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National and Regional Contests
China Contests
China National Olympiad
1992 China National Olympiad
1992 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
3
2
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China Mathematical Olympiad 1992 problem3
Given a
9
×
9
9\times 9
9
×
9
grid, we assign either
+
1
+1
+
1
or
−
1
-1
−
1
to each square on the grid. We define an adjustment as follow: for each square on the grid, we make a product of all numbers of those squares which share a common side with the square (excluding itself).Then we have
81
81
81
products. Next we replace all the squares’ values with their corresponding products. Determine if we can make all values in the grid equal to
1
1
1
through finite adjustments.
China Mathematical Olympiad 1992 problem6
Let sequence
{
a
1
,
a
2
,
…
}
\{a_1,a_2,\dots \}
{
a
1
,
a
2
,
…
}
with integer terms satisfy the following conditions: 1)
a
n
+
1
=
3
a
n
−
3
a
n
−
1
+
a
n
−
2
,
n
=
2
,
3
,
…
a_{n+1}=3a_n-3a_{n-1}+a_{n-2}, n=2,3,\dots
a
n
+
1
=
3
a
n
−
3
a
n
−
1
+
a
n
−
2
,
n
=
2
,
3
,
…
; 2)
2
a
1
=
a
0
+
a
2
−
2
2a_1=a_0+a_2-2
2
a
1
=
a
0
+
a
2
−
2
; 3) for arbitrary natural number
m
m
m
, there exist
m
m
m
consecutive terms
a
k
,
a
k
−
1
,
…
,
a
k
+
m
−
1
a_k, a_{k-1}, \dots ,a_{k+m-1}
a
k
,
a
k
−
1
,
…
,
a
k
+
m
−
1
among the sequence such that all such
m
m
m
terms are perfect squares. Prove that all terms of the sequence
{
a
1
,
a
2
,
…
}
\{a_1,a_2,\dots \}
{
a
1
,
a
2
,
…
}
are perfect squares.
2
1
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China Mathematical Olympiad 1992 problem2
Given nonnegative real numbers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\dots ,x_n
x
1
,
x
2
,
…
,
x
n
, let
a
=
m
i
n
{
x
1
,
x
2
,
…
,
x
n
}
a=min\{x_1, x_2,\dots ,x_n\}
a
=
min
{
x
1
,
x
2
,
…
,
x
n
}
. Prove that the following inequality holds: \sum^{n}_{i=1}\dfrac{1+x_i}{1+x_{i+1}}\le n+\dfrac{1}{(1+a)^2}\sum^{n}_{i=1}(x_i-a)^2 (x_{n+1}=x_1), and equality occurs if and only if
x
1
=
x
2
=
⋯
=
x
n
x_1=x_2=\dots =x_n
x
1
=
x
2
=
⋯
=
x
n
.
1
2
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China Mathematical Olympiad 1992 problem1
Let equation
x
n
+
a
n
−
1
x
n
−
1
+
a
n
−
2
x
n
−
2
+
⋯
+
a
1
x
+
a
0
=
0
x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots +a_1x+a_0=0
x
n
+
a
n
−
1
x
n
−
1
+
a
n
−
2
x
n
−
2
+
⋯
+
a
1
x
+
a
0
=
0
with real coefficients satisfy
0
<
a
0
≤
a
1
≤
a
2
≤
⋯
≤
a
n
−
1
≤
1
0<a_0\le a_1\le a_2\le \dots \le a_{n-1}\le 1
0
<
a
0
≤
a
1
≤
a
2
≤
⋯
≤
a
n
−
1
≤
1
. Suppose that
λ
\lambda
λ
(
∣
λ
∣
>
1
|\lambda|>1
∣
λ
∣
>
1
) is a complex root of the equation, prove that
λ
n
+
1
=
1
\lambda^{n+1}=1
λ
n
+
1
=
1
.
China Mathematical Olympiad 1992 problem4
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle with center
O
O
O
. The diagonals
A
C
AC
A
C
,
B
D
BD
B
D
of
A
B
C
D
ABCD
A
BC
D
meet at
P
P
P
. Circumcircles of
△
A
B
P
\triangle ABP
△
A
BP
and
△
C
D
P
\triangle CDP
△
C
D
P
meet at
P
P
P
and
Q
Q
Q
(
O
,
P
,
Q
O,P,Q
O
,
P
,
Q
are pairwise distinct). Show that
∠
O
Q
P
=
9
0
∘
\angle OQP=90^{\circ}
∠
OQP
=
9
0
∘
.