MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2014 China National Olympiad
2014 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
1
2
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Circumcentre is Incentre
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
>
A
C
AB>AC
A
B
>
A
C
. Let
D
D
D
be the foot of the internal angle bisector of
A
A
A
. Points
F
F
F
and
E
E
E
are on
A
C
,
A
B
AC,AB
A
C
,
A
B
respectively such that
B
,
C
,
F
,
E
B,C,F,E
B
,
C
,
F
,
E
are concyclic. Prove that the circumcentre of
D
E
F
DEF
D
EF
is the incentre of
A
B
C
ABC
A
BC
if and only if
B
E
+
C
F
=
B
C
BE+CF=BC
BE
+
CF
=
BC
.
No. of distinct prime factors and no. of prime factors
Let
n
=
p
1
a
1
p
2
a
2
⋯
p
t
a
t
n=p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}
n
=
p
1
a
1
p
2
a
2
⋯
p
t
a
t
be the prime factorisation of
n
n
n
. Define
ω
(
n
)
=
t
\omega(n)=t
ω
(
n
)
=
t
and
Ω
(
n
)
=
a
1
+
a
2
+
…
+
a
t
\Omega(n)=a_1+a_2+\ldots+a_t
Ω
(
n
)
=
a
1
+
a
2
+
…
+
a
t
. Prove or disprove: For any fixed positive integer
k
k
k
and positive reals
α
,
β
\alpha,\beta
α
,
β
, there exists a positive integer
n
>
1
n>1
n
>
1
such that i)
ω
(
n
+
k
)
ω
(
n
)
>
α
\frac{\omega(n+k)}{\omega(n)}>\alpha
ω
(
n
)
ω
(
n
+
k
)
>
α
ii)
Ω
(
n
+
k
)
Ω
(
n
)
<
β
\frac{\Omega(n+k)}{\Omega(n)}<\beta
Ω
(
n
)
Ω
(
n
+
k
)
<
β
.
2
2
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Difference of factors of integers
For the integer
n
>
1
n>1
n
>
1
, define
D
(
n
)
=
{
a
−
b
∣
a
b
=
n
,
a
>
b
>
0
,
a
,
b
∈
N
}
D(n)=\{ a-b\mid ab=n, a>b>0, a,b\in\mathbb{N} \}
D
(
n
)
=
{
a
−
b
∣
ab
=
n
,
a
>
b
>
0
,
a
,
b
∈
N
}
. Prove that for any integer
k
>
1
k>1
k
>
1
, there exists pairwise distinct positive integers
n
1
,
n
2
,
…
,
n
k
n_1,n_2,\ldots,n_k
n
1
,
n
2
,
…
,
n
k
such that
n
1
,
…
,
n
k
>
1
n_1,\ldots,n_k>1
n
1
,
…
,
n
k
>
1
and
∣
D
(
n
1
)
∩
D
(
n
2
)
∩
⋯
∩
D
(
n
k
)
∣
≥
2
|D(n_1)\cap D(n_2)\cap\cdots\cap D(n_k)|\geq 2
∣
D
(
n
1
)
∩
D
(
n
2
)
∩
⋯
∩
D
(
n
k
)
∣
≥
2
.
relation between x and f(x)
Let
f
:
X
→
X
f:X\rightarrow X
f
:
X
→
X
, where
X
=
{
1
,
2
,
…
,
100
}
X=\{1,2,\ldots ,100\}
X
=
{
1
,
2
,
…
,
100
}
, be a function satisfying: 1)
f
(
x
)
≠
x
f(x)\neq x
f
(
x
)
=
x
for all
x
=
1
,
2
,
…
,
100
x=1,2,\ldots,100
x
=
1
,
2
,
…
,
100
; 2) for any subset
A
A
A
of
X
X
X
such that
∣
A
∣
=
40
|A|=40
∣
A
∣
=
40
, we have
A
∩
f
(
A
)
≠
∅
A\cap f(A)\neq\emptyset
A
∩
f
(
A
)
=
∅
. Find the minimum
k
k
k
such that for any such function
f
f
f
, there exist a subset
B
B
B
of
X
X
X
, where
∣
B
∣
=
k
|B|=k
∣
B
∣
=
k
, such that
B
∪
f
(
B
)
=
X
B\cup f(B)=X
B
∪
f
(
B
)
=
X
.
3
2
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China Mathematical Olympiad P3
Prove that: there exists only one function
f
:
N
∗
→
N
∗
f:\mathbb{N^*}\to\mathbb{N^*}
f
:
N
∗
→
N
∗
satisfying: i)
f
(
1
)
=
f
(
2
)
=
1
f(1)=f(2)=1
f
(
1
)
=
f
(
2
)
=
1
; ii)
f
(
n
)
=
f
(
f
(
n
−
1
)
)
+
f
(
n
−
f
(
n
−
1
)
)
f(n)=f(f(n-1))+f(n-f(n-1))
f
(
n
)
=
f
(
f
(
n
−
1
))
+
f
(
n
−
f
(
n
−
1
))
for
n
≥
3
n\ge 3
n
≥
3
. For each integer
m
≥
2
m\ge 2
m
≥
2
, find the value of
f
(
2
m
)
f(2^m)
f
(
2
m
)
.
Addition of subsets
For non-empty number sets
S
,
T
S, T
S
,
T
, define the sets
S
+
T
=
{
s
+
t
∣
s
∈
S
,
t
∈
T
}
S+T=\{s+t\mid s\in S, t\in T\}
S
+
T
=
{
s
+
t
∣
s
∈
S
,
t
∈
T
}
and
2
S
=
{
2
s
∣
s
∈
S
}
2S=\{2s\mid s\in S\}
2
S
=
{
2
s
∣
s
∈
S
}
. Let
n
n
n
be a positive integer, and
A
,
B
A, B
A
,
B
be two non-empty subsets of
{
1
,
2
…
,
n
}
\{1,2\ldots,n\}
{
1
,
2
…
,
n
}
. Show that there exists a subset
D
D
D
of
A
+
B
A+B
A
+
B
such that 1)
D
+
D
⊆
2
(
A
+
B
)
D+D\subseteq 2(A+B)
D
+
D
⊆
2
(
A
+
B
)
, 2)
∣
D
∣
≥
∣
A
∣
⋅
∣
B
∣
2
n
|D|\geq\frac{|A|\cdot|B|}{2n}
∣
D
∣
≥
2
n
∣
A
∣
⋅
∣
B
∣
, where
∣
X
∣
|X|
∣
X
∣
is the number of elements of the finite set
X
X
X
.