MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
1988 China Team Selection Test
1988 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(4)
4
2
Hide problems
elements can be pairwise distinguished
Let
k
∈
N
,
k \in \mathbb{N},
k
∈
N
,
S
k
=
{
(
a
,
b
)
∣
a
,
b
=
1
,
2
,
…
,
k
}
.
S_k = \{(a, b) | a, b = 1, 2, \ldots, k \}.
S
k
=
{(
a
,
b
)
∣
a
,
b
=
1
,
2
,
…
,
k
}
.
Any two elements
(
a
,
b
)
(a, b)
(
a
,
b
)
,
(
c
,
d
)
(c, d)
(
c
,
d
)
∈
S
k
\in S_k
∈
S
k
are called "undistinguishing" in
S
k
S_k
S
k
if
a
−
c
≡
0
a - c \equiv 0
a
−
c
≡
0
or
±
1
(
m
o
d
k
)
\pm 1 \pmod{k}
±
1
(
mod
k
)
and
b
−
d
≡
0
b - d \equiv 0
b
−
d
≡
0
or
±
1
(
m
o
d
k
)
\pm 1 \pmod{k}
±
1
(
mod
k
)
; otherwise, we call them "distinguishing". For example,
(
1
,
1
)
(1, 1)
(
1
,
1
)
and
(
2
,
5
)
(2, 5)
(
2
,
5
)
are undistinguishing in
S
5
S_5
S
5
. Considering the subset
A
A
A
of
S
k
S_k
S
k
such that the elements of
A
A
A
are pairwise distinguishing. Let
r
k
r_k
r
k
be the maximum possible number of elements of
A
A
A
. (i) Find
r
5
r_5
r
5
. (ii) Find
r
7
r_7
r
7
. (iii) Find
r
k
r_k
r
k
for
k
∈
N
k \in \mathbb{N}
k
∈
N
.
broken computer does not allow for all operations
There is a broken computer such that only three primitive data
c
c
c
,
1
1
1
and
−
1
-1
−
1
are reserved. Only allowed operation may take
u
u
u
and
v
v
v
and output
u
⋅
v
+
v
.
u \cdot v + v.
u
⋅
v
+
v
.
At the beginning,
u
,
v
∈
{
c
,
1
,
−
1
}
.
u,v \in \{c, 1, -1\}.
u
,
v
∈
{
c
,
1
,
−
1
}
.
After then, it can also take the value of the previous step (only one step back) besides
{
c
,
1
,
−
1
}
\{c, 1, -1\}
{
c
,
1
,
−
1
}
. Prove that for any polynomial
P
n
(
x
)
=
a
0
⋅
x
n
+
a
1
⋅
x
n
−
1
+
…
+
a
n
P_{n}(x) = a_0 \cdot x^n + a_1 \cdot x^{n-1} + \ldots + a_n
P
n
(
x
)
=
a
0
⋅
x
n
+
a
1
⋅
x
n
−
1
+
…
+
a
n
with integer coefficients, the value of
P
n
(
c
)
P_n(c)
P
n
(
c
)
can be computed using this computer after only finite operation.
3
2
Hide problems
OI is perpendicular to DE
In triangle
A
B
C
ABC
A
BC
,
∠
C
=
3
0
∘
\angle C = 30^{\circ}
∠
C
=
3
0
∘
,
O
O
O
and
I
I
I
are the circumcenter and incenter respectively, Points
D
∈
A
C
D \in AC
D
∈
A
C
and
E
∈
B
C
E \in BC
E
∈
BC
, such that
A
D
=
B
E
=
A
B
AD = BE = AB
A
D
=
BE
=
A
B
. Prove that
O
I
=
D
E
OI = DE
O
I
=
D
E
and
O
I
⊥
D
E
OI \bot DE
O
I
⊥
D
E
.
n+1 points in the plane with integer differences
A polygon
∏
\prod
∏
is given in the
O
X
Y
OXY
OX
Y
plane and its area exceeds
n
.
n.
n
.
