MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2003 China Second Round Olympiad
2003 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(3)
3
1
Hide problems
Space quadrilateral
Let a space figure consist of
n
n
n
vertices and
l
l
l
lines connecting these vertices, with
n
=
q
2
+
q
+
1
n=q^2+q+1
n
=
q
2
+
q
+
1
,
l
≥
q
2
(
q
+
1
)
2
+
1
l\ge q^2(q+1)^2+1
l
≥
q
2
(
q
+
1
)
2
+
1
,
q
≥
2
q\ge2
q
≥
2
,
q
∈
N
q\in\mathbb{N}
q
∈
N
. Suppose the figure satisfies the following conditions: every four vertices are non-coplaner, every vertex is connected by at least one line, and there is a vertex connected by at least
p
+
2
p+2
p
+
2
lines. Prove that there exists a space quadrilateral in the figure, i.e. a quadrilateral with four vertices
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
and four lines
A
B
,
B
C
,
C
D
,
D
A
AB, BC, CD, DA
A
B
,
BC
,
C
D
,
D
A
in the figure.
2
1
Hide problems
Minimum Perimeter of a Triangle
Let the three sides of a triangle be
ℓ
,
m
,
n
\ell, m, n
ℓ
,
m
,
n
, respectively, satisfying
ℓ
>
m
>
n
\ell>m>n
ℓ
>
m
>
n
and
{
3
ℓ
1
0
4
}
=
{
3
m
1
0
4
}
=
{
3
n
1
0
4
}
\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}
{
1
0
4
3
ℓ
}
=
{
1
0
4
3
m
}
=
{
1
0
4
3
n
}
, where
{
x
}
=
x
−
⌊
x
⌋
\{x\}=x-\lfloor{x}\rfloor
{
x
}
=
x
−
⌊
x
⌋
and
⌊
x
⌋
\lfloor{x}\rfloor
⌊
x
⌋
denotes the integral part of the number
x
x
x
. Find the minimum perimeter of such a triangle.
1
1
Hide problems
Equal Angles
From point
P
P
P
outside a circle draw two tangents to the circle touching at
A
A
A
and
B
B
B
. Draw a secant line intersecting the circle at points
C
C
C
and
D
D
D
, with
C
C
C
between
P
P
P
and
D
D
D
. Choose point
Q
Q
Q
on the chord
C
D
CD
C
D
such that
∠
D
A
Q
=
∠
P
B
C
\angle DAQ=\angle PBC
∠
D
A
Q
=
∠
PBC
. Prove that
∠
D
B
Q
=
∠
P
A
C
\angle DBQ=\angle PAC
∠
D
BQ
=
∠
P
A
C
.