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Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2004 China Girls Math Olympiad
2004 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(8)
8
1
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A 10*11 chessboard
When the unit squares at the four corners are removed from a three by three squares, the resulting shape is called a cross. What is the maximum number of non-overlapping crosses placed within the boundary of a
10
×
11
10\times 11
10
×
11
chessboard? (Each cross covers exactly five unit squares on the board.)
7
1
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Determine the number of integers...
Let
p
p
p
and
q
q
q
be two coprime positive integers, and
n
n
n
be a non-negative integer. Determine the number of integers that can be written in the form ip \plus{} jq, where
i
i
i
and
j
j
j
are non-negative integers with i \plus{} j \leq n.
6
1
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Constant value
Given an acute triangle
A
B
C
ABC
A
BC
with
O
O
O
as its circumcenter. Line
A
O
AO
A
O
intersects
B
C
BC
BC
at
D
D
D
. Points
E
E
E
,
F
F
F
are on
A
B
AB
A
B
,
A
C
AC
A
C
respectively such that
A
A
A
,
E
E
E
,
D
D
D
,
F
F
F
are concyclic. Prove that the length of the projection of line segment
E
F
EF
EF
on side
B
C
BC
BC
does not depend on the positions of
E
E
E
and
F
F
F
.
5
1
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Find the smallest value of u + v + w
Let
u
,
v
,
w
u, v, w
u
,
v
,
w
be positive real numbers such that u\sqrt {vw} \plus{} v\sqrt {wu} \plus{} w\sqrt {uv} \geq 1. Find the smallest value of u \plus{} v \plus{} w.
4
1
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Value of the hands
A deck of
32
32
32
cards has
2
2
2
different jokers each of which is numbered
0
0
0
. There are
10
10
10
red cards numbered
1
1
1
through
10
10
10
and similarly for blue and green cards. One chooses a number of cards from the deck. If a card in hand is numbered
k
k
k
, then the value of the card is
2
k
2^k
2
k
, and the value of the hand is sum of the values of the cards in hand. Determine the number of hands having the value
2004
2004
2004
.
3
1
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Triangle inscribed in a circle radius 1
Let
A
B
C
ABC
A
BC
be an obtuse inscribed in a circle of radius
1
1
1
. Prove that
△
A
B
C
\triangle ABC
△
A
BC
can be covered by an isosceles right-angled triangle with hypotenuse of length \sqrt {2} \plus{} 1.
2
1
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Determine the minimum value
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive reals. Find the smallest value of \frac {a \plus{} 3c}{a \plus{} 2b \plus{} c} \plus{} \frac {4b}{a \plus{} b \plus{} 2c} \minus{} \frac {8c}{a \plus{} b \plus{} 3c}.
1
1
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Determine all the 'good' numbers
We say a positive integer
n
n
n
is good if there exists a permutation
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
of
1
,
2
,
…
,
n
1, 2, \ldots, n
1
,
2
,
…
,
n
such that k \plus{} a_k is perfect square for all
1
≤
k
≤
n
1\le k\le n
1
≤
k
≤
n
. Determine all the good numbers in the set
{
11
,
13
,
15
,
17
,
19
}
\{11, 13, 15, 17, 19\}
{
11
,
13
,
15
,
17
,
19
}
.