MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
Taiwan National Olympiad
2000 Taiwan National Olympiad
2000 Taiwan National Olympiad
Part of
Taiwan National Olympiad
Subcontests
(3)
2
1
Hide problems
Connected intersections of k subsets
Let
n
n
n
be a positive integer and
A
=
{
1
,
2
,
…
,
n
}
A=\{ 1,2,\ldots ,n\}
A
=
{
1
,
2
,
…
,
n
}
. A subset of
A
A
A
is said to be connected if it consists of one element or several consecutive elements. Determine the maximum
k
k
k
for which there exist
k
k
k
distinct subsets of
A
A
A
such that the intersection of any two of them is connected.
3
2
Hide problems
The subset M evading all labels
Consider the set
S
=
{
1
,
2
,
…
,
100
}
S=\{ 1,2,\ldots ,100\}
S
=
{
1
,
2
,
…
,
100
}
and the family
P
=
{
T
⊂
S
∣
∣
T
∣
=
49
}
\mathcal{P}=\{ T\subset S\mid |T|=49\}
P
=
{
T
⊂
S
∣
∣
T
∣
=
49
}
. Each
T
∈
P
T\in\mathcal{P}
T
∈
P
is labelled by an arbitrary number from
S
S
S
. Prove that there exists a subset
M
M
M
of
S
S
S
with
∣
M
∣
=
50
|M|=50
∣
M
∣
=
50
such that for each
x
∈
M
x\in M
x
∈
M
, the set
M
\
{
x
}
M\backslash\{ x\}
M
\
{
x
}
is not labelled by
x
x
x
.
Determine f(2000)
Define a function
f
:
N
→
N
0
f:\mathbb{N}\rightarrow\mathbb{N}_0
f
:
N
→
N
0
by
f
(
1
)
=
0
f(1)=0
f
(
1
)
=
0
and f(n)=\max_j\{ f(j)+f(n-j)+j\} \forall\, n\ge 2 Determine
f
(
2000
)
f(2000)
f
(
2000
)
.
1
2
Hide problems
Exponential x,y equation
Find all pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of positive integers such that
y
x
2
=
x
y
+
2
y^{x^2}=x^{y+2}
y
x
2
=
x
y
+
2
.
(m,n)>1 if Euler function of 5^m-1 is 5^n-1
Suppose that for some
m
,
n
∈
N
m,n\in\mathbb{N}
m
,
n
∈
N
we have
φ
(
5
m
−
1
)
=
5
n
−
1
\varphi (5^m-1)=5^n-1
φ
(
5
m
−
1
)
=
5
n
−
1
, where
φ
\varphi
φ
denotes the Euler function. Show that
(
m
,
n
)
>
1
(m,n)>1
(
m
,
n
)
>
1
.