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Contests
National and Regional Contests
Singapore Contests
Singapore Senior Math Olympiad
2019 Singapore Senior Math Olympiad
2019 Singapore Senior Math Olympiad
Part of
Singapore Senior Math Olympiad
Subcontests
(5)
5
1
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Singapore Combinatorics
Determine all integer
n
≥
2
n \ge 2
n
≥
2
such that it is possible to construct an
n
∗
n
n * n
n
∗
n
array where each entry is either
−
1
,
0
,
1
-1, 0, 1
−
1
,
0
,
1
so that the sums of elements in every row and every column are distinct
4
1
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2019 Singapore MO senior section P4
Positive integers
m
,
n
,
k
m,n,k
m
,
n
,
k
satisfy
1
+
2
+
3
+
+
.
.
.
+
n
=
m
k
1+2+3++...+n=mk
1
+
2
+
3
+
+
...
+
n
=
mk
and
m
≥
n
m \ge n
m
≥
n
. Show that we can partite
{
1
,
2
,
3
,
.
.
.
,
n
}
\{1,2,3,...,n \}
{
1
,
2
,
3
,
...
,
n
}
into
k
k
k
subsets (Every element belongs to exact one of these
k
k
k
subsets), such that the sum of elements in each subset is equal to
m
m
m
.
2
1
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Graph G has n vertices and mn edges, where $n>2m$, show that there exists
Graph
G
G
G
has
n
n
n
vertices and
m
n
mn
mn
edges, where
n
>
2
m
n>2m
n
>
2
m
, show that there exists a path with
m
+
1
m+1
m
+
1
vertices. (A path is an open walk without repeating vertices )
1
1
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Angle Bisector in a Parallelogram
In a parallelogram
A
B
C
D
ABCD
A
BC
D
, the bisector of
∠
A
\angle A
∠
A
intersects
B
C
BC
BC
at
M
M
M
and the extension of
D
C
DC
D
C
at
N
N
N
. Let
O
O
O
be the circumcircle of the triangle
M
C
N
MCN
MCN
. Prove that
∠
O
B
C
=
∠
O
D
C
\angle OBC = \angle ODC
∠
OBC
=
∠
O
D
C
3
1
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Pairwise LCMs greater than 4000
Let
a
1
,
a
2
,
⋯
,
a
2000
a_1,a_2,\cdots,a_{2000}
a
1
,
a
2
,
⋯
,
a
2000
be distinct positive integers such that
1
≤
a
1
<
a
2
<
⋯
<
a
2000
<
4000
1 \leq a_1 < a_2 < \cdots < a_{2000} < 4000
1
≤
a
1
<
a
2
<
⋯
<
a
2000
<
4000
such that the LCM (least common multiple) of any two of them is
≥
4000
\geq 4000
≥
4000
. Show that
a
1
≥
1334
a_1 \geq 1334
a
1
≥
1334