MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
Taiwan National Olympiad
2002 Taiwan National Olympiad
2002 Taiwan National Olympiad
Part of
Taiwan National Olympiad
Subcontests
(6)
2
1
Hide problems
lattice point visible from the origin
A lattice point
X
X
X
in the plane is said to be visible from the origin
O
O
O
if the line segment
O
X
OX
OX
does not contain any other lattice points. Show that for any positive integer
n
n
n
, there is square
A
B
C
D
ABCD
A
BC
D
of area
n
2
n^{2}
n
2
such that none of the lattice points inside the square is visible from the origin.
6
1
Hide problems
simson's line
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
be fixed points in the plane , and
D
D
D
be a variable point on the circle
A
B
C
ABC
A
BC
, distinct from
A
,
B
,
C
A,B,C
A
,
B
,
C
. Let
I
A
,
I
B
,
I
C
,
I
D
I_{A},I_{B},I_{C},I_{D}
I
A
,
I
B
,
I
C
,
I
D
be the Simson lines of
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
with respect to triangles
B
C
D
,
A
C
D
,
A
B
D
,
A
B
C
BCD,ACD,ABD,ABC
BC
D
,
A
C
D
,
A
B
D
,
A
BC
respectively. Find the locus of the intersection points of the four lines
I
A
,
I
B
,
I
C
,
I
D
I_{A},I_{B},I_{C},I_{D}
I
A
,
I
B
,
I
C
,
I
D
when point
D
D
D
varies.
1
1
Hide problems
find n and n nonnegative integers
Find all natural numbers
n
n
n
and nonnegative integers
x
1
,
x
2
,
.
.
.
,
x
n
x_{1},x_{2},...,x_{n}
x
1
,
x
2
,
...
,
x
n
such that
∑
i
=
1
n
x
i
2
=
1
+
4
4
n
+
1
(
∑
i
=
1
n
x
i
)
2
\sum_{i=1}^{n}x_{i}^{2}=1+\frac{4}{4n+1}(\sum_{i=1}^{n}x_{i})^{2}
∑
i
=
1
n
x
i
2
=
1
+
4
n
+
1
4
(
∑
i
=
1
n
x
i
)
2
.
3
1
Hide problems
max and min
Suppose
x
,
y
,
,
a
,
b
,
c
,
d
,
e
,
f
x,y,,a,b,c,d,e,f
x
,
y
,,
a
,
b
,
c
,
d
,
e
,
f
are real numbers satifying i)
max
(
a
,
0
)
+
max
(
b
,
0
)
<
x
+
a
y
+
b
z
<
1
+
min
(
a
,
0
)
+
min
(
b
,
0
)
\max{(a,0)}+\max{(b,0)}<x+ay+bz<1+\min{(a,0)}+\min{(b,0)}
max
(
a
,
0
)
+
max
(
b
,
0
)
<
x
+
a
y
+
b
z
<
1
+
min
(
a
,
0
)
+
min
(
b
,
0
)
, and ii)
max
(
c
,
0
)
+
max
(
d
,
0
)
<
c
x
+
y
+
d
z
<
1
+
min
(
c
,
0
)
+
min
(
d
,
0
)
\max{(c,0)}+\max{(d,0)}<cx+y+dz<1+\min{(c,0)}+\min{(d,0)}
max
(
c
,
0
)
+
max
(
d
,
0
)
<
c
x
+
y
+
d
z
<
1
+
min
(
c
,
0
)
+
min
(
d
,
0
)
, and iii)
max
(
e
,
0
)
+
max
(
f
,
0
)
<
e
x
+
f
y
+
z
<
1
+
min
(
e
,
0
)
+
min
(
f
,
0
)
\max{(e,0)}+\max{(f,0)}<ex+fy+z<1+\min{(e,0)}+\min{(f,0)}
max
(
e
,
0
)
+
max
(
f
,
0
)
<
e
x
+
f
y
+
z
<
1
+
min
(
e
,
0
)
+
min
(
f
,
0
)
. Prove that
0
<
x
,
y
,
z
<
1
0<x,y,z<1
0
<
x
,
y
,
z
<
1
.
4
1
Hide problems
$0<x_1,x_2,x_3,x_4\leq\frac{1}{2}$
Let
0
<
x
1
,
x
2
,
x
3
,
x
4
≤
1
2
0<x_{1},x_{2},x_{3},x_{4}\leq\frac{1}{2}
0
<
x
1
,
x
2
,
x
3
,
x
4
≤
2
1
are real numbers. Prove that
x
1
x
2
x
3
x
4
(
1
−
x
1
)
(
1
−
x
2
)
(
1
−
x
3
)
(
1
−
x
4
)
≤
x
1
4
+
x
2
4
+
x
3
4
+
x
4
4
(
1
−
x
1
)
4
+
(
1
−
x
2
)
4
+
(
1
−
x
3
)
4
+
(
1
−
x
4
)
4
\frac{x_{1}x_{2}x_{3}x_{4}}{(1-x_{1})(1-x_{2})(1-x_{3})(1-x_{4})}\leq\frac{x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}}{(1-x_{1})^{4}+(1-x_{2})^{4}+(1-x_{3})^{4}+(1-x_{4})^{4}}
(
1
−
x
1
)
(
1
−
x
2
)
(
1
−
x
3
)
(
1
−
x
4
)
x
1
x
2
x
3
x
4
≤
(
1
−
x
1
)
4
+
(
1
−
x
2
)
4
+
(
1
−
x
3
)
4
+
(
1
−
x
4
)
4
x
1
4
+
x
2
4
+
x
3
4
+
x
4
4
.
5
1
Hide problems
vlue of a sum
Suppose that the real numbers
a
1
,
a
2
,
.
.
.
,
a
2002
a_{1},a_{2},...,a_{2002}
a
1
,
a
2
,
...
,
a
2002
satisfying
a
1
2
+
a
2
3
+
.
.
.
+
a
2002
2003
=
4
3
\frac{a_{1}}{2}+\frac{a_{2}}{3}+...+\frac{a_{2002}}{2003}=\frac{4}{3}
2
a
1
+
3
a
2
+
...
+
2003
a
2002
=
3
4
a
1
3
+
a
2
4
+
.
.
.
+
a
2002
2004
=
4
5
\frac{a_{1}}{3}+\frac{a_{2}}{4}+...+\frac{a_{2002}}{2004}=\frac{4}{5}
3
a
1
+
4
a
2
+
...
+
2004
a
2002
=
5
4
.
.
.
...
...
a
1
2003
+
a
2
2004
+
.
.
.
+
a
2002
4004
=
4
4005
\frac{a_{1}}{2003}+\frac{a_{2}}{2004}+...+\frac{a_{2002}}{4004}=\frac{4}{4005}
2003
a
1
+
2004
a
2
+
...
+
4004
a
2002
=
4005
4
Evaluate the sum
a
1
3
+
a
2
5
+
.
.
.
+
a
2002
4005
\frac{a_{1}}{3}+\frac{a_{2}}{5}+...+\frac{a_{2002}}{4005}
3
a
1
+
5
a
2
+
...
+
4005
a
2002
.