MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2015 China Second Round Olympiad
2015 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(4)
4
2
Hide problems
Find k not satisfying divisibility condition
Find all positive integers
k
k
k
such that for any positive integer
n
n
n
,
2
(
k
−
1
)
n
+
1
2^{(k-1)n+1}
2
(
k
−
1
)
n
+
1
does not divide
(
k
n
)
!
n
!
\frac{(kn)!}{n!}
n
!
(
kn
)!
.
Number of good sequences
Given positive integers
m
,
n
(
2
≤
m
≤
n
)
m,n(2\le m\le n)
m
,
n
(
2
≤
m
≤
n
)
, let
a
1
,
a
2
,
…
,
a
m
a_1,a_2,\ldots ,a_m
a
1
,
a
2
,
…
,
a
m
be a permutation of any
m
m
m
pairwise distinct numbers taken from
1
,
2
,
…
,
n
1,2,\ldots ,n
1
,
2
,
…
,
n
. If there exist
k
∈
{
1
,
2
,
…
,
m
}
k\in\{1,2,\ldots ,m\}
k
∈
{
1
,
2
,
…
,
m
}
such that
a
k
+
k
a_k+k
a
k
+
k
is odd, or there exist positive integers
k
,
l
(
1
≤
k
<
l
≤
m
)
k,l(1\le k<l\le m)
k
,
l
(
1
≤
k
<
l
≤
m
)
such that
a
k
>
a
l
a_k>a_l
a
k
>
a
l
, then call
a
1
,
a
2
,
…
,
a
m
a_1,a_2,\ldots ,a_m
a
1
,
a
2
,
…
,
a
m
a good sequence. Find the number of good sequences.
2
2
Hide problems
Element belonging to at least n/k subsets
Let
S
=
{
A
1
,
A
2
,
…
,
A
n
}
S=\{A_1,A_2,\ldots ,A_n\}
S
=
{
A
1
,
A
2
,
…
,
A
n
}
, where
A
1
,
A
2
,
…
,
A
n
A_1,A_2,\ldots ,A_n
A
1
,
A
2
,
…
,
A
n
are
n
n
n
pairwise distinct finite sets
(
n
≥
2
)
(n\ge 2)
(
n
≥
2
)
, such that for any
A
i
,
A
j
∈
S
A_i,A_j\in S
A
i
,
A
j
∈
S
,
A
i
∪
A
j
∈
S
A_i\cup A_j\in S
A
i
∪
A
j
∈
S
. If
k
=
min
1
≤
i
≤
n
∣
A
i
∣
≥
2
k= \min_{1\le i\le n}|A_i|\ge 2
k
=
min
1
≤
i
≤
n
∣
A
i
∣
≥
2
, prove that there exist
x
∈
⋃
i
=
1
n
A
i
x\in \bigcup_{i=1}^n A_i
x
∈
⋃
i
=
1
n
A
i
, such that
x
x
x
is in at least
n
k
\frac{n}{k}
k
n
of the sets
A
1
,
A
2
,
…
,
A
n
A_1,A_2,\ldots ,A_n
A
1
,
A
2
,
…
,
A
n
(Here
∣
X
∣
|X|
∣
X
∣
denotes the number of elements in finite set
X
X
X
).
Prove CD^2=BDxCE
In isoceles
△
A
B
C
\triangle ABC
△
A
BC
,
A
B
=
A
C
AB=AC
A
B
=
A
C
,
I
I
I
is its incenter,
D
D
D
is a point inside
△
A
B
C
\triangle ABC
△
A
BC
such that
I
,
B
,
C
,
D
I,B,C,D
I
,
B
,
C
,
D
are concyclic. The line through
C
C
C
parallel to
B
D
BD
B
D
meets
A
D
AD
A
D
at
E
E
E
. Prove that
C
D
2
=
B
D
⋅
C
E
CD^2=BD\cdot CE
C
D
2
=
B
D
⋅
CE
.
3
2
Hide problems
Angle is twice of another
P
P
P
is a point on arc \overarc{BC} of the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
not containing
A
A
A
,
K
K
K
lies on segment
A
P
AP
A
P
such that
B
K
BK
B
K
bisects
∠
A
B
C
\angle ABC
∠
A
BC
. The circumcircle of
△
K
P
C
\triangle KPC
△
K
PC
meets
A
C
,
B
D
AC,BD
A
C
,
B
D
at
D
,
E
D,E
D
,
E
respectively.
P
E
PE
PE
meets
A
B
AB
A
B
at
F
F
F
. Prove that
∠
A
B
C
=
2
∠
F
C
B
\angle ABC=2\angle FCB
∠
A
BC
=
2∠
FCB
.
Infinitely many triplets
Prove that there exist infinitely many positive integer triples
(
a
,
b
,
c
)
(
a
,
b
,
c
>
2015
)
(a,b,c)(a,b,c>2015)
(
a
,
b
,
c
)
(
a
,
b
,
c
>
2015
)
such that
a
∣
b
c
−
1
,
b
∣
a
c
+
1
,
c
∣
a
b
+
1.
a|bc-1, b|ac+1, c|ab+1.
a
∣
b
c
−
1
,
b
∣
a
c
+
1
,
c
∣
ab
+
1.
1
2
Hide problems
China Second Round Olympiad 2015 Test 2 Q1
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
be real numbers.Prove that you can select
ε
1
,
ε
2
,
…
,
ε
n
∈
{
−
1
,
1
}
\varepsilon _1, \varepsilon _2, \ldots, \varepsilon _n\in\{-1,1\}
ε
1
,
ε
2
,
…
,
ε
n
∈
{
−
1
,
1
}
such that
(
∑
i
=
1
n
a
i
)
2
+
(
∑
i
=
1
n
ε
i
a
i
)
2
≤
(
n
+
1
)
(
∑
i
=
1
n
a
i
2
)
.
\left( \sum_{i=1}^{n}a_{i}\right)^2 +\left( \sum_{i=1}^{n}\varepsilon _ia_{i}\right)^2 \leq(n+1)\left( \sum_{i=1}^{n}a^2_{i}\right).
(
i
=
1
∑
n
a
i
)
2
+
(
i
=
1
∑
n
ε
i
a
i
)
2
≤
(
n
+
1
)
(
i
=
1
∑
n
a
i
2
)
.
China Second Round Olympiad 2015 （B) Test 2 ，Q1
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be nonnegative real numbers.Prove that
(
a
−
b
c
)
2
+
(
b
−
c
a
)
2
+
(
c
−
a
b
)
2
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
a
)
2
≥
1
2
.
\frac{(a-bc)^2+(b-ca)^2+(c-ab)^2}{(a-b)^2+(b-c)^2+(c-a)^2}\geq\frac{1}{2}.
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
a
)
2
(
a
−
b
c
)
2
+
(
b
−
c
a
)
2
+
(
c
−
ab
)
2
≥
2
1
.