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Contests
National and Regional Contests
China Contests
China National Olympiad
2008 China National Olympiad
2008 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
3
2
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China Mathematics Olympiads (National Round) 2008 Problem 3
Given a positive integer
n
n
n
and
x
1
≤
x
2
≤
…
≤
x
n
,
y
1
≥
y
2
≥
…
≥
y
n
x_1 \leq x_2 \leq \ldots \leq x_n, y_1 \geq y_2 \geq \ldots \geq y_n
x
1
≤
x
2
≤
…
≤
x
n
,
y
1
≥
y
2
≥
…
≥
y
n
, satisfying
∑
i
=
1
n
i
x
i
=
∑
i
=
1
n
i
y
i
\displaystyle\sum_{i = 1}^{n} ix_i = \displaystyle\sum_{i = 1}^{n} iy_i
i
=
1
∑
n
i
x
i
=
i
=
1
∑
n
i
y
i
Show that for any real number
α
\alpha
α
, we have
∑
i
=
1
n
x
i
[
i
α
]
≥
∑
i
=
1
n
y
i
[
i
α
]
\displaystyle\sum_{i =1}^{n} x_i[i\alpha] \geq \displaystyle\sum_{i =1}^{n} y_i[i\alpha]
i
=
1
∑
n
x
i
[
i
α
]
≥
i
=
1
∑
n
y
i
[
i
α
]
Here
[
β
]
[\beta]
[
β
]
denotes the greastest integer not larger than
β
\beta
β
.
China Mathematics Olympiads (National Round) 2008 Problem 6
Find all triples
(
p
,
q
,
n
)
(p,q,n)
(
p
,
q
,
n
)
that satisfy q^{n+2} \equiv 3^{n+2} (\mod p^n) , p^{n+2} \equiv 3^{n+2} (\mod q^n) where
p
,
q
p,q
p
,
q
are odd primes and
n
n
n
is an positive integer.
1
2
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China Mathematics Olympiads (National Round) 2008 Problem 1
Suppose
△
A
B
C
\triangle ABC
△
A
BC
is scalene.
O
O
O
is the circumcenter and
A
′
A'
A
′
is a point on the extension of segment
A
O
AO
A
O
such that
∠
B
A
′
A
=
∠
C
A
′
A
\angle BA'A = \angle CA'A
∠
B
A
′
A
=
∠
C
A
′
A
. Let point
A
1
A_1
A
1
and
A
2
A_2
A
2
be foot of perpendicular from
A
′
A'
A
′
onto
A
B
AB
A
B
and
A
C
AC
A
C
.
H
A
H_{A}
H
A
is the foot of perpendicular from
A
A
A
onto
B
C
BC
BC
. Denote
R
A
R_{A}
R
A
to be the radius of circumcircle of
△
H
A
A
1
A
2
\triangle H_{A}A_1A_2
△
H
A
A
1
A
2
. Similiarly we can define
R
B
R_{B}
R
B
and
R
C
R_{C}
R
C
. Show that:
1
R
A
+
1
R
B
+
1
R
C
=
2
R
\frac{1}{R_{A}} + \frac{1}{R_{B}} + \frac{1}{R_{C}} = \frac{2}{R}
R
A
1
+
R
B
1
+
R
C
1
=
R
2
where R is the radius of circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
.
China Mathematics Olympiads (National Round) 2008 Problem 4
Let
A
A
A
be an infinite subset of
N
\mathbb{N}
N
, and
n
n
n
a fixed integer. For any prime
p
p
p
not dividing
n
n
n
, There are infinitely many elements of
A
A
A
not divisible by
p
p
p
. Show that for any integer
m
>
1
,
(
m
,
n
)
=
1
m >1, (m,n) =1
m
>
1
,
(
m
,
n
)
=
1
, There exist finitely many elements of
A
A
A
, such that their sum is congruent to 1 modulo
m
m
m
and congruent to 0 modulo
n
n
n
.
2
2
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partition into subsets not containing convex sequences
Given an integer
n
≥
3
n\ge3
n
≥
3
, prove that the set
X
=
{
1
,
2
,
3
,
…
,
n
2
−
n
}
X=\{1,2,3,\ldots,n^2-n\}
X
=
{
1
,
2
,
3
,
…
,
n
2
−
n
}
can be divided into two non-intersecting subsets such that neither of them contains
n
n
n
elements
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
with
a
1
<
a
2
<
…
<
a
n
a_1<a_2<\ldots<a_n
a
1
<
a
2
<
…
<
a
n
and
a
k
≤
a
k
−
1
+
a
k
+
1
2
a_k\le\frac{a_{k-1}+a_{k+1}}2
a
k
≤
2
a
k
−
1
+
a
k
+
1
for all
k
=
2
,
…
,
n
−
1
k=2,\ldots,n-1
k
=
2
,
…
,
n
−
1
.
China Mathematics Olympiads (National Round) 2008 Problem 5
Find the smallest integer
n
n
n
satisfying the following condition: regardless of how one colour the vertices of a regular
n
n
n
-gon with either red, yellow or blue, one can always find an isosceles trapezoid whose vertices are of the same colour.