MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2003 China National Olympiad
2003 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
3
2
Hide problems
If product of tan x_i is 2^(n/2) then inequality holds
Given a positive integer
n
n
n
, find the least
λ
>
0
\lambda>0
λ
>
0
such that for any
x
1
,
…
x
n
∈
(
0
,
π
2
)
x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right)
x
1
,
…
x
n
∈
(
0
,
2
π
)
, the condition
∏
i
=
1
n
tan
x
i
=
2
n
2
\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}}
∏
i
=
1
n
tan
x
i
=
2
2
n
implies
∑
i
=
1
n
cos
x
i
≤
λ
\sum_{i=1}^{n}\cos x_i\le\lambda
∑
i
=
1
n
cos
x
i
≤
λ
.Huang Yumin
Inequality with reals a,b,c,d and x_i, y_i
Suppose
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are positive reals such that
a
b
+
c
d
=
1
ab+cd=1
ab
+
c
d
=
1
and
x
i
,
y
i
x_i,y_i
x
i
,
y
i
are real numbers such that
x
i
2
+
y
i
2
=
1
x_i^2+y_i^2=1
x
i
2
+
y
i
2
=
1
for
i
=
1
,
2
,
3
,
4
i=1,2,3,4
i
=
1
,
2
,
3
,
4
. Prove that
(
a
x
1
+
b
x
2
+
c
x
3
+
d
x
4
)
2
+
(
a
y
4
+
b
y
3
+
c
y
2
+
d
y
1
)
2
≤
2
(
a
2
+
b
2
a
b
+
c
2
+
d
2
c
d
)
.
(ax_1+bx_2+cx_3+dx_4)^2+(ay_4+by_3+cy_2+dy_1)^2\le 2\left(\frac{a^2+b^2}{ab}+\frac{c^2+d^2}{cd}\right).
(
a
x
1
+
b
x
2
+
c
x
3
+
d
x
4
)
2
+
(
a
y
4
+
b
y
3
+
c
y
2
+
d
y
1
)
2
≤
2
(
ab
a
2
+
b
2
+
c
d
c
2
+
d
2
)
.
Li Shenghong
2
2
Hide problems
Maximal size set of S with 3 conditions
Determine the maximal size of the set
S
S
S
such that: i) all elements of
S
S
S
are natural numbers not exceeding
100
100
100
; ii) for any two elements
a
,
b
a,b
a
,
b
in
S
S
S
, there exists
c
c
c
in
S
S
S
such that
(
a
,
c
)
=
(
b
,
c
)
=
1
(a,c)=(b,c)=1
(
a
,
c
)
=
(
b
,
c
)
=
1
; iii) for any two elements
a
,
b
a,b
a
,
b
in
S
S
S
, there exists
d
d
d
in
S
S
S
such that
(
a
,
d
)
>
1
,
(
b
,
d
)
>
1
(a,d)>1,(b,d)>1
(
a
,
d
)
>
1
,
(
b
,
d
)
>
1
.Yao Jiangang
10 candidates for a job are interviewed
Ten people apply for a job. The manager decides to interview the candidates one by one according to the following conditions: i) the first three candidates will not be employed; ii) from the fourth candidates onwards, if a candidate's comptence surpasses the competence of all those who preceded him, then that candidate is employed; iii) if the first nine candidates are not employed, then the tenth candidate will be employed. We assume that none of the
10
10
10
applicants have the same competence, and these competences can be ranked from the first to tenth. Let
P
k
P_k
P
k
represent the probability that the
k
k
k
th-ranked applicant in competence is employed. Prove that: i)
P
1
>
P
2
>
…
>
P
8
=
P
9
=
P
10
P_1>P_2>\ldots>P_8=P_9=P_{10}
P
1
>
P
2
>
…
>
P
8
=
P
9
=
P
10
; ii)
P
1
+
P
2
+
P
3
>
0.7
P_1+P_2+P_3>0.7
P
1
+
P
2
+
P
3
>
0.7
iii)
P
8
+
P
9
+
P
10
≤
0.1
P_8+P_9+P_{10}\le 0.1
P
8
+
P
9
+
P
10
≤
0.1
.Su Chun
1
2
Hide problems
A, I and T are collinear iff [BKS]=[CKR]
Let
I
I
I
and
H
H
H
be the incentre and orthocentre of triangle
A
B
C
ABC
A
BC
respectively. Let
P
,
Q
P,Q
P
,
Q
be the midpoints of
A
B
,
A
C
AB,AC
A
B
,
A
C
. The rays
P
I
,
Q
I
PI,QI
P
I
,
Q
I
intersect
A
C
,
A
B
AC,AB
A
C
,
A
B
at
R
,
S
R,S
R
,
S
respectively. Suppose that
T
T
T
is the circumcentre of triangle
B
H
C
BHC
B
H
C
. Let
R
S
RS
RS
intersect
B
C
BC
BC
at
K
K
K
. Prove that
A
,
I
A,I
A
,
I
and
T
T
T
are collinear if and only if
[
B
K
S
]
=
[
C
K
R
]
[BKS]=[CKR]
[
B
K
S
]
=
[
C
K
R
]
.Shen Wunxuan
a^m+1 divides a^n + 203
Find all integer triples
(
a
,
m
,
n
)
(a,m,n)
(
a
,
m
,
n
)
such that
a
m
+
1
∣
a
n
+
203
a^m+1|a^n+203
a
m
+
1∣
a
n
+
203
where
a
,
m
>
1
a,m>1
a
,
m
>
1
.Chen Yonggao