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Contests
National and Regional Contests
China Contests
China National Olympiad
1988 China National Olympiad
1988 China National Olympiad
Part of
China National Olympiad
Subcontests
(6)
5
1
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China Mathematical Olympiad 1988 problem5
Given three tetrahedrons
A
i
B
i
C
i
D
i
A_iB_i C_i D_i
A
i
B
i
C
i
D
i
(
i
=
1
,
2
,
3
i=1,2,3
i
=
1
,
2
,
3
), planes
α
i
,
β
i
,
γ
i
\alpha _i,\beta _i,\gamma _i
α
i
,
β
i
,
γ
i
(
i
=
1
,
2
,
3
i=1,2,3
i
=
1
,
2
,
3
) are drawn through
B
i
,
C
i
,
D
i
B_i ,C_i ,D_i
B
i
,
C
i
,
D
i
respectively, and they are perpendicular to edges
A
i
B
i
,
A
i
C
i
,
A
i
D
i
A_i B_i, A_i C_i, A_i D_i
A
i
B
i
,
A
i
C
i
,
A
i
D
i
(
i
=
1
,
2
,
3
i=1,2,3
i
=
1
,
2
,
3
) respectively. Suppose that all nine planes
α
i
,
β
i
,
γ
i
\alpha _i,\beta _i,\gamma _i
α
i
,
β
i
,
γ
i
(
i
=
1
,
2
,
3
i=1,2,3
i
=
1
,
2
,
3
) meet at a point
E
E
E
, and points
A
1
,
A
2
,
A
3
A_1,A_2,A_3
A
1
,
A
2
,
A
3
lie on line
l
l
l
. Determine the intersection (shape and position) of the circumscribed spheres of the three tetrahedrons.
6
1
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China Mathematical Olympiad 1988 problem6
Let
n
n
n
(
n
≥
3
n\ge 3
n
≥
3
) be a natural number. Denote by
f
(
n
)
f(n)
f
(
n
)
the least natural number by which
n
n
n
is not divisible (e.g.
f
(
12
)
=
5
f(12)=5
f
(
12
)
=
5
). If
f
(
n
)
≥
3
f(n)\ge 3
f
(
n
)
≥
3
, we may have
f
(
f
(
n
)
)
f(f(n))
f
(
f
(
n
))
in the same way. Similarly, if
f
(
f
(
n
)
)
≥
3
f(f(n))\ge 3
f
(
f
(
n
))
≥
3
, we may have
f
(
f
(
f
(
n
)
)
)
f(f(f(n)))
f
(
f
(
f
(
n
)))
, and so on. If
f
(
f
(
…
f
⏟
k
times
(
n
)
…
)
)
=
2
\underbrace{f(f(\dots f}_{k\text{ times}}(n)\dots ))=2
k
times
f
(
f
(
…
f
(
n
)
…
))
=
2
, we call
k
k
k
the “length” of
n
n
n
(also we denote by
l
n
l_n
l
n
the “length” of
n
n
n
). For arbitrary natural number
n
n
n
(
n
≥
3
n\ge 3
n
≥
3
), find
l
n
l_n
l
n
with proof.
4
1
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China Mathematical Olympiad 1988 problem4
(1) Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers satisfying
(
a
2
+
b
2
+
c
2
)
2
>
2
(
a
4
+
b
4
+
c
4
)
(a^2+b^2+c^2)^2>2(a^4+b^4+c^4)
(
a
2
+
b
2
+
c
2
)
2
>
2
(
a
4
+
b
4
+
c
4
)
. Prove that
a
,
b
,
c
a,b,c
a
,
b
,
c
can be the lengths of three sides of a triangle respectively. (2) Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots ,a_n
a
1
,
a
2
,
…
,
a
n
be
n
n
n
(
n
>
3
n>3
n
>
3
) positive real numbers satisfying
(
a
1
2
+
a
2
2
+
⋯
+
a
n
2
)
2
>
(
n
−
1
)
(
a
1
4
+
a
2
4
+
⋯
+
a
n
4
)
(a_1^2+a_2^2+\dots +a_n^2)^2>(n-1)(a_1^4+ a_2^4+\dots +a_n^4)
(
a
1
2
+
a
2
2
+
⋯
+
a
n
2
)
2
>
(
n
−
1
)
(
a
1
4
+
a
2
4
+
⋯
+
a
n
4
)
. Prove that any three of
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots ,a_n
a
1
,
a
2
,
…
,
a
n
can be the lengths of three sides of a triangle respectively.
