MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2000 Canada National Olympiad
2000 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
Hide problems
Max of the sum of the square.
Suppose that the real numbers
a
1
,
a
2
,
…
,
a
100
a_1, a_2, \ldots, a_{100}
a
1
,
a
2
,
…
,
a
100
satisfy \begin{eqnarray*} 0 \leq a_{100} \leq a_{99} \leq \cdots \leq a_2 &\leq& a_1 , \\ a_1+a_2 & \leq & 100 \\ a_3+a_4+\cdots+a_{100} &\leq & 100. \end{eqnarray*} Determine the maximum possible value of
a
1
2
+
a
2
2
+
⋯
+
a
100
2
a_1^2 + a_2^2 + \cdots + a_{100}^2
a
1
2
+
a
2
2
+
⋯
+
a
100
2
, and find all possible sequences
a
1
,
a
2
,
…
,
a
100
a_1, a_2, \ldots , a_{100}
a
1
,
a
2
,
…
,
a
100
which achieve this maximum.
4
1
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Prove quadrilateral is kite
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
∠
C
B
D
=
2
∠
A
D
B
\angle CBD = 2 \angle ADB
∠
CB
D
=
2∠
A
D
B
,
∠
A
B
D
=
2
∠
C
D
B
\angle ABD = 2 \angle CDB
∠
A
B
D
=
2∠
C
D
B
and
A
B
=
C
B
AB = CB
A
B
=
CB
. Prove that
A
D
=
C
D
AD = CD
A
D
=
C
D
.
3
1
Hide problems
Sum
Let
A
=
(
a
1
,
a
2
,
⋯
,
a
2000
)
A = (a_1, a_2, \cdots ,a_{2000})
A
=
(
a
1
,
a
2
,
⋯
,
a
2000
)
be a sequence of integers each lying in the interval
[
−
1000
,
1000
]
[-1000,1000]
[
−
1000
,
1000
]
. Suppose that the entries in A sum to
1
1
1
. Show that some nonempty subsequence of
A
A
A
sums to zero.
2
1
Hide problems
Permutation of 100 numbers
A permutation of the integers
1901
,
1902
,
⋯
,
2000
1901, 1902, \cdots, 2000
1901
,
1902
,
⋯
,
2000
is a sequence
a
1
,
a
2
,
⋯
,
a
100
a_1, a_2, \cdots, a_{100}
a
1
,
a
2
,
⋯
,
a
100
in which each of those integers appears exactly once. Given such a permutation, we form the sequence of partial sums
s
1
=
a
1
,
s
2
=
a
1
+
a
2
,
s
3
=
a
1
+
a
2
+
a
3
,
…
,
s
100
=
a
1
+
a
2
+
⋯
+
a
100
.
s_1 = a_1,\;\;s_2 = a_1 + a_2,\;\;s_3 = a_1 + a_2 + a_3, \; \ldots\;, \; s_{100} = a_1 + a_2 + \cdots + a_{100}.
s
1
=
a
1
,
s
2
=
a
1
+
a
2
,
s
3
=
a
1
+
a
2
+
a
3
,
…
,
s
100
=
a
1
+
a
2
+
⋯
+
a
100
.
How many of these permutations will have no terms of the sequence
s
1
,
…
,
s
100
s_1, \ldots, s_{100}
s
1
,
…
,
s
100
divisible by three?
1
1
Hide problems
Three joggers
At 12:00 noon, Anne, Beth and Carmen begin running laps around a circular track of length
300
300
300
meters, all starting from the same point on the track. Each jogger maintains a constant speed in one of the two possible directions for an indefinite period of time. Show that if Anne's speed is different from the other two speeds, then at some later time Anne will be at least
100
100
100
meters from each of the other runners. (Here, distance is measured along the shorter of the two arcs separating two runners.)