MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2018 China Second Round Olympiad
2018 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(4)
4
2
Hide problems
Coprime to sum of terms before it
Define sequence
{
a
n
}
\{a_n\}
{
a
n
}
:
a
1
a_1
a
1
is any positive integer, and for any positive integer
n
≥
1
n\ge 1
n
≥
1
,
a
n
+
1
a_{n+1}
a
n
+
1
is the smallest positive integer coprime to
∑
i
=
1
n
a
i
\sum_{i=1}^{n} a_i
∑
i
=
1
n
a
i
and not equal to
a
1
,
…
,
a
n
a_1,\ldots, a_n
a
1
,
…
,
a
n
. Prove that every positive integer appears in the sequence
{
a
n
}
\{a_n\}
{
a
n
}
.
a^k+n are all composite numbers
Prove that for any integer
a
≥
2
a \ge 2
a
≥
2
and positive integer
n
,
n,
n
,
there exist positive integer
k
k
k
such that
a
k
+
1
,
a
k
+
2
,
…
,
a
k
+
n
a^k+1,a^k+2,\ldots,a^k+n
a
k
+
1
,
a
k
+
2
,
…
,
a
k
+
n
are all composite numbers.
3
2
Hide problems
Expressing integers as difference of two elements
Let
n
,
k
,
m
n,k,m
n
,
k
,
m
be positive integers, where
k
≥
2
k\ge 2
k
≥
2
and
n
≤
m
<
2
k
−
1
k
n
n\le m < \frac{2k-1}{k}n
n
≤
m
<
k
2
k
−
1
n
. Let
A
A
A
be a subset of
{
1
,
2
,
…
,
m
}
\{1,2,\ldots ,m\}
{
1
,
2
,
…
,
m
}
with
n
n
n
elements. Prove that every integer in the range
(
0
,
n
k
−
1
)
\left(0,\frac{n}{k-1}\right)
(
0
,
k
−
1
n
)
can be expressed as
a
−
b
a-b
a
−
b
, where
a
,
b
∈
A
a,b\in A
a
,
b
∈
A
.
A=\{1,2,\ldots,n\} maxX>minY
Let set
A
=
{
1
,
2
,
…
,
n
}
,
A=\{1,2,\ldots,n\} ,
A
=
{
1
,
2
,
…
,
n
}
,
and
X
,
Y
X,Y
X
,
Y
be two subsets (not necessarily distinct) of
A
.
A.
A
.
Define that
max
X
\textup{max} X
max
X
and
min
Y
\textup{min} Y
min
Y
represent the greatest element of
X
X
X
and the least element of
Y
,
Y,
Y
,
respectively. Determine the number of two-tuples
(
X
,
Y
)
(X,Y)
(
X
,
Y
)
which satisfies
max
X
>
min
Y
.
\textup{max} X>\textup{min} Y.
max
X
>
min
Y
.
2
2
Hide problems
Equal lengths in incircle configuration
In triangle
△
A
B
C
\triangle ABC
△
A
BC
,
A
B
<
A
C
AB<AC
A
B
<
A
C
,
M
,
D
,
E
M,D,E
M
,
D
,
E
are the midpoints of
B
C
BC
BC
, the arcs
B
A
C
BAC
B
A
C
and
B
C
BC
BC
of the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
respectively. The incircle of
△
A
B
C
\triangle ABC
△
A
BC
touches
A
B
AB
A
B
at
F
F
F
,
A
E
AE
A
E
meets
B
C
BC
BC
at
G
G
G
, and the perpendicular to
A
B
AB
A
B
at
B
B
B
meets segment
E
F
EF
EF
at
N
N
N
. If
B
N
=
E
M
BN=EM
BN
=
EM
, prove that
D
F
DF
D
F
is perpendicular to
F
G
FG
FG
.
\frac{AD}{DC}=\frac{BC}{2CE}
In triangle
△
A
B
C
,
A
B
=
A
C
.
\triangle ABC, AB=AC.
△
A
BC
,
A
B
=
A
C
.
Let
D
D
D
be on segment
A
C
AC
A
C
and
E
E
E
be a point on the extended line
B
C
BC
BC
such that
C
C
C
is located between
B
B
B
and
E
E
E
and
A
D
D
C
=
B
C
2
C
E
\frac{AD}{DC}=\frac{BC}{2CE}
D
C
A
D
=
2
CE
BC
. Let
ω
\omega
ω
be the circle with diameter
A
B
,
AB,
A
B
,
and
ω
\omega
ω
intersects segment
D
E
DE
D
E
at
F
.
F.
F
.
Prove that
B
,
C
,
F
,
D
B,C,F,D
B
,
C
,
F
,
D
are concyclic.
1
2
Hide problems
China Second Round Olympiad 2018Test 2 Q1
Let
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
,
A
,
B
a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n,A,B
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
,
A
,
B
are positive reals such that
a
i
≤
b
i
,
a
i
≤
A
a_i\leq b_i,a_i\leq A
a
i
≤
b
i
,
a
i
≤
A
(
i
=
1
,
2
,
⋯
,
n
)
(i=1,2,\cdots,n)
(
i
=
1
,
2
,
⋯
,
n
)
and
b
1
b
2
⋯
b
n
a
1
a
2
⋯
a
n
≤
B
A
.
\frac{b_1 b_2 \cdots b_n}{a_1 a_2 \cdots a_n}\leq \frac{B}{A}.
a
1
a
2
⋯
a
n
b
1
b
2
⋯
b
n
≤
A
B
.
Prove that
(
b
1
+
1
)
(
b
2
+
1
)
⋯
(
b
n
+
1
)
(
a
1
+
1
)
(
a
2
+
1
)
⋯
(
a
n
+
1
)
≤
B
+
1
A
+
1
.
\frac{(b_1+1) (b_2+1) \cdots (b_n+1)}{(a_1+1) (a_2+1) \cdots (a_n+1)}\leq \frac{B+1}{A+1}.
(
a
1
+
1
)
(
a
2
+
1
)
⋯
(
a
n
+
1
)
(
b
1
+
1
)
(
b
2
+
1
)
⋯
(
b
n
+
1
)
≤
A
+
1
B
+
1
.
f(x)=ax+b+\frac{9}{x}
Let
a
,
b
∈
R
,
f
(
x
)
=
a
x
+
b
+
9
x
.
a,b \in \mathbb R,f(x)=ax+b+\frac{9}{x}.
a
,
b
∈
R
,
f
(
x
)
=
a
x
+
b
+
x
9
.
Prove that there exists
x
0
∈
[
1
,
9
]
,
x_0 \in \left[1,9 \right],
x
0
∈
[
1
,
9
]
,
such that
∣
f
(
x
0
)
∣
≥
2.
|f(x_0)| \ge 2.
∣
f
(
x
0
)
∣
≥
2.