Prove that there exist
n
+
1
n+1
n
+
1
points
P
1
(
x
1
,
y
1
)
,
P
2
(
x
2
,
y
2
)
,
…
,
P
n
+
1
(
x
n
+
1
,
y
n
+
1
)
P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})
P
1
(
x
1
,
y
1
)
,
P
2
(
x
2
,
y
2
)
,
…
,
P
n
+
1
(
x
n
+
1
,
y
n
+
1
)
in
∏
\prod
∏
such that
∀
i
,
j
∈
{
1
,
2
,
…
,
n
+
1
}
\forall i,j \in \{1, 2, \ldots, n+1\}
∀
i
,
j
∈
{
1
,
2
,
…
,
n
+
1
}
,
x
j
−
x
i
x_j - x_i
x
j
−
x
i
and
y
j
−
y
i
y_j - y_i
y
j
−
y
i
are all integers.
2
2
Hide problems
sum equals product in function
Find all functions
f
:
Q
↦
C
f: \mathbb{Q} \mapsto \mathbb{C}
f
:
Q
↦
C
satisfying (i) For any
x
1
,
x
2
,
…
,
x
1988
∈
Q
x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}
x
1
,
x
2
,
…
,
x
1988
∈
Q
,
f
(
x
1
+
x
2
+
…
+
x
1988
)
=
f
(
x
1
)
f
(
x
2
)
…
f
(
x
1988
)
f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})
f
(
x
1
+
x
2
+
…
+
x
1988
)
=
f
(
x
1
)
f
(
x
2
)
…
f
(
x
1988
)
. (ii)
f
(
1988
)
‾
f
(
x
)
=
f
(
1988
)
f
(
x
)
‾
\overline{f(1988)}f(x) = f(1988)\overline{f(x)}
f
(
1988
)
f
(
x
)
=
f
(
1988
)
f
(
x
)
for all
x
∈
Q
x \in \mathbb{Q}
x
∈
Q
.
quadrilateral PEFG is maximum
Let
A
B
C
D
ABCD
A
BC
D
be a trapezium
A
B
/
/
C
D
,
AB // CD,
A
B
//
C
D
,
M
M
M
and
N
N
N
are fixed points on
A
B
,
AB,
A
B
,
P
P
P
is a variable point on
C
D
CD
C
D
.
E
=
D
N
∩
A
P
E = DN \cap AP
E
=
D
N
∩
A
P
,
F
=
D
N
∩
M
C
F = DN \cap MC
F
=
D
N
∩
MC
,
G
=
M
C
∩
P
B
G = MC \cap PB
G
=
MC
∩
PB
,
D
P
=
λ
⋅
C
D
DP = \lambda \cdot CD
D
P
=
λ
⋅
C
D
. Find the value of
λ
\lambda
λ
for which the area of quadrilateral
P
E
F
G
PEFG
PEFG
is maximum.
1
2
Hide problems
A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) >= 0
Suppose real numbers
A
,
B
,
C
A,B,C
A
,
B
,
C
such that for all real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
the following inequality holds:
A
(
x
−
y
)
(
x
−
z
)
+
B
(
y
−
z
)
(
y
−
x
)
+
C
(
z
−
x
)
(
z
−
y
)
≥
0.
A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.
A
(
x
−
y
)
(
x
−
z
)
+
B
(
y
−
z
)
(
y
−
x
)
+
C
(
z
−
x
)
(
z
−
y
)
≥
0.
Find the necessary and sufficient condition
A
,
B
,
C
A,B,C
A
,
B
,
C
must satisfy (expressed by means of an equality or an inequality).
f(x) = 3x + 2 iterated 100 times
Let
f
(
x
)
=
3
x
+
2.
f(x) = 3x + 2.
f
(
x
)
=
3
x
+
2.
Prove that there exists
m
∈
N
m \in \mathbb{N}
m
∈
N
such that
f
100
(
m
)
f^{100}(m)
f
100
(
m
)
is divisible by
1988
1988
1988
.