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Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2003 China Girls Math Olympiad
2003 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(8)
8
1
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Elements in Sn have their last digits equal to 3
Let
n
n
n
be a positive integer, and
S
n
,
S_n,
S
n
,
be the set of all positive integer divisors of
n
n
n
(including 1 and itself). Prove that at most half of the elements in
S
n
S_n
S
n
have their last digits equal to 3.
7
1
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Prove that angle BAC is greater than 90°
Let the sides of a scalene triangle
△
A
B
C
\triangle ABC
△
A
BC
be AB \equal{} c, BC \equal{} a, CA \equal{}b, and
D
,
E
,
F
D, E , F
D
,
E
,
F
be points on
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
such that
A
D
,
B
E
,
C
F
AD, BE, CF
A
D
,
BE
,
CF
are angle bisectors of the triangle, respectively. Assume that DE \equal{} DF. Prove that (1) \frac{a}{b\plus{}c} \equal{} \frac{b}{c\plus{}a} \plus{} \frac{c}{a\plus{}b} (2)
∠
B
A
C
>
9
0
∘
.
\angle BAC > 90^{\circ}.
∠
B
A
C
>
9
0
∘
.
6
1
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Find the largest real number lambda
Let
n
≥
2
n \geq 2
n
≥
2
be an integer. Find the largest real number
λ
\lambda
λ
such that the inequality a^2_n \geq \lambda \sum^{n\minus{}1}_{i\equal{}1} a_i \plus{} 2 \cdot a_n. holds for any positive integers
a
1
,
a
2
,
…
a
n
a_1, a_2, \ldots a_n
a
1
,
a
2
,
…
a
n
satisfying
a
1
<
a
2
<
…
<
a
n
.
a_1 < a_2 < \ldots < a_n.
a
1
<
a
2
<
…
<
a
n
.
5
1
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1 - 2003 to the power of 2003 less than 1
Let
{
a
n
}
1
∞
\{a_n\}^{\infty}_1
{
a
n
}
1
∞
be a sequence of real numbers such that a_1 \equal{} 2, and a_{n\plus{}1} \equal{} a^2_n \minus{} a_n \plus{} 1, \forall n \in \mathbb{N}. Prove that 1 \minus{} \frac{1}{2003^{2003}} < \sum^{2003}_{i\equal{}1} \frac{1}{a_i} < 1.
4
1
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For any arrangement of these numbers around a circle
(1) Prove that there exist five nonnegative real numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
and
e
e
e
with their sum equal to 1 such that for any arrangement of these numbers around a circle, there are always two neighboring numbers with their product not less than
1
9
.
\frac{1}{9}.
9
1
.
(2) Prove that for any five nonnegative real numbers with their sum equal to 1 , it is always possible to arrange them around a circle such that there are two neighboring numbers with their product not greater than
1
9
.
\frac{1}{9}.
9
1
.
3
1
Hide problems
ABCD is inscribed in a circle with AC as its diameter
As shown in the figure, quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle with
A
C
AC
A
C
as its diameter,
B
D
⊥
A
C
,
BD \perp AC,
B
D
⊥
A
C
,
and
E
E
E
the intersection of
A
C
AC
A
C
and
B
D
.
BD.
B
D
.
Extend line segment
D
A
DA
D
A
and
B
A
BA
B
A
through
A
A
A
to
F
F
F
and
G
G
G
respectively, such that
D
G
∥
B
F
.
DG \parallel{} BF.
D
G
∥
BF
.
Extend
G
F
GF
GF
to
H
H
H
such that
C
H
⊥
G
H
.
CH \perp GH.
C
H
⊥
G
H
.
Prove that points
B
,
E
,
F
B, E, F
B
,
E
,
F
and
H
H
H
lie on one circle.[asy] defaultpen(linewidth(0.8)+fontsize(10));size(150); real a=4, b=6.5, c=9, d=a*c/b, g=14, f=sqrt(a^2+b^2)*sqrt(a^2+d^2)/g; pair E=origin, A=(0,a), B=(-b,0), C=(0,-c), D=(d,0), G=A+g*dir(B--A), F=A+f*dir(D--A), M=midpoint(G--C); path c1=circumcircle(A,B,C), c2=Circle(M, abs(M-G)); pair Hf=F+10*dir(G--F), H=intersectionpoint(F--Hf, c2); dot(A^^B^^C^^D^^E^^F^^G^^H); draw(c1^^c2^^G--D--C--A--G--F--D--B--A^^F--H--C--B--F); draw(H--B^^F--E^^G--C, linetype("2 2")); pair point= E; label("
A
A
A
", A, dir(point--A)); label("
B
B
B
", B, dir(point--B)); label("
C
C
C
", C, dir(point--C)); label("
D
D
D
", D, dir(point--D)); label("
F
F
F
", F, dir(point--F)); label("
G
G
G
", G, dir(point--G)); label("
H
H
H
", H, dir(point--H)); label("
E
E
E
", E, NE);[/asy]
2
1
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There are 47 students in a classroom with seats
There are 47 students in a classroom with seats arranged in 6 rows
×
\times
×
8 columns, and the seat in the
i
i
i
-th row and
j
j
j
-th column is denoted by
(
i
,
j
)
.
(i,j).
(
i
,
j
)
.
Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat
(
i
,
j
)
,
(i,j),
(
i
,
j
)
,
if his/her new seat is
(
m
,
n
)
,
(m,n),
(
m
,
n
)
,
we say that the student is moved by [a, b] \equal{} [i \minus{} m, j \minus{} n] and define the position value of the student as a\plus{}b. Let
S
S
S
denote the sum of the position values of all the students. Determine the difference between the greatest and smallest possible values of
S
.
S.
S
.
1
1
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Third square root inequality
Let
A
B
C
ABC
A
BC
be a triangle. Points
D
D
D
and
E
E
E
are on sides
A
B
AB
A
B
and
A
C
,
AC,
A
C
,
respectively, and point
F
F
F
is on line segment
D
E
.
DE.
D
E
.
Let \frac {AD}{AB} \equal{} x, \frac {AE}{AC} \equal{} y, \frac {DF}{DE} \equal{} z. Prove that (1) S_{\triangle BDF} \equal{} (1 \minus{} x)y S_{\triangle ABC} and S_{\triangle CEF} \equal{} x(1 \minus{} y) (1 \minus{} z)S_{\triangle ABC}; (2) \sqrt [3]{S_{\triangle BDF}} \plus{} \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.