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Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2008 China Girls Math Olympiad
2008 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(8)
8
1
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Infinitely many odd numbers
For positive integers
n
n
n
, f_n \equal{} \lfloor2^n\sqrt {2008}\rfloor \plus{} \lfloor2^n\sqrt {2009}\rfloor. Prove there are infinitely many odd numbers and infinitely many even numbers in the sequence
f
1
,
f
2
,
…
f_1,f_2,\ldots
f
1
,
f
2
,
…
.
7
1
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2800 x 2800 chessboard
On a given
2008
×
2008
2008 \times 2008
2008
×
2008
chessboard, each unit square is colored in a different color. Every unit square is filled with one of the letters C, G, M, O. The resulting board is called harmonic if every
2
×
2
2 \times 2
2
×
2
subsquare contains all four different letters. How many harmonic boards are there?
6
1
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Sequence is monotonically decreasing
Let
(
x
1
,
x
2
,
⋯
)
(x_1,x_2,\cdots)
(
x
1
,
x
2
,
⋯
)
be a sequence of positive numbers such that (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8 and x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots Determine real number
a
a
a
such that if
x
1
>
a
x_1 > a
x
1
>
a
, then the sequence is monotonically decreasing, and if
0
<
x
1
<
a
0 < x_1 < a
0
<
x
1
<
a
, then the sequence is not monotonic.
5
1
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Corresponding vertices are in the same orientation
In convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, AB \equal{} BC and AD \equal{} DC. Point
E
E
E
lies on segment
A
B
AB
A
B
and point
F
F
F
lies on segment
A
D
AD
A
D
such that
B
B
B
,
E
E
E
,
F
F
F
,
D
D
D
lie on a circle. Point
P
P
P
is such that triangles
D
P
E
DPE
D
PE
and
A
D
C
ADC
A
D
C
are similar and the corresponding vertices are in the same orientation (clockwise or counterclockwise). Point
Q
Q
Q
is such that triangles
B
Q
F
BQF
BQF
and
A
B
C
ABC
A
BC
are similar and the corresponding vertices are in the same orientation. Prove that points
A
A
A
,
P
P
P
,
Q
Q
Q
are collinear.
4
1
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Maximum value of (XY + ZW)/(AC + BD)
Equilateral triangles
A
B
Q
ABQ
A
BQ
,
B
C
R
BCR
BCR
,
C
D
S
CDS
C
D
S
,
D
A
P
DAP
D
A
P
are erected outside of the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Let
X
X
X
,
Y
Y
Y
,
Z
Z
Z
,
W
W
W
be the midpoints of the segments
P
Q
PQ
PQ
,
Q
R
QR
QR
,
R
S
RS
RS
,
S
P
SP
SP
, respectively. Determine the maximum value of \frac {XZ\plus{}YW}{AC \plus{} BD}.
3
1
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Area ratio between two of the triangles
Determine the least real number
a
a
a
greater than
1
1
1
such that for any point
P
P
P
in the interior of the square
A
B
C
D
ABCD
A
BC
D
, the area ratio between two of the triangles
P
A
B
PAB
P
A
B
,
P
B
C
PBC
PBC
,
P
C
D
PCD
PC
D
,
P
D
A
PDA
P
D
A
lies in the interval
[
1
a
,
a
]
\left[\frac {1}{a},a\right]
[
a
1
,
a
]
.
2
1
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Let ax^3 + bx^2 + cx + d be a polynomial
Let \varphi(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d be a polynomial with real coefficients. Given that
φ
(
x
)
\varphi(x)
φ
(
x
)
has three positive real roots and that
φ
(
0
)
<
0
\varphi(0) < 0
φ
(
0
)
<
0
, prove that 2b^3 \plus{} 9a^2d \minus{} 7abc \leq 0.
1
1
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Partitioned into 32 sets of equal size and equal sum
(a) Determine if the set
{
1
,
2
,
…
,
96
}
\{1,2,\ldots,96\}
{
1
,
2
,
…
,
96
}
can be partitioned into 32 sets of equal size and equal sum. (b) Determine if the set
{
1
,
2
,
…
,
99
}
\{1,2,\ldots,99\}
{
1
,
2
,
…
,
99
}
can be partitioned into 33 sets of equal size and equal sum.