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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2014 Canada National Olympiad
2014 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
Hide problems
n integers written on a blackboard
Fix positive integers
n
n
n
and
k
≥
2
k\ge 2
k
≥
2
. A list of
n
n
n
integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add
1
1
1
to all of them or subtract
1
1
1
from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least
n
−
k
+
2
n-k+2
n
−
k
+
2
of the numbers on the blackboard are all simultaneously divisible by
k
k
k
.
4
1
Hide problems
Point P in interior of ABCD
The quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle. The point
P
P
P
lies in the interior of
A
B
C
D
ABCD
A
BC
D
, and
∠
P
A
B
=
∠
P
B
C
=
∠
P
C
D
=
∠
P
D
A
\angle P AB = \angle P BC = \angle P CD = \angle P DA
∠
P
A
B
=
∠
PBC
=
∠
PC
D
=
∠
P
D
A
. The lines
A
D
AD
A
D
and
B
C
BC
BC
meet at
Q
Q
Q
, and the lines
A
B
AB
A
B
and
C
D
CD
C
D
meet at
R
R
R
. Prove that the lines
P
Q
P Q
PQ
and
P
R
P R
PR
form the same angle as the diagonals of
A
B
C
D
ABCD
A
BC
D
.
3
1
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Determine the number of "good" p-tuples
Let
p
p
p
be a fixed odd prime. A
p
p
p
-tuple
(
a
1
,
a
2
,
a
3
,
…
,
a
p
)
(a_1,a_2,a_3,\ldots,a_p)
(
a
1
,
a
2
,
a
3
,
…
,
a
p
)
of integers is said to be good if [*] (i)
0
≤
a
i
≤
p
−
1
0\le a_i\le p-1
0
≤
a
i
≤
p
−
1
for all
i
i
i
, and [*] (ii)
a
1
+
a
2
+
a
3
+
⋯
+
a
p
a_1+a_2+a_3+\cdots+a_p
a
1
+
a
2
+
a
3
+
⋯
+
a
p
is not divisible by
p
p
p
, and [*] (iii)
a
1
a
2
+
a
2
a
3
+
a
3
a
4
+
⋯
+
a
p
a
1
a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1
a
1
a
2
+
a
2
a
3
+
a
3
a
4
+
⋯
+
a
p
a
1
is divisible by
p
p
p
.Determine the number of good
p
p
p
-tuples.
1
1
Hide problems
Sequence a with product 1
Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
be positive real numbers whose product is
1
1
1
. Show that the sum
a
1
1
+
a
1
+
a
2
(
1
+
a
1
)
(
1
+
a
2
)
+
a
3
(
1
+
a
1
)
(
1
+
a
2
)
(
1
+
a
3
)
+
⋯
+
a
n
(
1
+
a
1
)
(
1
+
a
2
)
⋯
(
1
+
a
n
)
\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}
1
+
a
1
a
1
+
(
1
+
a
1
)
(
1
+
a
2
)
a
2
+
(
1
+
a
1
)
(
1
+
a
2
)
(
1
+
a
3
)
a
3
+
⋯
+
(
1
+
a
1
)
(
1
+
a
2
)
⋯
(
1
+
a
n
)
a
n
is greater than or equal to
2
n
−
1
2
n
\frac{2^n-1}{2^n}
2
n
2
n
−
1
.
2
1
Hide problems
Red/Blue Dominated Columns
Let
m
m
m
and
n
n
n
be odd positive integers. Each square of an
m
m
m
by
n
n
n
board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of
m
m
m
and
n
n
n
.