MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2005 China Second Round Olympiad
2005 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(3)
3
1
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summation
For each positive integer, define a function
f
(
n
)
=
{
0
,
if n is the square of an integer
⌊
1
{
n
}
⌋
,
if n is not the square of an integer
.
f(n)=\begin{cases}0, &\text{if n is the square of an integer}\\ \\ \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}.
f
(
n
)
=
⎩
⎨
⎧
0
,
⌊
{
n
}
1
⌋
,
if n is the square of an integer
if n is not the square of an integer
.
Find the value of
∑
k
=
1
200
f
(
k
)
\sum_{k=1}^{200} f(k)
∑
k
=
1
200
f
(
k
)
.
2
1
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Minimum value of function
Assume that positive numbers
a
,
b
,
c
,
x
,
y
,
z
a, b, c, x, y, z
a
,
b
,
c
,
x
,
y
,
z
satisfy
c
y
+
b
z
=
a
cy + bz = a
cy
+
b
z
=
a
,
a
z
+
c
x
=
b
az + cx = b
a
z
+
c
x
=
b
, and
b
x
+
a
y
=
c
bx + ay = c
b
x
+
a
y
=
c
. Find the minimum value of the function
f
(
x
,
y
,
z
)
=
x
2
x
+
1
+
y
2
y
+
1
+
z
2
z
+
1
.
f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}.
f
(
x
,
y
,
z
)
=
x
+
1
x
2
+
y
+
1
y
2
+
z
+
1
z
2
.
1
1
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Lines pass through incenter and excenter
In
△
A
B
C
\triangle ABC
△
A
BC
,
A
B
>
A
C
AB>AC
A
B
>
A
C
,
l
l
l
is a tangent line of the circumscribed circle of
△
A
B
C
\triangle ABC
△
A
BC
, passing through
A
A
A
. The circle, centered at
A
A
A
with radius
A
C
AC
A
C
, intersects
A
B
AB
A
B
at
D
D
D
, and line
l
l
l
at
E
,
F
E, F
E
,
F
. Prove that lines
D
E
,
D
F
DE, DF
D
E
,
D
F
pass through the incenter and an excenter of
△
A
B
C
\triangle ABC
△
A
BC
respectively.