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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2019 Canada National Olympiad
2019 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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A Game where Construction of an odd cycle loses
A 2-player game is played on
n
≥
3
n\geq 3
n
≥
3
points, where no 3 points are collinear. Each move consists of selecting 2 of the points and drawing a new line segment connecting them. The first player to draw a line segment that creates an odd cycle loses. (An odd cycle must have all its vertices among the
n
n
n
points from the start, so the vertices of the cycle cannot be the intersections of the lines drawn.) Find all
n
n
n
such that the player to move first wins.
1
1
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Prove that the distance between circumcentres is > 2019
Points
A
,
B
,
C
A,B,C
A
,
B
,
C
are on a plane such that
A
B
=
B
C
=
C
A
=
6
AB=BC=CA=6
A
B
=
BC
=
C
A
=
6
. At any step, you may choose any three existing points and draw that triangle's circumcentre. Prove that you can draw a point such that its distance from an previously drawn point is:
(
a
)
(a)
(
a
)
greater than 7
(
b
)
(b)
(
b
)
greater than 2019
4
1
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Absolute Value Inequality for all k
Prove that for
n
>
1
n>1
n
>
1
and real numbers
a
0
,
a
1
,
…
,
a
n
,
k
a_0,a_1,\dots, a_n,k
a
0
,
a
1
,
…
,
a
n
,
k
with
a
1
=
a
n
−
1
=
0
a_1=a_{n-1}=0
a
1
=
a
n
−
1
=
0
,
∣
a
0
∣
−
∣
a
n
∣
≤
∑
i
=
0
n
−
2
∣
a
i
−
k
a
i
+
1
−
a
i
+
2
∣
.
|a_0|-|a_n|\leq \sum_{i=0}^{n-2}|a_i-ka_{i+1}-a_{i+2}|.
∣
a
0
∣
−
∣
a
n
∣
≤
i
=
0
∑
n
−
2
∣
a
i
−
k
a
i
+
1
−
a
i
+
2
∣.
3
1
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Ways to Place Counters on 2mx2n board
You have a
2
m
2m
2
m
by
2
n
2n
2
n
grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place
m
n
mn
mn
counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.
2
1
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Prove a^2+3ab+3b^2-1|a+b^3 implies c^3|a^2+3ab+3b^2-1
Let
a
,
b
a,b
a
,
b
be positive integers such that
a
+
b
3
a+b^3
a
+
b
3
is divisible by
a
2
+
3
a
b
+
3
b
2
−
1
a^2+3ab+3b^2-1
a
2
+
3
ab
+
3
b
2
−
1
. Prove that
a
2
+
3
a
b
+
3
b
2
−
1
a^2+3ab+3b^2-1
a
2
+
3
ab
+
3
b
2
−
1
is divisible by the cube of an integer greater than 1.