MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2001 China Second Round Olympiad
2001 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(3)
3
1
Hide problems
Divide rectangle into squares
An
m
×
n
(
m
,
n
∈
N
∗
)
m\times n(m,n\in \mathbb{N}^*)
m
×
n
(
m
,
n
∈
N
∗
)
rectangle is divided into some smaller squares. The sides of each square are all parallel to the corresponding sides of the rectangle, and the length of each side is integer. Determine the minimum of the sum of the sides of these squares.
1
1
Hide problems
An old geometry
Let
O
,
H
O,H
O
,
H
be the circumcenter and orthocenter of
△
A
B
C
,
\triangle ABC,
△
A
BC
,
respectively. Line
A
H
AH
A
H
and
B
C
BC
BC
intersect at
D
,
D,
D
,
Line
B
H
BH
B
H
and
A
C
AC
A
C
intersect at
E
,
E,
E
,
Line
C
H
CH
C
H
and
A
B
AB
A
B
intersect at
F
,
F,
F
,
Line
A
B
AB
A
B
and
E
D
ED
E
D
intersect at
M
,
M,
M
,
A
C
AC
A
C
and
F
D
FD
F
D
intersect at
N
.
N.
N
.
Prove that
(
1
)
O
B
⊥
D
F
,
O
C
⊥
D
E
;
(1)OB\perp DF,OC\perp DE;
(
1
)
OB
⊥
D
F
,
OC
⊥
D
E
;
(
2
)
O
H
⊥
M
N
.
(2)OH\perp MN.
(
2
)
O
H
⊥
MN
.
2
1
Hide problems
Find extremes of \sum x_i
If nonnegative reals
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots, x_n
x
1
,
x
2
,
…
,
x
n
satisfy
∑
i
=
1
n
x
i
2
+
2
∑
1
≤
k
<
j
≤
n
k
j
x
k
x
j
=
1
\sum_{i=1}^n x_i^2 + 2\sum_{1 \leq k < j \leq n} \sqrt{\frac{k}{j}}x_kx_j = 1
i
=
1
∑
n
x
i
2
+
2
1
≤
k
<
j
≤
n
∑
j
k
x
k
x
j
=
1
what are the minimum and maximum values of
∑
i
=
1
n
x
i
\sum_{i=1}^n x_i
∑
i
=
1
n
x
i
?