MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
1990 China Team Selection Test
1990 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(4)
4
2
Hide problems
double difference square and summed up
Number
a
a
a
is such that
∀
a
1
,
a
2
,
a
3
,
a
4
∈
R
\forall a_1, a_2, a_3, a_4 \in \mathbb{R}
∀
a
1
,
a
2
,
a
3
,
a
4
∈
R
, there are integers
k
1
,
k
2
,
k
3
,
k
4
k_1, k_2, k_3, k_4
k
1
,
k
2
,
k
3
,
k
4
such that
∑
1
≤
i
<
j
≤
4
(
(
a
i
−
k
i
)
−
(
a
j
−
k
j
)
)
2
≤
a
\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a
∑
1
≤
i
<
j
≤
4
((
a
i
−
k
i
)
−
(
a
j
−
k
j
)
)
2
≤
a
. Find the minimum of
a
a
a
.
maximum number of circles that can be drawn
There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.
3
2
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Let's define an operation
In set
S
S
S
, there is an operation
′
′
∘
′
′
'' \circ ''
′′
∘
′′
such that
∀
a
,
b
∈
S
\forall a,b \in S
∀
a
,
b
∈
S
, a unique
a
∘
b
∈
S
a \circ b \in S
a
∘
b
∈
S
exists. And (i)
∀
a
,
b
,
c
∈
S
\forall a,b,c \in S
∀
a
,
b
,
c
∈
S
,
(
a
∘
b
)
∘
c
=
a
∘
(
b
∘
c
)
(a \circ b) \circ c = a \circ (b \circ c)
(
a
∘
b
)
∘
c
=
a
∘
(
b
∘
c
)
. (ii)
a
∘
b
≠
b
∘
a
a \circ b \neq b \circ a
a
∘
b
=
b
∘
a
when
a
≠
b
a \neq b
a
=
b
. Prove that: a.)
∀
a
,
b
,
c
∈
S
\forall a,b,c \in S
∀
a
,
b
,
c
∈
S
,
(
a
∘
b
)
∘
c
=
a
∘
c
(a \circ b) \circ c = a \circ c
(
a
∘
b
)
∘
c
=
a
∘
c
. b.) If
S
=
{
1
,
2
,
…
,
1990
}
S = \{1,2, \ldots, 1990\}
S
=
{
1
,
2
,
…
,
1990
}
, try to define an operation
′
′
∘
′
′
'' \circ ''
′′
∘
′′
in
S
S
S
with the above properties.
multiple of it with all digits not zero
Prove that for every integer power of 2, there exists a multiple of it with all digits (in decimal expression) not zero.
2
2
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polygon is cut by at least one of these m lines
Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin
O
O
O
that cuts them both, then these polygons are called "properly placed". Find the least
m
∈
N
m \in \mathbb{N}
m
∈
N
, such that for any group of properly placed polygons,
m
m
m
lines can drawn through
O
O
O
and every polygon is cut by at least one of these
m
m
m
lines.
Find three functions
Find all functions
f
,
g
,
h
:
R
↦
R
f,g,h: \mathbb{R} \mapsto \mathbb{R}
f
,
g
,
h
:
R
↦
R
such that
f
(
x
)
−
g
(
y
)
=
(
x
−
y
)
⋅
h
(
x
+
y
)
f(x) - g(y) = (x-y) \cdot h(x+y)
f
(
x
)
−
g
(
y
)
=
(
x
−
y
)
⋅
h
(
x
+
y
)
for
x
,
y
∈
R
.
x,y \in \mathbb{R}.
x
,
y
∈
R
.
1
2
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m people have exactly one common friend
In a wagon, every
m
≥
3
m \geq 3
m
≥
3
people have exactly one common friend. (When
A
A
A
is
B
B
B
's friend,
B
B
B
is also
A
A
A
's friend. No one was considered as his own friend.) Find the number of friends of the person who has the most friends.
Inequality for a triangle with C >= 60°
Given a triangle
A
B
C
ABC
A
BC
with angle
C
≥
6
0
∘
C \geq 60^{\circ}
C
≥
6
0
∘
. Prove that: \left(a \plus{} b\right) \cdot \left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \right) \geq 4 \plus{} \frac {1}{\sin\left(\frac {C}{2}\right)}.