MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2004 China National Olympiad
2004 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
3
2
Hide problems
Minimum value of the number of points in the set M
Let
M
M
M
be a set consisting of
n
n
n
points in the plane, satisfying: i) there exist
7
7
7
points in
M
M
M
which constitute the vertices of a convex heptagon; ii) if for any
5
5
5
points in
M
M
M
which constitute the vertices of a convex pentagon, then there is a point in
M
M
M
which lies in the interior of the pentagon. Find the minimum value of
n
n
n
.Leng Gangsong
Only a finite number of integers cannot be represented
Prove that every positive integer
n
n
n
, except a finite number of them, can be represented as a sum of
2004
2004
2004
positive integers:
n
=
a
1
+
a
2
+
⋯
+
a
2004
n=a_1+a_2+\cdots +a_{2004}
n
=
a
1
+
a
2
+
⋯
+
a
2004
, where
1
≤
a
1
<
a
2
<
⋯
<
a
2004
1\le a_1<a_2<\cdots <a_{2004}
1
≤
a
1
<
a
2
<
⋯
<
a
2004
, and
a
i
∣
a
i
+
1
a_i \mid a_{i+1}
a
i
∣
a
i
+
1
for all
1
≤
i
≤
2003
1\le i\le 2003
1
≤
i
≤
2003
.Chen Yonggao
1
2
Hide problems
Exactly one sequence x_i
For a given real number
a
a
a
and a positive integer
n
n
n
, prove that: i) there exists exactly one sequence of real numbers
x
0
,
x
1
,
…
,
x
n
,
x
n
+
1
x_0,x_1,\ldots,x_n,x_{n+1}
x
0
,
x
1
,
…
,
x
n
,
x
n
+
1
such that
{
x
0
=
x
n
+
1
=
0
,
1
2
(
x
i
+
x
i
+
1
)
=
x
i
+
x
i
3
−
a
3
,
i
=
1
,
2
,
…
,
n
.
\begin{cases} x_0=x_{n+1}=0,\\ \frac{1}{2}(x_i+x_{i+1})=x_i+x_i^3-a^3,\ i=1,2,\ldots,n.\end{cases}
{
x
0
=
x
n
+
1
=
0
,
2
1
(
x
i
+
x
i
+
1
)
=
x
i
+
x
i
3
−
a
3
,
i
=
1
,
2
,
…
,
n
.
ii) the sequence
x
0
,
x
1
,
…
,
x
n
,
x
n
+
1
x_0,x_1,\ldots,x_n,x_{n+1}
x
0
,
x
1
,
…
,
x
n
,
x
n
+
1
in i) satisfies
∣
x
i
∣
≤
∣
a
∣
|x_i|\le |a|
∣
x
i
∣
≤
∣
a
∣
where
i
=
0
,
1
,
…
,
n
+
1
i=0,1,\ldots,n+1
i
=
0
,
1
,
…
,
n
+
1
.Liang Yengde
Find expression for F_1C/CG_1 in terms of lambda
Let
E
F
G
H
,
A
B
C
D
EFGH,ABCD
EFG
H
,
A
BC
D
and
E
1
F
1
G
1
H
1
E_1F_1G_1H_1
E
1
F
1
G
1
H
1
be three convex quadrilaterals satisfying:i) The points
E
,
F
,
G
E,F,G
E
,
F
,
G
and
H
H
H
lie on the sides
A
B
,
B
C
,
C
D
AB,BC,CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
respectively, and
A
E
E
B
⋅
B
F
F
C
⋅
C
G
G
D
⋅
D
H
H
A
=
1
\frac{AE}{EB}\cdot\frac{BF}{FC}\cdot \frac{CG}{GD}\cdot \frac{DH}{HA}=1
EB
A
E
⋅
FC
BF
⋅
G
D
CG
⋅
H
A
DH
=
1
; ii) The points
A
,
B
,
C
A,B,C
A
,
B
,
C
and
D
D
D
lie on sides
H
1
E
1
,
E
1
F
1
,
F
1
,
G
1
H_1E_1,E_1F_1,F_1,G_1
H
1
E
1
,
E
1
F
1
,
F
1
,
G
1
and
G
1
H
1
G_1H_1
G
1
H
1
respectively, and
E
1
F
1
∣
∣
E
F
,
F
1
G
1
∣
∣
F
G
,
G
1
H
1
∣
∣
G
H
,
H
1
E
1
∣
∣
H
E
E_1F_1||EF,F_1G_1||FG,G_1H_1||GH,H_1E_1||HE
E
1
F
1
∣∣
EF
,
F
1
G
1
∣∣
FG
,
G
1
H
1
∣∣
G
H
,
H
1
E
1
∣∣
H
E
.Suppose that
E
1
A
A
H
1
=
λ
\frac{E_1A}{AH_1}=\lambda
A
H
1
E
1
A
=
λ
. Find an expression for
F
1
C
C
G
1
\frac{F_1C}{CG_1}
C
G
1
F
1
C
in terms of
λ
\lambda
λ
.Xiong Bin
2
2
Hide problems
Expression for x_n in terms of n and x_1
Let
c
c
c
be a positive integer. Consider the sequence
x
1
,
x
2
,
…
x_1,x_2,\ldots
x
1
,
x
2
,
…
which satisfies
x
1
=
c
x_1=c
x
1
=
c
and, for
n
≥
2
n\ge 2
n
≥
2
,
x
n
=
x
n
−
1
+
⌊
2
x
n
−
1
−
(
n
+
2
)
n
⌋
+
1
x_n=x_{n-1}+\left\lfloor\frac{2x_{n-1}-(n+2)}{n}\right\rfloor+1
x
n
=
x
n
−
1
+
⌊
n
2
x
n
−
1
−
(
n
+
2
)
⌋
+
1
where
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
denotes the largest integer not greater than
x
x
x
. Determine an expression for
x
n
x_n
x
n
in terms of
n
n
n
and
c
c
c
. Huang Yumin
Inequality for the n positive integers a_i
For a given positive integer
n
≥
2
n\ge 2
n
≥
2
, suppose positive integers
a
i
a_i
a
i
where
1
≤
i
≤
n
1\le i\le n
1
≤
i
≤
n
satisfy
a
1
<
a
2
<
…
<
a
n
a_1<a_2<\ldots <a_n
a
1
<
a
2
<
…
<
a
n
and
∑
i
=
1
n
1
a
i
≤
1
\sum_{i=1}^n \frac{1}{a_i}\le 1
∑
i
=
1
n
a
i
1
≤
1
. Prove that, for any real number
x
x
x
, the following inequality holds
(
∑
i
=
1
n
1
a
i
2
+
x
2
)
2
≤
1
2
⋅
1
a
1
(
a
1
−
1
)
+
x
2
\left(\sum_{i=1}^n\frac{1}{a_i^2+x^2}\right)^2\le\frac{1}{2}\cdot\frac{1}{a_1(a_1-1)+x^2}
(
i
=
1
∑
n
a
i
2
+
x
2
1
)
2
≤
2
1
⋅
a
1
(
a
1
−
1
)
+
x
2
1
Li Shenghong