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Contests
National and Regional Contests
China Contests
China National Olympiad
1986 China National Olympiad
1986 China National Olympiad
Part of
China National Olympiad
Subcontests
(6)
6
1
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China Mathematical Olympiad 1986 problem6
Suppose that each point on the plane is colored either white or black. Show that there exists an equilateral triangle with the side length equal to
1
1
1
or
3
\sqrt{3}
3
whose three vertices are in the same color.
5
1
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China Mathematical Olympiad 1986 problem5
Given a sequence
1
,
1
,
2
,
2
,
3
,
3
,
…
,
1986
,
1986
1,1,2,2,3,3,\ldots,1986,1986
1
,
1
,
2
,
2
,
3
,
3
,
…
,
1986
,
1986
, determine, with proof, if we can rearrange the sequence so that for any integer
1
≤
k
≤
1986
1\le k \le 1986
1
≤
k
≤
1986
there are exactly
k
k
k
numbers between the two “
k
k
k
”s.
4
1
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China Mathematical Olympiad 1986 problem4
Given a
△
A
B
C
\triangle ABC
△
A
BC
with its area equal to
1
1
1
, suppose that the vertices of quadrilateral
P
1
P
2
P
3
P
4
P_1P_2P_3P_4
P
1
P
2
P
3
P
4
all lie on the sides of
△
A
B
C
\triangle ABC
△
A
BC
. Show that among the four triangles
△
P
1
P
2
P
3
,
△
P
1
P
2
P
4
,
△
P
1
P
3
P
4
,
△
P
2
P
3
P
4
\triangle P_1P_2P_3, \triangle P_1P_2P_4, \triangle P_1P_3P_4, \triangle P_2P_3P_4
△
P
1
P
2
P
3
,
△
P
1
P
2
P
4
,
△
P
1
P
3
P
4
,
△
P
2
P
3
P
4
there is at least one whose area is not larger than
1
/
4
1/4
1/4
.
3
1
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China Mathematical Olympiad 1986 problem3
Let
Z
1
,
Z
2
,
⋯
,
Z
n
Z_1,Z_2,\cdots ,Z_n
Z
1
,
Z
2
,
⋯
,
Z
n
be complex numbers satisfying
∣
Z
1
∣
+
∣
Z
2
∣
+
⋯
+
∣
Z
n
∣
=
1
|Z_1|+|Z_2|+\cdots +|Z_n|=1
∣
Z
1
∣
+
∣
Z
2
∣
+
⋯
+
∣
Z
n
∣
=
1
. Show that there exist some among the
n
n
n
complex numbers such that the modulus of the sum of these complex numbers is not less than
1
/
6
1/6
1/6
.
2
1
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China Mathematical Olympiad 1986 problem2
In
△
A
B
C
\triangle ABC
△
A
BC
, the length of altitude
A
D
AD
A
D
is
12
12
12
, and the bisector
A
E
AE
A
E
of
∠
A
\angle A
∠
A
is
13
13
13
. Denote by
m
m
m
the length of median
A
F
AF
A
F
. Find the range of
m
m
m
when
∠
A
\angle A
∠
A
is acute, orthogonal and obtuse respectively.
1
1
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China Mathematical Olympiad 1986 problem1
We are given
n
n
n
reals
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots , a_n
a
1
,
a
2
,
⋯
,
a
n
such that the sum of any two of them is non-negative. Prove that the following statement and its converse are both true: if
n
n
n
non-negative reals
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots ,x_n
x
1
,
x
2
,
⋯
,
x
n
satisfy
x
1
+
x
2
+
⋯
+
x
n
=
1
x_1+x_2+\cdots +x_n=1
x
1
+
x
2
+
⋯
+
x
n
=
1
, then the inequality
a
1
x
1
+
a
2
x
2
+
⋯
+
a
n
x
n
≥
a
1
x
1
2
+
a
2
x
2
2
+
⋯
+
a
n
x
n
2
a_1x_1+a_2x_2+\cdots +a_nx_n\ge a_1x^2_1+ a_2x^2_2+\cdots + a_nx^2_n
a
1
x
1
+
a
2
x
2
+
⋯
+
a
n
x
n
≥
a
1
x
1
2
+
a
2
x
2
2
+
⋯
+
a
n
x
n
2
holds.