MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
1993 China National Olympiad
1993 China National Olympiad
Part of
China National Olympiad
Subcontests
(6)
6
1
Hide problems
China Mathematical Olympiad 1993 problem6
Let
f
:
(
0
,
+
∞
)
→
(
0
,
+
∞
)
f: (0,+\infty)\rightarrow (0,+\infty)
f
:
(
0
,
+
∞
)
→
(
0
,
+
∞
)
be a function satisfying the following condition: for arbitrary positive real numbers
x
x
x
and
y
y
y
, we have
f
(
x
y
)
≤
f
(
x
)
f
(
y
)
f(xy)\le f(x)f(y)
f
(
x
y
)
≤
f
(
x
)
f
(
y
)
. Show that for arbitrary positive real number
x
x
x
and natural number
n
n
n
, inequality
f
(
x
n
)
≤
f
(
x
)
f
(
x
2
)
1
2
…
f
(
x
n
)
1
n
f(x^n)\le f(x)f(x^2)^{\dfrac{1}{2}}\dots f(x^n)^{\dfrac{1}{n}}
f
(
x
n
)
≤
f
(
x
)
f
(
x
2
)
2
1
…
f
(
x
n
)
n
1
holds.
5
1
Hide problems
China Mathematical Olympiad 1993 problem5
10
10
10
students bought some books in a bookstore. It is known that every student bought exactly three kinds of books, and any two of them shared at least one kind of book. Determine, with proof, how many students bought the most popular book at least? (Note: the most popular book means most students bought this kind of book)
4
1
Hide problems
China Mathematical Olympiad 1993 problem4
We are given a set
S
=
{
z
1
,
z
2
,
⋯
,
z
1993
}
S=\{z_1,z_2,\cdots ,z_{1993}\}
S
=
{
z
1
,
z
2
,
⋯
,
z
1993
}
, where
z
1
,
z
2
,
⋯
,
z
1993
z_1,z_2,\cdots ,z_{1993}
z
1
,
z
2
,
⋯
,
z
1993
are nonzero complex numbers (also viewed as nonzero vectors in the plane). Prove that we can divide
S
S
S
into some groups such that the following conditions are satisfied: (1) Each element in
S
S
S
belongs and only belongs to one group; (2) For any group
p
p
p
, if we use
T
(
p
)
T(p)
T
(
p
)
to denote the sum of all memebers in
p
p
p
, then for any memeber
z
i
(
1
≤
i
≤
1993
)
z_i (1\le i \le 1993)
z
i
(
1
≤
i
≤
1993
)
of
p
p
p
, the angle between
z
i
z_i
z
i
and
T
(
p
)
T(p)
T
(
p
)
does not exceed
9
0
∘
90^{\circ}
9
0
∘
; (3) For any two groups
p
p
p
and
q
q
q
, the angle between
T
(
p
)
T(p)
T
(
p
)
and
T
(
q
)
T(q)
T
(
q
)
exceeds
9
0
∘
90^{\circ}
9
0
∘
(use the notation introduced in (2)).
3
1
Hide problems
China Mathematical Olympiad 1993 problem3
Let
K
,
K
1
K, K_1
K
,
K
1
be two circles with the same center and their radii equal to
R
R
R
and
R
1
(
R
1
>
R
)
R_1 (R_1>R)
R
1
(
R
1
>
R
)
respectively. Quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in circle
K
K
K
. Quadrilateral
A
1
B
1
C
1
D
1
A_1B_1C_1D_1
A
1
B
1
C
1
D
1
is inscribed in circle
K
1
K_1
K
1
where
A
1
,
B
1
,
C
1
,
D
1
A_1,B_1,C_1,D_1
A
1
,
B
1
,
C
1
,
D
1
lie on rays
C
D
,
D
A
,
A
B
,
B
C
CD,DA,AB,BC
C
D
,
D
A
,
A
B
,
BC
respectively. Show that
S
A
1
B
1
C
1
D
1
S
A
B
C
D
≥
R
1
2
R
2
\dfrac{S_{A_1B_1C_1D_1}}{S_{ABCD}}\ge \dfrac{R^2_1}{R^2}
S
A
BC
D
S
A
1
B
1
C
1
D
1
≥
R
2
R
1
2
.
2
1
Hide problems
China Mathematical Olympiad 1993 problem2
Given a natural number
k
k
k
and a real number
a
(
a
>
0
)
a (a>0)
a
(
a
>
0
)
, find the maximal value of
a
k
1
+
a
k
2
+
⋯
+
a
k
r
a^{k_1}+a^{k_2}+\cdots +a^{k_r}
a
k
1
+
a
k
2
+
⋯
+
a
k
r
, where
k
1
+
k
2
+
⋯
+
k
r
=
k
k_1+k_2+\cdots +k_r=k
k
1
+
k
2
+
⋯
+
k
r
=
k
(
k
i
∈
N
,
1
≤
r
≤
k
k_i\in \mathbb{N} ,1\le r \le k
k
i
∈
N
,
1
≤
r
≤
k
).
1
1
Hide problems
China Mathematical Olympiad 1993 problem1
Given an odd
n
n
n
, prove that there exist
2
n
2n
2
n
integers
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots ,a_n
a
1
,
a
2
,
⋯
,
a
n
;
b
1
,
b
2
,
⋯
,
b
n
b_1,b_2,\cdots ,b_n
b
1
,
b
2
,
⋯
,
b
n
, such that for any integer
k
k
k
(
0
<
k
<
n
0<k<n
0
<
k
<
n
), the following
3
n
3n
3
n
integers:
a
i
+
a
i
+
1
,
a
i
+
b
i
,
b
i
+
b
i
+
k
a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}
a
i
+
a
i
+
1
,
a
i
+
b
i
,
b
i
+
b
i
+
k
(
i
=
1
,
2
,
⋯
,
n
;
a
n
+
1
=
a
1
,
b
n
+
j
=
b
j
,
0
<
j
<
n
i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n
i
=
1
,
2
,
⋯
,
n
;
a
n
+
1
=
a
1
,
b
n
+
j
=
b
j
,
0
<
j
<
n
) are of different remainders on division by
3
n
3n
3
n
.