MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
Taiwan National Olympiad
1998 Taiwan National Olympiad
1998 Taiwan National Olympiad
Part of
Taiwan National Olympiad
Subcontests
(6)
4
1
Hide problems
distance
Let
I
I
I
be the incenter of triangle
A
B
C
ABC
A
BC
. Lines
A
I
AI
A
I
,
B
I
BI
B
I
,
C
I
CI
C
I
meet the sides of
△
A
B
C
\triangle ABC
△
A
BC
at
D
D
D
,
E
E
E
,
F
F
F
respectively. Let
X
X
X
,
Y
Y
Y
,
Z
Z
Z
be arbitrary points on segments
E
F
EF
EF
,
F
D
FD
F
D
,
D
E
DE
D
E
, respectively. Prove that
d
(
X
,
A
B
)
+
d
(
Y
,
B
C
)
+
d
(
Z
,
C
A
)
≤
X
Y
+
Y
Z
+
Z
X
d(X, AB) + d(Y, BC) + d(Z, CA) \leq XY + YZ + ZX
d
(
X
,
A
B
)
+
d
(
Y
,
BC
)
+
d
(
Z
,
C
A
)
≤
X
Y
+
Y
Z
+
ZX
, where
d
(
X
,
ℓ
)
d(X, \ell)
d
(
X
,
ℓ
)
denotes the distance from a point
X
X
X
to a line
ℓ
\ell
ℓ
.
6
1
Hide problems
m signals
In a group of
n
≥
4
n\geq 4
n
≥
4
persons, every three who know each other have a common signal. Assume that these signals are not repeatad and that there are
m
≥
1
m\geq 1
m
≥
1
signals in total. For any set of four persons in which there are three having a common signal, the fourth person has a common signal with at most one of them. Show that there three persons who have a common signal, such that the number of persons having no signal with anyone of them does not exceed
[
n
+
3
−
18
m
n
]
[n+3-\frac{18m}{n}]
[
n
+
3
−
n
18
m
]
.
5
1
Hide problems
Find the smallest positive tinteger $k$
For a positive integer
n
n
n
, let
ω
(
n
)
\omega(n)
ω
(
n
)
denote the number of positive prime divisors of
n
n
n
. Find the smallest positive tinteger
k
k
k
such that
2
ω
(
n
)
≤
k
n
4
∀
n
∈
N
2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}
2
ω
(
n
)
≤
k
4
n
∀
n
∈
N
.
3
1
Hide problems
family of $m$-element subsets [Erdös-Ko-Rado theorem]
Let
m
,
n
m,n
m
,
n
be positive integers, and let
F
F
F
be a family of
m
m
m
-element subsets of
{
1
,
2
,
.
.
.
,
n
}
\{1,2,...,n\}
{
1
,
2
,
...
,
n
}
satisfying A\cap B \not \equal{} \emptyset for all
A
,
B
∈
F
A,B\in F
A
,
B
∈
F
. Determine the maximum possible number of elements in
F
F
F
.
2
1
Hide problems
$x^2+y^2+z^2+u^2+v^2=xyzuv-65$
Does there exist a solution
(
x
,
y
,
z
,
u
,
v
)
(x,y,z,u,v)
(
x
,
y
,
z
,
u
,
v
)
in integers greater than
1998
1998
1998
to the equation
x
2
+
y
2
+
z
2
+
u
2
+
v
2
=
x
y
z
u
v
−
65
x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65
x
2
+
y
2
+
z
2
+
u
2
+
v
2
=
x
yz
uv
−
65
?
1
1
Hide problems
[x] and function gcd
Let
m
,
n
m,n
m
,
n
are positive integers. a)Prove that
(
m
,
n
)
=
2
∑
k
=
0
m
−
1
[
k
n
m
]
+
m
+
n
−
m
n
(m,n)=2\sum_{k=0}^{m-1}[\frac{kn}{m}]+m+n-mn
(
m
,
n
)
=
2
∑
k
=
0
m
−
1
[
m
kn
]
+
m
+
n
−
mn
. b)If
m
,
n
≥
2
m,n\geq 2
m
,
n
≥
2
, prove that
∑
k
=
0
m
−
1
[
k
n
m
]
=
∑
k
=
0
n
−
1
[
k
m
n
]
\sum_{k=0}^{m-1}[\frac{kn}{m}]=\sum_{k=0}^{n-1}[\frac{km}{n}]
∑
k
=
0
m
−
1
[
m
kn
]
=
∑
k
=
0
n
−
1
[
n
km
]
.