MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2015 China National Olympiad
2015 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
2
2
Hide problems
Graph with no K_5
Given
30
30
30
students such that each student has at most
5
5
5
friends and for every
5
5
5
students there is a pair of students that are not friends, determine the maximum
k
k
k
such that for all such possible configurations, there exists
k
k
k
students who are all not friends.
Two ratio are equal
Let
A
,
B
,
D
,
E
,
F
,
C
A, B, D, E, F, C
A
,
B
,
D
,
E
,
F
,
C
be six points lie on a circle (in order) satisfy
A
B
=
A
C
AB=AC
A
B
=
A
C
. Let
P
=
A
D
∩
B
E
,
R
=
A
F
∩
C
E
,
Q
=
B
F
∩
C
D
,
S
=
A
D
∩
B
F
,
T
=
A
F
∩
C
D
P=AD \cap BE, R=AF \cap CE, Q=BF \cap CD, S=AD \cap BF, T=AF \cap CD
P
=
A
D
∩
BE
,
R
=
A
F
∩
CE
,
Q
=
BF
∩
C
D
,
S
=
A
D
∩
BF
,
T
=
A
F
∩
C
D
. Let
K
K
K
be a point lie on
S
T
ST
ST
satisfy
∠
Q
K
S
=
∠
E
C
A
\angle QKS=\angle ECA
∠
Q
K
S
=
∠
EC
A
.Prove that
S
K
K
T
=
P
Q
Q
R
\frac{SK}{KT}=\frac{PQ}{QR}
K
T
S
K
=
QR
PQ
3
2
Hide problems
x+y in B iff x,y in A
Let
n
≥
5
n \geq 5
n
≥
5
be a positive integer and let
A
A
A
and
B
B
B
be sets of integers satisfying the following conditions:i)
∣
A
∣
=
n
|A| = n
∣
A
∣
=
n
,
∣
B
∣
=
m
|B| = m
∣
B
∣
=
m
and
A
A
A
is a subset of
B
B
B
ii) For any distinct
x
,
y
∈
B
x,y \in B
x
,
y
∈
B
,
x
+
y
∈
B
x+y \in B
x
+
y
∈
B
iff
x
,
y
∈
A
x,y \in A
x
,
y
∈
A
Determine the minimum value of
m
m
m
.
Erdos Discrepancy Problem Variation
Let
a
1
,
a
2
,
.
.
.
a_1,a_2,...
a
1
,
a
2
,
...
be a sequence of non-negative integers such that for any
m
,
n
m,n
m
,
n
∑
i
=
1
2
m
a
i
n
≤
m
.
\sum_{i=1}^{2m} a_{in} \leq m.
i
=
1
∑
2
m
a
in
≤
m
.
Show that there exist
k
,
d
k,d
k
,
d
such that
∑
i
=
1
2
k
a
i
d
=
k
−
2014.
\sum_{i=1}^{2k} a_{id} = k-2014.
i
=
1
∑
2
k
a
i
d
=
k
−
2014.
1
2
Hide problems
Inequality on Complex Numbers
Let
z
1
,
z
2
,
.
.
.
,
z
n
z_1,z_2,...,z_n
z
1
,
z
2
,
...
,
z
n
be complex numbers satisfying
∣
z
i
−
1
∣
≤
r
|z_i - 1| \leq r
∣
z
i
−
1∣
≤
r
for some
r
r
r
in
(
0
,
1
)
(0,1)
(
0
,
1
)
. Show that
∣
∑
i
=
1
n
z
i
∣
⋅
∣
∑
i
=
1
n
1
z
i
∣
≥
n
2
(
1
−
r
2
)
.
\left | \sum_{i=1}^n z_i \right | \cdot \left | \sum_{i=1}^n \frac{1}{z_i} \right | \geq n^2(1-r^2).
i
=
1
∑
n
z
i
⋅
i
=
1
∑
n
z
i
1
≥
n
2
(
1
−
r
2
)
.
Divisibility condition on Binomial coefficient
Determine all integers
k
k
k
such that there exists infinitely many positive integers
n
n
n
not satisfying
n
+
k
∣
(
2
n
n
)
n+k |\binom{2n}{n}
n
+
k
∣
(
n
2
n
)