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Contests
National and Regional Contests
China Contests
China National Olympiad
1994 China National Olympiad
1994 China National Olympiad
Part of
China National Olympiad
Subcontests
(6)
6
1
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China Mathematical Olympiad 1994 problem6
Let
M
M
M
be a point which has coordinates
(
p
×
1994
,
7
p
×
1994
)
(p\times 1994,7p\times 1994)
(
p
×
1994
,
7
p
×
1994
)
in the Cartesian plane (
p
p
p
is a prime). Find the number of right-triangles satisfying the following conditions: (1) all vertexes of the triangle are lattice points, moreover
M
M
M
is on the right-angled corner of the triangle; (2) the origin (
0
,
0
0,0
0
,
0
) is the incenter of the triangle.
5
1
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China Mathematical Olympiad 1994 problem5
For arbitrary natural number
n
n
n
, prove that
∑
k
=
0
n
C
n
k
2
k
C
n
−
k
[
(
n
−
k
)
/
2
]
=
C
2
n
+
1
n
\sum^n_{k=0}C^k_n2^kC^{[(n-k)/2]}_{n-k}=C^n_{2n+1}
∑
k
=
0
n
C
n
k
2
k
C
n
−
k
[(
n
−
k
)
/2
]
=
C
2
n
+
1
n
, where
C
0
0
=
1
C^0_0=1
C
0
0
=
1
and
[
n
−
k
2
]
[\dfrac{n-k}{2}]
[
2
n
−
k
]
denotes the integer part of
n
−
k
2
\dfrac{n-k}{2}
2
n
−
k
.
4
1
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China Mathematical Olympiad 1994 problem4
Let
f
(
z
)
=
c
0
z
n
+
c
1
z
n
−
1
+
c
2
z
n
−
2
+
⋯
+
c
n
−
1
z
+
c
n
f(z)=c_0z^n+c_1z^{n-1}+ c_2z^{n-2}+\cdots +c_{n-1}z+c_n
f
(
z
)
=
c
0
z
n
+
c
1
z
n
−
1
+
c
2
z
n
−
2
+
⋯
+
c
n
−
1
z
+
c
n
be a polynomial with complex coefficients. Prove that there exists a complex number
z
0
z_0
z
0
such that
∣
f
(
z
0
)
∣
≥
∣
c
0
∣
+
∣
c
n
∣
|f(z_0)|\ge |c_0|+|c_n|
∣
f
(
z
0
)
∣
≥
∣
c
0
∣
+
∣
c
n
∣
, where
∣
z
0
∣
≤
1
|z_0|\le 1
∣
z
0
∣
≤
1
.
3
1
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China Mathematical Olympiad 1994 problem3
Find all functions
f
:
[
1
,
∞
)
→
[
1
,
∞
)
f:[1,\infty )\rightarrow [1,\infty)
f
:
[
1
,
∞
)
→
[
1
,
∞
)
satisfying the following conditions: (1)
f
(
x
)
≤
2
(
x
+
1
)
f(x)\le 2(x+1)
f
(
x
)
≤
2
(
x
+
1
)
; (2)
f
(
x
+
1
)
=
1
x
[
(
f
(
x
)
)
2
−
1
]
f(x+1)=\dfrac{1}{x}[(f(x))^2-1]
f
(
x
+
1
)
=
x
1
[(
f
(
x
)
)
2
−
1
]
.
2
1
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China Mathematical Olympiad 1994 problem2
There are
m
m
m
pieces of candy held in
n
n
n
trays(
n
,
m
≥
4
n,m\ge 4
n
,
m
≥
4
). An operation is defined as follow: take out one piece of candy from any two trays respectively, then put them in a third tray. Determine, with proof, if we can move all candies to a single tray by finite operations.
1
1
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China Mathematical Olympiad 1994 problem1
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with
A
B
∥
C
D
AB\parallel CD
A
B
∥
C
D
. Points
E
,
F
E,F
E
,
F
lie on segments
A
B
,
C
D
AB,CD
A
B
,
C
D
respectively. Segments
C
E
,
B
F
CE,BF
CE
,
BF
meet at
H
H
H
, and segments
E
D
,
A
F
ED,AF
E
D
,
A
F
meet at
G
G
G
. Show that
S
E
H
F
G
≤
1
4
S
A
B
C
D
S_{EHFG}\le \dfrac{1}{4}S_{ABCD}
S
E
H
FG
≤
4
1
S
A
BC
D
. Determine, with proof, if the conclusion still holds when
A
B
C
D
ABCD
A
BC
D
is just any convex quadrilateral.