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Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2007 China Girls Math Olympiad
2007 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(8)
8
1
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Chess tournament
In a round robin chess tournament each player plays every other player exactly once. The winner of each game gets
1
1
1
point and the loser gets
0
0
0
points. If the game is tied, each player gets
0.5
0.5
0.5
points. Given a positive integer
m
m
m
, a tournament is said to have property
P
(
m
)
P(m)
P
(
m
)
if the following holds: among every set
S
S
S
of
m
m
m
players, there is one player who won all her games against the other m\minus{}1 players in
S
S
S
and one player who lost all her games against the other m \minus{} 1 players in
S
S
S
. For a given integer
m
≥
4
m \ge 4
m
≥
4
, determine the minimum value of
n
n
n
(as a function of
m
m
m
) such that the following holds: in every
n
n
n
-player round robin chess tournament with property
P
(
m
)
P(m)
P
(
m
)
, the final scores of the
n
n
n
players are all distinct.
7
1
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CGMO 2007 Problem 7
Let
a
a
a
,
b
b
b
,
c
c
c
be integers each with absolute value less than or equal to
10
10
10
. The cubic polynomial f(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c satisfies the property \Big|f\left(2 \plus{} \sqrt 3\right)\Big| < 0.0001. Determine if 2 \plus{} \sqrt 3 is a root of
f
f
f
.
5
1
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Perpendicularity
Point
D
D
D
lies inside triangle
A
B
C
ABC
A
BC
such that
∠
D
A
C
=
∠
D
C
A
=
3
0
∘
\angle DAC = \angle DCA = 30^{\circ}
∠
D
A
C
=
∠
D
C
A
=
3
0
∘
and
∠
D
B
A
=
6
0
∘
\angle DBA = 60^{\circ}
∠
D
B
A
=
6
0
∘
. Point
E
E
E
is the midpoint of segment
B
C
BC
BC
. Point
F
F
F
lies on segment
A
C
AC
A
C
with
A
F
=
2
F
C
AF = 2FC
A
F
=
2
FC
. Prove that
D
E
⊥
E
F
DE \perp EF
D
E
⊥
EF
.
3
1
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Determine the minimum value
Let
n
n
n
be an integer greater than
3
3
3
, and let
a
1
,
a
2
,
⋯
,
a
n
a_1, a_2, \cdots, a_n
a
1
,
a
2
,
⋯
,
a
n
be non-negative real numbers with a_1 \plus{} a_2 \plus{} \cdots \plus{} a_n \equal{} 2. Determine the minimum value of \frac{a_1}{a_2^2 \plus{} 1}\plus{} \frac{a_2}{a^2_3 \plus{} 1}\plus{} \cdots \plus{} \frac{a_n}{a^2_1 \plus{} 1}.
1
1
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Good numbers
A positive integer
m
m
m
is called good if there is a positive integer
n
n
n
such that
m
m
m
is the quotient of
n
n
n
by the number of positive integer divisors of
n
n
n
(including
1
1
1
and
n
n
n
itself). Prove that
1
,
2
,
…
,
17
1, 2, \ldots, 17
1
,
2
,
…
,
17
are good numbers and that
18
18
18
is not a good number.
2
1
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Prove ABC is an isoscles triangle
Let
A
B
C
ABC
A
BC
be an acute triangle. Points
D
D
D
,
E
E
E
, and
F
F
F
lie on segments
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
, respectively, and each of the three segments
A
D
AD
A
D
,
B
E
BE
BE
, and
C
F
CF
CF
contains the circumcenter of
A
B
C
ABC
A
BC
. Prove that if any two of the ratios
B
D
D
C
\frac{BD}{DC}
D
C
B
D
,
C
E
E
A
\frac{CE}{EA}
E
A
CE
,
A
F
F
B
\frac{AF}{FB}
FB
A
F
,
B
F
F
A
\frac{BF}{FA}
F
A
BF
,
A
E
E
C
\frac{AE}{EC}
EC
A
E
,
C
D
D
B
\frac{CD}{DB}
D
B
C
D
are integers, then triangle
A
B
C
ABC
A
BC
is isosceles.
4
1
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axis of symmetry number
The set
S
S
S
consists of
n
>
2
n > 2
n
>
2
points in the plane. The set
P
P
P
consists of
m
m
m
lines in the plane such that every line in
P
P
P
is an axis of symmetry for
S
S
S
. Prove that
m
≤
n
m\leq n
m
≤
n
, and determine when equality holds.
6
1
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non-symmetric ineq (for girls)
For
a
,
b
,
c
≥
0
a,b,c\geq 0
a
,
b
,
c
≥
0
with a\plus{}b\plus{}c\equal{}1, prove that \sqrt{a\plus{}\frac{(b\minus{}c)^2}{4}}\plus{}\sqrt{b}\plus{}\sqrt{c}\leq \sqrt{3}