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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2006 Canada National Olympiad
2006 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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prove that triangle $DEF$ is equilateral
The vertices of a right triangle
A
B
C
ABC
A
BC
inscribed in a circle divide the circumference into three arcs. The right angle is at
A
A
A
, so that the opposite arc
B
C
BC
BC
is a semicircle while arc
B
C
BC
BC
and arc
A
C
AC
A
C
are supplementary. To each of three arcs, we draw a tangent such that its point of tangency is the mid point of that portion of the tangent intercepted by the extended lines
A
B
,
A
C
AB,AC
A
B
,
A
C
. More precisely, the point
D
D
D
on arc
B
C
BC
BC
is the midpoint of the segment joining the points
D
′
D'
D
′
and
D
′
′
D''
D
′′
where tangent at
D
D
D
intersects the extended lines
A
B
,
A
C
AB,AC
A
B
,
A
C
. Similarly for
E
E
E
on arc
A
C
AC
A
C
and
F
F
F
on arc
A
B
AB
A
B
. Prove that triangle
D
E
F
DEF
D
EF
is equilateral.
4
1
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round-robin tournament with $2n+1$ teams
Consider a round-robin tournament with
2
n
+
1
2n+1
2
n
+
1
teams, where each team plays each other team exactly one. We say that three teams
X
,
Y
X,Y
X
,
Y
and
Z
Z
Z
, form a cycle triplet if
X
X
X
beats
Y
Y
Y
,
Y
Y
Y
beats
Z
Z
Z
and
Z
Z
Z
beats
X
X
X
. There are no ties. a)Determine the minimum number of cycle triplets possible. b)Determine the maximum number of cycle triplets possible.
3
1
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rectangular array of nonnegative reals
In a rectangular array of nonnegative reals with
m
m
m
rows and
n
n
n
columns, each row and each column contains at least one positive element. Moreover, if a row and a column intersect in a positive element, then the sums of their elements are the same. Prove that
m
=
n
m=n
m
=
n
.
2
1
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intersections of the diagonals of rectangles
Let
A
B
C
ABC
A
BC
be acute triangle. Inscribe a rectangle
D
E
F
G
DEFG
D
EFG
in this triangle such that
D
∈
A
B
,
E
∈
A
C
,
F
∈
B
C
,
G
∈
B
C
D\in AB,E\in AC,F\in BC,G\in BC
D
∈
A
B
,
E
∈
A
C
,
F
∈
BC
,
G
∈
BC
. Describe the locus of (i.e., the curve occupied by) the intersections of the diagonals of all possible rectangles
D
E
F
G
DEFG
D
EFG
.
1
1
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number of ways of distributing $k$ candies to $n$ children
Let
f
(
n
,
k
)
f(n,k)
f
(
n
,
k
)
be the number of ways of distributing
k
k
k
candies to
n
n
n
children so that each child receives at most
2
2
2
candies. For example f(3,7) \equal{} 0,f(3,6) \equal{} 1,f(3,4) \equal{} 6. Determine the value of f(2006,1) \plus{} f(2006,4) \plus{} \ldots \plus{} f(2006,1000) \plus{} f(2006,1003) \plus{} \ldots \plus{} f(2006,4012).