MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2011 Singapore MO Open
2011 Singapore MO Open
Part of
Singapore MO Open
Subcontests
(5)
5
1
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Find all pairs of positive integers
Find all pairs of positive integers
(
m
,
n
)
(m,n)
(
m
,
n
)
such that
m
+
n
−
3
m
n
m
+
n
=
2011
3
.
m+n-\frac{3mn}{m+n}=\frac{2011}{3}.
m
+
n
−
m
+
n
3
mn
=
3
2011
.
4
1
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P(a) integer implies a integer
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients such that
P
(
a
)
∈
Z
implies that
a
∈
Z
.
P(a)\in\mathbb{Z}\ \ \ \text{implies that}\ \ \ a\in\mathbb{Z}.
P
(
a
)
∈
Z
implies that
a
∈
Z
.
3
1
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1/x+1/y+1/z<1/xyz
Let
x
,
y
,
z
>
0
x,y,z>0
x
,
y
,
z
>
0
such that
1
x
+
1
y
+
1
z
<
1
x
y
z
\frac1x+\frac1y+\frac1z<\frac{1}{xyz}
x
1
+
y
1
+
z
1
<
x
yz
1
. Show that
2
x
1
+
x
2
+
2
y
1
+
y
2
+
2
z
1
+
z
2
<
3.
\frac{2x}{\sqrt{1+x^2}}+\frac{2y}{\sqrt{1+y^2}}+\frac{2z}{\sqrt{1+z^2}}<3.
1
+
x
2
2
x
+
1
+
y
2
2
y
+
1
+
z
2
2
z
<
3.
2
1
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9x9 board
If 46 squares are colored red in a
9
×
9
9\times 9
9
×
9
board, show that there is a
2
×
2
2\times 2
2
×
2
block on the board in which at least 3 of the squares are colored red.
1
1
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Angle ODQ=90
In the acute-angled non-isosceles triangle
A
B
C
ABC
A
BC
,
O
O
O
is its circumcenter,
H
H
H
is its orthocenter and
A
B
>
A
C
AB>AC
A
B
>
A
C
. Let
Q
Q
Q
be a point on
A
C
AC
A
C
such that the extension of
H
Q
HQ
H
Q
meets the extension of
B
C
BC
BC
at the point
P
P
P
. Suppose
B
D
=
D
P
BD=DP
B
D
=
D
P
, where
D
D
D
is the foot of the perpendicular from
A
A
A
onto
B
C
BC
BC
. Prove that
∠
O
D
Q
=
9
0
∘
\angle ODQ=90^{\circ}
∠
O
D
Q
=
9
0
∘
.