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Contests
National and Regional Contests
China Contests
China National Olympiad
1991 China National Olympiad
1991 China National Olympiad
Part of
China National Olympiad
Subcontests
(6)
6
1
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China Mathematical Olympiad 1991 problem6
A football is covered by some polygonal pieces of leather which are sewed up by three different colors threads. It features as follows: i) any edge of a polygonal piece of leather is sewed up with an equal-length edge of another polygonal piece of leather by a certain color thread; ii) each node on the ball is vertex to exactly three polygons, and the three threads joint at the node are of different colors. Show that we can assign to each node on the ball a complex number (not equal to
1
1
1
), such that the product of the numbers assigned to the vertices of any polygonal face is equal to
1
1
1
.
5
1
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China Mathematical Olympiad 1991 problem5
Find all natural numbers
n
n
n
, such that
min
k
∈
N
(
k
2
+
[
n
/
k
2
]
)
=
1991
\min_{k\in \mathbb{N}}(k^2+[n/k^2])=1991
min
k
∈
N
(
k
2
+
[
n
/
k
2
])
=
1991
. (
[
n
/
k
2
]
[n/k^2]
[
n
/
k
2
]
denotes the integer part of
n
/
k
2
n/k^2
n
/
k
2
.)
4
1
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China Mathematical Olympiad 1991 problem4
Find all positive integer solutions
(
x
,
y
,
z
,
n
)
(x,y,z,n)
(
x
,
y
,
z
,
n
)
of equation
x
2
n
+
1
−
y
2
n
+
1
=
x
y
z
+
2
2
n
+
1
x^{2n+1}-y^{2n+1}=xyz+2^{2n+1}
x
2
n
+
1
−
y
2
n
+
1
=
x
yz
+
2
2
n
+
1
, where
n
≥
2
n\ge 2
n
≥
2
and
z
≤
5
×
2
2
n
z \le 5\times 2^{2n}
z
≤
5
×
2
2
n
.
3
1
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China Mathematical Olympiad 1991 problem3
There are
10
10
10
birds on the ground. For any
5
5
5
of them, there are at least
4
4
4
birds on a circle. Determine the least possible number of birds on the circle with the most birds.
2
1
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China Mathematical Olympiad 1991 problem2
Given
I
=
[
0
,
1
]
I=[0,1]
I
=
[
0
,
1
]
and
G
=
{
(
x
,
y
)
∣
x
,
y
∈
I
}
G=\{(x,y)|x,y \in I\}
G
=
{(
x
,
y
)
∣
x
,
y
∈
I
}
, find all functions
f
:
G
→
I
f:G\rightarrow I
f
:
G
→
I
, such that
∀
x
,
y
,
z
∈
I
\forall x,y,z \in I
∀
x
,
y
,
z
∈
I
we have: i.
f
(
f
(
x
,
y
)
,
z
)
=
f
(
x
,
f
(
y
,
z
)
)
f(f(x,y),z)=f(x,f(y,z))
f
(
f
(
x
,
y
)
,
z
)
=
f
(
x
,
f
(
y
,
z
))
; ii.
f
(
x
,
1
)
=
x
,
f
(
1
,
y
)
=
y
f(x,1)=x, f(1,y)=y
f
(
x
,
1
)
=
x
,
f
(
1
,
y
)
=
y
; iii.
f
(
z
x
,
z
y
)
=
z
k
f
(
x
,
y
)
f(zx,zy)=z^kf(x,y)
f
(
z
x
,
zy
)
=
z
k
f
(
x
,
y
)
. (
k
k
k
is a positive real number irrelevant to
x
,
y
,
z
x,y,z
x
,
y
,
z
.)
1
1
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China Mathematical Olympiad 1991 problem1
We are given a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
in the plane. (i) If there exists a point
P
P
P
in the plane such that the areas of
△
A
B
P
,
△
B
C
P
,
△
C
D
P
,
△
D
A
P
\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP
△
A
BP
,
△
BCP
,
△
C
D
P
,
△
D
A
P
are equal, what condition must be satisfied by the quadrilateral
A
B
C
D
ABCD
A
BC
D
? (ii) Find (with proof) the maximum possible number of such point
P
P
P
which satisfies the condition in (i).