MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2018 China National Olympiad
2018 China National Olympiad
Part of
China National Olympiad
Subcontests
(5)
4
1
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Incenters and concyclic points
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral whose diagonals intersect at
P
P
P
. The circumcircle of
△
A
P
D
\triangle APD
△
A
P
D
meets segment
A
B
AB
A
B
at points
A
A
A
and
E
E
E
. The circumcircle of
△
B
P
C
\triangle BPC
△
BPC
meets segment
A
B
AB
A
B
at points
B
B
B
and
F
F
F
. Let
I
I
I
and
J
J
J
be the incenters of
△
A
D
E
\triangle ADE
△
A
D
E
and
△
B
C
F
\triangle BCF
△
BCF
, respectively. Segments
I
J
IJ
I
J
and
A
C
AC
A
C
meet at
K
K
K
. Prove that the points
A
,
I
,
K
,
E
A,I,K,E
A
,
I
,
K
,
E
are cyclic.
6
1
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China Mathematical Olympiad 2018 Q6
China Mathematical Olympiad 2018 Q6 Given the positive integer
n
,
k
n ,k
n
,
k
(
n
>
k
)
(n>k)
(
n
>
k
)
and
a
1
,
a
2
,
⋯
,
a
n
∈
(
k
−
1
,
k
)
a_1,a_2,\cdots ,a_n\in (k-1,k)
a
1
,
a
2
,
⋯
,
a
n
∈
(
k
−
1
,
k
)
,if positive number
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots ,x_n
x
1
,
x
2
,
⋯
,
x
n
satisfying:For any set
I
⊆
{
1
,
2
,
⋯
,
n
}
\mathbb{I} \subseteq \{1,2,\cdots,n\}
I
⊆
{
1
,
2
,
⋯
,
n
}
,
∣
I
∣
=
k
|\mathbb{I} |=k
∣
I
∣
=
k
,have
∑
i
∈
I
x
i
≤
∑
i
∈
I
a
i
\sum_{i\in \mathbb{I} }x_i\le \sum_{i\in \mathbb{I} }a_i
∑
i
∈
I
x
i
≤
∑
i
∈
I
a
i
, find the maximum value of
x
1
x
2
⋯
x
n
.
x_1x_2\cdots x_n.
x
1
x
2
⋯
x
n
.
3
1
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Simiultaneous Diophantine Approximation on Cube Roots
Let
q
q
q
be a positive integer which is not a perfect cube. Prove that there exists a positive constant
C
C
C
such that for all natural numbers
n
n
n
, one has
{
n
q
1
3
}
+
{
n
q
2
3
}
≥
C
n
−
1
2
\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}
{
n
q
3
1
}
+
{
n
q
3
2
}
≥
C
n
−
2
1
where
{
x
}
\{ x \}
{
x
}
denotes the fractional part of
x
x
x
.
2
1
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Coloring points of a lattice cube
Let
n
n
n
and
k
k
k
be positive integers and let
T
=
{
(
x
,
y
,
z
)
∈
N
3
∣
1
≤
x
,
y
,
z
≤
n
}
T = \{ (x,y,z) \in \mathbb{N}^3 \mid 1 \leq x,y,z \leq n \}
T
=
{(
x
,
y
,
z
)
∈
N
3
∣
1
≤
x
,
y
,
z
≤
n
}
be the length
n
n
n
lattice cube. Suppose that
3
n
2
−
3
n
+
1
+
k
3n^2 - 3n + 1 + k
3
n
2
−
3
n
+
1
+
k
points of
T
T
T
are colored red such that if
P
P
P
and
Q
Q
Q
are red points and
P
Q
PQ
PQ
is parallel to one of the coordinate axes, then the whole line segment
P
Q
PQ
PQ
consists of only red points. Prove that there exists at least
k
k
k
unit cubes of length
1
1
1
, all of whose vertices are colored red.
1
1
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Inequality on subsets of primes satisfying conditions
Let
n
n
n
be a positive integer. Let
A
n
A_n
A
n
denote the set of primes
p
p
p
such that there exists positive integers
a
,
b
a,b
a
,
b
satisfying
a
+
b
p
and
a
n
+
b
n
p
2
\frac{a+b}{p} \text{ and } \frac{a^n + b^n}{p^2}
p
a
+
b
and
p
2
a
n
+
b
n
are both integers that are relatively prime to
p
p
p
. If
A
n
A_n
A
n
is finite, let
f
(
n
)
f(n)
f
(
n
)
denote
∣
A
n
∣
|A_n|
∣
A
n
∣
.a) Prove that
A
n
A_n
A
n
is finite if and only if
n
≠
2
n \not = 2
n
=
2
.b) Let
m
,
k
m,k
m
,
k
be odd positive integers and let
d
d
d
be their gcd. Show that
f
(
d
)
≤
f
(
k
)
+
f
(
m
)
−
f
(
k
m
)
≤
2
f
(
d
)
.
f(d) \leq f(k) + f(m) - f(km) \leq 2 f(d).
f
(
d
)
≤
f
(
k
)
+
f
(
m
)
−
f
(
km
)
≤
2
f
(
d
)
.