MathDB
Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2002 China Girls Math Olympiad
2002 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(7)
8
1
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If the eight projections are pairwise disjoint
Assume that
A
1
,
A
2
,
…
,
A
8
A_1, A_2, \ldots, A_8
A
1
,
A
2
,
…
,
A
8
are eight points taken arbitrarily on a plane. For a directed line
l
l
l
taken arbitrarily on the plane, assume that projections of
A
1
,
A
2
,
…
,
A
8
A_1, A_2, \ldots, A_8
A
1
,
A
2
,
…
,
A
8
on the line are
P
1
,
P
2
,
…
,
P
8
P_1, P_2, \ldots, P_8
P
1
,
P
2
,
…
,
P
8
respectively. If the eight projections are pairwise disjoint, they can be arranged as
P
i
1
,
P
i
2
,
…
,
P
i
8
P_{i_1}, P_{i_2}, \ldots, P_{i_8}
P
i
1
,
P
i
2
,
…
,
P
i
8
according to the direction of line
l
.
l.
l
.
Thus we get one permutation for
1
,
2
,
…
,
8
,
1, 2, \ldots, 8,
1
,
2
,
…
,
8
,
namely,
i
1
,
i
2
,
…
,
i
8
.
i_1, i_2, \ldots, i_8.
i
1
,
i
2
,
…
,
i
8
.
In the figure, this permutation is
2
,
1
,
8
,
3
,
7
,
4
,
6
,
5.
2, 1, 8, 3, 7, 4, 6, 5.
2
,
1
,
8
,
3
,
7
,
4
,
6
,
5.
Assume that after these eight points are projected to every directed line on the plane, we get the number of different permutations as N_8 \equal{} N(A_1, A_2, \ldots, A_8). Find the maximal value of
N
8
.
N_8.
N
8
.
7
1
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Triangle DEF is not over half of perimeter of triangle ABC
An acute triangle
A
B
C
ABC
A
BC
has three heights
A
D
,
B
E
AD, BE
A
D
,
BE
and
C
F
CF
CF
respectively. Prove that the perimeter of triangle
D
E
F
DEF
D
EF
is not over half of the perimeter of triangle
A
B
C
.
ABC.
A
BC
.
5
1
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Sum of reciprocal consecutive permutation sum
There are
n
≥
2
n \geq 2
n
≥
2
permutations
P
1
,
P
2
,
…
,
P
n
P_1, P_2, \ldots, P_n
P
1
,
P
2
,
…
,
P
n
each being an arbitrary permutation of
{
1
,
…
,
n
}
.
\{1,\ldots,n\}.
{
1
,
…
,
n
}
.
Prove that \sum^{n\minus{}1}_{i\equal{}1} \frac{1}{P_i \plus{} P_{i\plus{}1}} > \frac{n\minus{}1}{n\plus{}2}.
4
1
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Prove that AH/HF = AC/CD
Circles
O
1
O_1
O
1
and
O
2
O_2
O
2
interest at two points
B
B
B
and
C
,
C,
C
,
and
B
C
BC
BC
is the diameter of circle
O
1
.
O_1.
O
1
.
Construct a tangent line of circle
O
1
O_1
O
1
at
C
C
C
and intersecting circle
O
2
O_2
O
2
at another point
A
.
A.
A
.
We join
A
B
AB
A
B
to intersect circle
O
1
O_1
O
1
at point
E
,
E,
E
,
then join
C
E
CE
CE
and extend it to intersect circle
O
2
O_2
O
2
at point
F
.
F.
F
.
Assume
H
H
H
is an arbitrary point on line segment
A
F
.
AF.
A
F
.
We join
H
E
HE
H
E
and extend it to intersect circle
O
1
O_1
O
1
at point
G
,
G,
G
,
and then join
B
G
BG
BG
and extend it to intersect the extend line of
A
C
AC
A
C
at point
D
.
D.
D
.
Prove that
A
H
H
F
=
A
C
C
D
.
\frac{AH}{HF} = \frac{AC}{CD}.
H
F
A
H
=
C
D
A
C
.
3
1
Hide problems
Triangle with k(ab + bc + ca) > 5(a^2 + b^2 + c^2)
Find all positive integers
k
k
k
such that for any positive numbers
a
,
b
a, b
a
,
b
and
c
c
c
satisfying the inequality k(ab \plus{} bc \plus{} ca) > 5(a^2 \plus{} b^2 \plus{} c^2), there must exist a triangle with
a
,
b
a, b
a
,
b
and
c
c
c
as the length of its three sides respectively.
2
1
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There were three girl students to be on duty every day
There are 3n, n \in \mathbb{Z}^\plus{} girl students who took part in a summer camp. There were three girl students to be on duty every day. When the summer camp ended, it was found that any two of the
3
n
3n
3
n
students had just one time to be on duty on the same day. (1) When n\equal{}3, is there any arrangement satisfying the requirement above. Prove yor conclusion. (2) Prove that
n
n
n
is an odd number.
1
1
Hide problems
20n+2 can divide 2003n + 2002
Find all positive integers
n
n
n
such 20n\plus{}2 can divide 2003n \plus{} 2002.