MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2016 Canada National Olympiad
2016 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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Concurrency on Circumcircle
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute-angled triangle with altitudes
A
D
AD
A
D
and
B
E
BE
BE
meeting at
H
H
H
. Let
M
M
M
be the midpoint of segment
A
B
AB
A
B
, and suppose that the circumcircles of
△
D
E
M
\triangle DEM
△
D
EM
and
△
A
B
H
\triangle ABH
△
A
B
H
meet at points
P
P
P
and
Q
Q
Q
with
P
P
P
on the same side of
C
H
CH
C
H
as
A
A
A
. Prove that the lines
E
D
,
P
H
,
ED, PH,
E
D
,
P
H
,
and
M
Q
MQ
MQ
all pass through a single point on the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
.
4
1
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Lavaman versus the Flea
Let
A
,
B
A, B
A
,
B
, and
F
F
F
be positive integers, and assume
A
<
B
<
2
A
A < B < 2A
A
<
B
<
2
A
. A flea is at the number
0
0
0
on the number line. The flea can move by jumping to the right by
A
A
A
or by
B
B
B
. Before the flea starts jumping, Lavaman chooses finitely many intervals
{
m
+
1
,
m
+
2
,
…
,
m
+
A
}
\{m+1, m+2, \ldots, m+A\}
{
m
+
1
,
m
+
2
,
…
,
m
+
A
}
consisting of
A
A
A
consecutive positive integers, and places lava at all of the integers in the intervals. The intervals must be chosen so that:(i) any two distinct intervals are disjoint and not adjacent; (ii) there are at least
F
F
F
positive integers with no lava between any two intervals; and (iii) no lava is placed at any integer less than
F
F
F
.Prove that the smallest
F
F
F
for which the flea can jump over all the intervals and avoid all the lava, regardless of what Lavaman does, is
F
=
(
n
−
1
)
A
+
B
F = (n-1)A + B
F
=
(
n
−
1
)
A
+
B
, where
n
n
n
is the positive integer such that
A
n
+
1
≤
B
−
A
<
A
n
\frac{A}{n+1} \le B-A < \frac{A}{n}
n
+
1
A
≤
B
−
A
<
n
A
.
3
1
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Polynomial Prime Values
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with integer coefficients such that
P
(
P
(
n
)
+
n
)
P(P(n) + n)
P
(
P
(
n
)
+
n
)
is a prime number for infinitely many integers
n
n
n
.
2
1
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System of Equations
Consider the following system of
10
10
10
equations in
10
10
10
real variables
v
1
,
…
,
v
10
v_1, \ldots, v_{10}
v
1
,
…
,
v
10
:
v
i
=
1
+
6
v
i
2
v
1
2
+
v
2
2
+
⋯
+
v
10
2
(
i
=
1
,
…
,
10
)
.
v_i = 1 + \frac{6v_i^2}{v_1^2 + v_2^2 + \cdots + v_{10}^2} \qquad (i = 1, \ldots, 10).
v
i
=
1
+
v
1
2
+
v
2
2
+
⋯
+
v
10
2
6
v
i
2
(
i
=
1
,
…
,
10
)
.
Find all
10
10
10
-tuples
(
v
1
,
v
2
,
…
,
v
10
)
(v_1, v_2, \ldots , v_{10})
(
v
1
,
v
2
,
…
,
v
10
)
that are solutions of this system.
1
1
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Averaging Sequneces
The integers
1
,
2
,
3
,
…
,
2016
1, 2, 3, \ldots, 2016
1
,
2
,
3
,
…
,
2016
are written on a board. You can choose any two numbers on the board and replace them with their average. For example, you can replace
1
1
1
and
2
2
2
with
1.5
1.5
1.5
, or you can replace
1
1
1
and
3
3
3
with a second copy of
2
2
2
. After
2015
2015
2015
replacements of this kind, the board will have only one number left on it.(a) Prove that there is a sequence of replacements that will make the final number equal to
2
2
2
.(b) Prove that there is a sequence of replacements that will make the final number equal to
1000
1000
1000
.