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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2003 Canada National Olympiad
2003 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
4
1
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three circles
Prove that when three circles share the same chord
A
B
AB
A
B
, every line through
A
A
A
different from
A
B
AB
A
B
determines the same ratio
X
Y
:
Y
Z
X Y : Y Z
X
Y
:
Y
Z
, where
X
X
X
is an arbitrary point different from
B
B
B
on the first circle while
Y
Y
Y
and
Z
Z
Z
are the points where AX intersects the other two circles (labeled so that
Y
Y
Y
is between
X
X
X
and
Z
Z
Z
).
5
1
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Distance between points
Let
S
S
S
be a set of
n
n
n
points in the plane such that any two points of
S
S
S
are at least
1
1
1
unit apart. Prove there is a subset
T
T
T
of
S
S
S
with at least
n
7
\frac{n}{7}
7
n
points such that any two points of
T
T
T
are at least
3
\sqrt{3}
3
units apart.
3
1
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2 equations, 3 variables
Find all real positive solutions (if any) to \begin{align*} x^3+y^3+z^3 &= x+y+z, \mbox{ and} \\ x^2+y^2+z^2 &= xyz. \end{align*}
2
1
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Last 3 digits
Find the last three digits of the number
200
3
2002
2001
2003^{{2002}^{2001}}
200
3
2002
2001
.
1
1
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Clock and angle
Consider a standard twelve-hour clock whose hour and minute hands move continuously. Let
m
m
m
be an integer, with
1
≤
m
≤
720
1 \leq m \leq 720
1
≤
m
≤
720
. At precisely
m
m
m
minutes after 12:00, the angle made by the hour hand and minute hand is exactly
1
∘
1^\circ
1
∘
. Determine all possible values of
m
m
m
.