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Problems
Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2009 South East Mathematical Olympiad
2009 South East Mathematical Olympiad
Part of
South East Mathematical Olympiad
Subcontests
(8)
8
1
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squares chart
In an
8
{8}
8
×
8
{8}
8
squares chart , we dig out
n
n
n
squares , then we cannot cut a "T"shaped-5-squares out of the surplus chart . Then find the mininum value of
n
n
n
.
7
1
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Find minimum value and maximum value of f(x,y,z)
Let
x
,
y
,
z
≥
0
x,y,z\geq0
x
,
y
,
z
≥
0
be real numbers such that
x
+
y
+
z
=
1
x+y+z=1
x
+
y
+
z
=
1
Define
f
(
x
,
y
,
z
)
f(x,y,z)
f
(
x
,
y
,
z
)
in this way :
f
(
x
,
y
,
z
)
=
x
(
2
y
−
z
)
1
+
x
+
3
y
+
y
(
2
z
−
x
)
1
+
y
+
3
z
+
z
(
2
x
−
y
)
1
+
z
+
3
x
f(x,y,z)=\frac{x(2y-z)}{1+x+3y}+\frac{y(2z-x)}{1+y+3z}+\frac{z(2x-y)}{1+z+3x}
f
(
x
,
y
,
z
)
=
1
+
x
+
3
y
x
(
2
y
−
z
)
+
1
+
y
+
3
z
y
(
2
z
−
x
)
+
1
+
z
+
3
x
z
(
2
x
−
y
)
Find the minimum value and maximum value of
f
(
x
,
y
,
z
)
f(x,y,z)
f
(
x
,
y
,
z
)
.
6
1
Hide problems
same circumcircle and inscribed circles
Let
⊙
O
\odot O
⊙
O
,
⊙
I
\odot I
⊙
I
be the circumcircle and inscribed circles of triangle
A
B
C
ABC
A
BC
. Prove that : From every point
D
D
D
on
⊙
O
\odot O
⊙
O
,we can construct a triangle
D
E
F
DEF
D
EF
such that
A
B
C
ABC
A
BC
and
D
E
F
DEF
D
EF
have the same circumcircle and inscribed circles
5
1
Hide problems
find │M│
Let
X
=
(
x
1
,
x
2
,
.
.
.
.
.
.
,
x
9
)
X=(x_1,x_2,......,x_9)
X
=
(
x
1
,
x
2
,
......
,
x
9
)
be a permutation of the set
{
1
,
2
,
…
,
9
}
\{1,2,\ldots,9\}
{
1
,
2
,
…
,
9
}
and let
A
A
A
be the set of all such
X
X
X
. For any
X
∈
A
X \in A
X
∈
A
, denote
f
(
X
)
=
x
1
+
2
x
2
+
⋯
+
9
x
9
f(X)=x_1+2x_2+\cdots+9x_9
f
(
X
)
=
x
1
+
2
x
2
+
⋯
+
9
x
9
and
M
=
{
f
(
X
)
∣
X
∈
A
}
M=\{f(X)|X \in A \}
M
=
{
f
(
X
)
∣
X
∈
A
}
. Find
∣
M
∣
|M|
∣
M
∣
. (
∣
S
∣
|S|
∣
S
∣
denotes number of members of the set
S
S
S
.)
4
1
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combinatoric geometry
Given 12 red points on a circle , find the mininum value of
n
n
n
such that there exists
n
n
n
triangles whose vertex are the red points . Satisfies: every chord whose points are the red points is the edge of one of the
n
n
n
triangles .
3
1
Hide problems
Inequality for six variables-China South East Olympiad 2009
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive reals such that
a
=
x
(
y
−
z
)
2
\sqrt{a}=x(y-z)^2
a
=
x
(
y
−
z
)
2
,
b
=
y
(
z
−
x
)
2
\sqrt{b}=y(z-x)^2
b
=
y
(
z
−
x
)
2
and
c
=
z
(
x
−
y
)
2
\sqrt{c}=z(x-y)^2
c
=
z
(
x
−
y
)
2
. Prove that
a
2
+
b
2
+
c
2
≥
2
(
a
b
+
b
c
+
c
a
)
a^2+b^2+c^2 \geq 2(ab+bc+ca)
a
2
+
b
2
+
c
2
≥
2
(
ab
+
b
c
+
c
a
)
2
1
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Convex pentagon - A, B, C, D are concyclic
In the convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
we know that
A
B
=
D
E
,
B
C
=
E
A
AB=DE, BC=EA
A
B
=
D
E
,
BC
=
E
A
but
A
B
≠
E
A
AB \neq EA
A
B
=
E
A
.
B
,
C
,
D
,
E
B,C,D,E
B
,
C
,
D
,
E
are concyclic . Prove that
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
are concyclic if and only if
A
C
=
A
D
.
AC=AD.
A
C
=
A
D
.
1
1
Hide problems
x^2-2xy+126y^2=2009 - China South East Olympiad 2009
Find all pairs (
x
,
y
x,y
x
,
y
) of integers such that
x
2
−
2
x
y
+
126
y
2
=
2009
x^2-2xy+126y^2=2009
x
2
−
2
x
y
+
126
y
2
=
2009
.