MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2013 China National Olympiad
2013 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
3
2
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help: Find all positive real numbers
Find all positive real numbers
t
t
t
with the following property: there exists an infinite set
X
X
X
of real numbers such that the inequality
max
{
∣
x
−
(
a
−
d
)
∣
,
∣
y
−
a
∣
,
∣
z
−
(
a
+
d
)
∣
}
>
t
d
\max\{|x-(a-d)|,|y-a|,|z-(a+d)|\}>td
max
{
∣
x
−
(
a
−
d
)
∣
,
∣
y
−
a
∣
,
∣
z
−
(
a
+
d
)
∣
}
>
t
d
holds for all (not necessarily distinct)
x
,
y
,
z
∈
X
x,y,z\in X
x
,
y
,
z
∈
X
, all real numbers
a
a
a
and all positive real numbers
d
d
d
.
Find the minimum positive integer
Let
m
,
n
m,n
m
,
n
be positive integers. Find the minimum positive integer
N
N
N
which satisfies the following condition. If there exists a set
S
S
S
of integers that contains a complete residue system module
m
m
m
such that
∣
S
∣
=
N
| S | = N
∣
S
∣
=
N
, then there exists a nonempty set
A
⊆
S
A \subseteq S
A
⊆
S
so that
n
∣
∑
x
∈
A
x
n\mid {\sum\limits_{x \in A} x }
n
∣
x
∈
A
∑
x
.
1
2
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CA, AP and PE are the side lengths of a right triangle
Two circles
K
1
K_1
K
1
and
K
2
K_2
K
2
of different radii intersect at two points
A
A
A
and
B
B
B
, let
C
C
C
and
D
D
D
be two points on
K
1
K_1
K
1
and
K
2
K_2
K
2
, respectively, such that
A
A
A
is the midpoint of the segment
C
D
CD
C
D
. The extension of
D
B
DB
D
B
meets
K
1
K_1
K
1
at another point
E
E
E
, the extension of
C
B
CB
CB
meets
K
2
K_2
K
2
at another point
F
F
F
. Let
l
1
l_1
l
1
and
l
2
l_2
l
2
be the perpendicular bisectors of
C
D
CD
C
D
and
E
F
EF
EF
, respectively. i) Show that
l
1
l_1
l
1
and
l
2
l_2
l
2
have a unique common point (denoted by
P
P
P
). ii) Prove that the lengths of
C
A
CA
C
A
,
A
P
AP
A
P
and
P
E
PE
PE
are the side lengths of a right triangle.
Find the minimum
Let
n
⩾
2
n \geqslant 2
n
⩾
2
be an integer. There are
n
n
n
finite sets
A
1
,
A
2
,
…
,
A
n
{A_1},{A_2},\ldots,{A_n}
A
1
,
A
2
,
…
,
A
n
which satisfy the condition \left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \forall i,j \in \left\{ {1,2,...,n} \right\}. Find the minimum of
∑
i
=
1
n
∣
A
i
∣
\sum\limits_{i = 1}^n {\left| {{A_i}} \right|}
i
=
1
∑
n
∣
A
i
∣
.
2
2
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Divisible
For any positive integer
n
n
n
and
0
⩽
i
⩽
n
0 \leqslant i \leqslant n
0
⩽
i
⩽
n
, denote
C
n
i
≡
c
(
n
,
i
)
(
m
o
d
2
)
C_n^i \equiv c(n,i)\pmod{2}
C
n
i
≡
c
(
n
,
i
)
(
mod
2
)
, where
c
(
n
,
i
)
∈
{
0
,
1
}
c(n,i) \in \left\{ {0,1} \right\}
c
(
n
,
i
)
∈
{
0
,
1
}
. Define
f
(
n
,
q
)
=
∑
i
=
0
n
c
(
n
,
i
)
q
i
f(n,q) = \sum\limits_{i = 0}^n {c(n,i){q^i}}
f
(
n
,
q
)
=
i
=
0
∑
n
c
(
n
,
i
)
q
i
where
m
,
n
,
q
m,n,q
m
,
n
,
q
are positive integers and
q
+
1
≠
2
α
q + 1 \ne {2^\alpha }
q
+
1
=
2
α
for any
α
∈
N
\alpha \in \mathbb N
α
∈
N
. Prove that if
f
(
m
,
q
)
∣
f
(
n
,
q
)
f(m,q)\left| {f(n,q)} \right.
f
(
m
,
q
)
∣
f
(
n
,
q
)
, then
f
(
m
,
r
)
∣
f
(
n
,
r
)
f(m,r)\left| {f(n,r)} \right.
f
(
m
,
r
)
∣
f
(
n
,
r
)
for any positive integer
r
r
r
.
Determine set
Find all nonempty sets
S
S
S
of integers such that
3
m
−
2
n
∈
S
3m-2n \in S
3
m
−
2
n
∈
S
for all (not necessarily distinct)
m
,
n
∈
S
m,n \in S
m
,
n
∈
S
.