square in the grid is stranded if there is some square to its left in the same row that is white and there is some square above it in the same column that is white. Find a closed formula for the number of
2
×
n
2\times n
2
×
n
grids with no stranded black square. Note that
n
n
n
is any natural number and the formula must be in terms of
n
n
n
with no other variables.