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National and Regional Contests
Canada Contests
Canada National Olympiad
2020 Canada National Olympiad
2020 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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2020 Canadian MO P5
Simple graph
G
G
G
has
19998
19998
19998
vertices. For any subgraph
G
ˉ
\bar G
G
ˉ
of
G
G
G
with
9999
9999
9999
vertices,
G
ˉ
\bar G
G
ˉ
has at least
9999
9999
9999
edges. Find the minimum number of edges in
G
G
G
4
1
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2020 Canadian MO P4
S
=
{
1
,
4
,
8
,
9
,
16
,
.
.
.
}
S= \{1,4,8,9,16,...\}
S
=
{
1
,
4
,
8
,
9
,
16
,
...
}
is the set of perfect integer power. (
S
=
{
n
k
∣
n
,
k
∈
Z
,
k
≥
2
}
S=\{ n^k| n, k \in Z, k \ge 2 \}
S
=
{
n
k
∣
n
,
k
∈
Z
,
k
≥
2
}
. )We arrange the elements in
S
S
S
into an increasing sequence
{
a
i
}
\{a_i\}
{
a
i
}
. Show that there are infinite many
n
n
n
, such that
9999
∣
a
n
+
1
−
a
n
9999|a_{n+1}-a_n
9999∣
a
n
+
1
−
a
n
3
1
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2020 Canadian MO P3.
There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exactly
2020
2020
2020
different ways to select the coins in his purse and the sum of these selected coins is
2020
2020
2020
?
2
1
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2020 Canadian MO P2
A
B
C
D
ABCD
A
BC
D
is a fixed rhombus. Segment
P
Q
PQ
PQ
is tangent to the inscribed circle of
A
B
C
D
ABCD
A
BC
D
, where
P
P
P
is on side
A
B
AB
A
B
,
Q
Q
Q
is on side
A
D
AD
A
D
. Show that, when segment
P
Q
PQ
PQ
is moving, the area of
Δ
C
P
Q
\Delta CPQ
Δ
CPQ
is a constant.
1
1
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2020 Canadian MO P1
There are
n
≥
3
n \ge 3
n
≥
3
distinct positive real numbers. Show that there are at most
n
−
2
n-2
n
−
2
different integer power of three that can be written as the sum of three distinct elements from these
n
n
n
numbers.