MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2007 China Second Round Olympiad
2007 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(3)
3
1
Hide problems
f(m,k) run over N
For positive integers
k
,
m
k,m
k
,
m
, where
1
≤
k
≤
5
1\le k\le 5
1
≤
k
≤
5
, define the function
f
(
m
,
k
)
f(m,k)
f
(
m
,
k
)
as
f
(
m
,
k
)
=
∑
i
=
1
5
[
m
k
+
1
i
+
1
]
f(m,k)=\sum_{i=1}^{5}\left[m\sqrt{\frac{k+1}{i+1}}\right]
f
(
m
,
k
)
=
i
=
1
∑
5
[
m
i
+
1
k
+
1
]
where
[
x
]
[x]
[
x
]
denotes the greatest integer not exceeding
x
x
x
. Prove that for any positive integer
n
n
n
, there exist positive integers
k
,
m
k,m
k
,
m
, where
1
≤
k
≤
5
1\le k\le 5
1
≤
k
≤
5
, such that
f
(
m
,
k
)
=
n
f(m,k)=n
f
(
m
,
k
)
=
n
.
2
1
Hide problems
no 5 adjacent stones
In a
7
×
8
7\times 8
7
×
8
chessboard,
56
56
56
stones are placed in the squares. Now we have to remove some of the stones such that after the operation, there are no five adjacent stones horizontally, vertically or diagonally. Find the minimal number of stones that have to be removed.
1
1
Hide problems
a condition for 4 points to be concyclic
In an acute triangle
A
B
C
ABC
A
BC
,
A
B
<
A
C
AB<AC
A
B
<
A
C
.
A
D
AD
A
D
is the altitude dropped onto
B
C
BC
BC
and
P
P
P
is a point on
A
D
AD
A
D
. Let
P
E
⊥
A
C
PE\perp AC
PE
⊥
A
C
at
E
E
E
,
P
F
⊥
A
B
PF\perp AB
PF
⊥
A
B
at
F
F
F
and let
J
,
K
J,K
J
,
K
be the circumcentres of triangles
B
D
F
,
C
D
E
BDF, CDE
B
D
F
,
C
D
E
respectively. Prove that
J
,
K
,
E
,
F
J,K,E,F
J
,
K
,
E
,
F
are concyclic if and only if
P
P
P
is the orthocentre of triangle
A
B
C
ABC
A
BC
.