3
1
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China Mathematical Olympiad 1988 problem3
Given a finite sequence of real numbers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots ,a_n
a
1
,
a
2
,
…
,
a
n
(
∗
\ast
∗
), we call a segment
a
k
,
…
,
a
k
+
l
−
1
a_k,\dots ,a_{k+l-1}
a
k
,
…
,
a
k
+
l
−
1
of the sequence (
∗
\ast
∗
) a “long”(Chinese dragon) and
a
k
a_k
a
k
“head” of the “long” if the arithmetic mean of
a
k
,
…
,
a
k
+
l
−
1
a_k,\dots ,a_{k+l-1}
a
k
,
…
,
a
k
+
l
−
1
is greater than
1988
1988
1988
. (especially if a single item
a
m
>
1988
a_m>1988
a
m
>
1988
, we still regard
a
m
a_m
a
m
as a “long”). Suppose that there is at least one “long” among the sequence (
∗
\ast
∗
), show that the arithmetic mean of all those items of sequence (
∗
\ast
∗
) that could be “head” of a certain “long” individually is greater than
1988
1988
1988
.
2
1
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China Mathematical Olympiad 1988 problem2
Given two circles
C
1
,
C
2
C_1,C_2
C
1
,
C
2
with common center, the radius of
C
2
C_2
C
2
is twice the radius of
C
1
C_1
C
1
. Quadrilateral
A
1
A
2
A
3
A
4
A_1A_2A_3A_4
A
1
A
2
A
3
A
4
is inscribed in
C
1
C_1
C
1
. The extension of
A
4
A
1
A_4A_1
A
4
A
1
meets
C
2
C_2
C
2
at
B
1
B_1
B
1
; the extension of
A
1
A
2
A_1A_2
A
1
A
2
meets
C
2
C_2
C
2
at
B
2
B_2
B
2
; the extension of
A
2
A
3
A_2A_3
A
2
A
3
meets
C
2
C_2
C
2
at
B
3
B_3
B
3
; the extension of
A
3
A
4
A_3A_4
A
3
A
4
meets
C
2
C_2
C
2
at
B
4
B_4
B
4
. Prove that
P
(
B
1
B
2
B
3
B
4
)
≥
2
P
(
A
1
A
2
A
3
A
4
)
P(B_1B_2B_3B_4)\ge 2P(A_1A_2A_3A_4)
P
(
B
1
B
2
B
3
B
4
)
≥
2
P
(
A
1
A
2
A
3
A
4
)
, and in what case the equality holds? (
P
(
X
)
P(X)
P
(
X
)
denotes the perimeter of quadrilateral
X
X
X
)
1
1
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China Mathematical Olympiad 1988 problem1
Let
r
1
,
r
2
,
…
,
r
n
r_1,r_2,\dots ,r_n
r
1
,
r
2
,
…
,
r
n
be real numbers. Given
n
n
n
reals
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots ,a_n
a
1
,
a
2
,
…
,
a
n
that are not all equal to
0
0
0
, suppose that inequality
r
1
(
x
1
−
a
1
)
+
r
2
(
x
2
−
a
2
)
+
⋯
+
r
n
(
x
n
−
a
n
)
≤
x
1
2
+
x
2
2
+
⋯
+
x
n
2
−
a
1
2
+
a
2
2
+
⋯
+
a
n
2
r_1(x_1-a_1)+ r_2(x_2-a_2)+\dots + r_n(x_n-a_n)\leq\sqrt{x_1^2+ x_2^2+\dots + x_n^2}-\sqrt{a_1^2+a_2^2+\dots +a_n^2}
r
1
(
x
1
−
a
1
)
+
r
2
(
x
2
−
a
2
)
+
⋯
+
r
n
(
x
n
−
a
n
)
≤
x
1
2
+
x
2
2
+
⋯
+
x
n
2
−
a
1
2
+
a
2
2
+
⋯
+
a
n
2
holds for arbitrary reals
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\dots ,x_n
x
1
,
x
2
,
…
,
x
n
. Find the values of
r
1
,
r
2
,
…
,
r
n
r_1,r_2,\dots ,r_n
r
1
,
r
2
,
…
,
r
n
.