MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
Taiwan National Olympiad
1999 Taiwan National Olympiad
1999 Taiwan National Olympiad
Part of
Taiwan National Olympiad
Subcontests
(6)
6
1
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eight different symbols
There are eight different symbols designed on
n
≥
2
n\geq 2
n
≥
2
different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any
k
k
k
symbols
(
1
≤
k
≤
7
)
(1\leq k\leq 7)
(
1
≤
k
≤
7
)
the number of shirts containing at least one of the
k
k
k
symbols is even. Determine the value of
n
n
n
.
5
1
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$BN>CN$
Let
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
be the altitudes of an acute triangle
A
B
C
ABC
A
BC
with
A
B
>
A
C
AB>AC
A
B
>
A
C
. Line
E
F
EF
EF
meets
B
C
BC
BC
at
P
P
P
, and line through
D
D
D
parallel to
E
F
EF
EF
meets
A
C
AC
A
C
and
A
B
AB
A
B
at
Q
Q
Q
and
R
R
R
, respectively. Let
N
N
N
be any poin on side
B
C
BC
BC
such that
N
Q
P
^
+
N
R
P
^
<
18
0
0
\widehat{NQP}+\widehat{NRP}<180^{0}
NQP
+
NRP
<
18
0
0
. Prove that
B
N
>
C
N
BN>CN
BN
>
CN
.
4
1
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set of primes less than $10000$
Let
P
∗
P^{*}
P
∗
be the set of primes less than
10000
10000
10000
. Find all possible primes
p
∈
P
∗
p\in P^{*}
p
∈
P
∗
such that for each subset
S
=
{
p
1
,
p
2
,
.
.
.
,
p
k
}
S=\{p_{1},p_{2},...,p_{k}\}
S
=
{
p
1
,
p
2
,
...
,
p
k
}
of
P
∗
P^{*}
P
∗
with
k
≥
2
k\geq 2
k
≥
2
and each
p
∉
S
p\not\in S
p
∈
S
, there is a
q
∈
P
∗
−
S
q\in P^{*}-S
q
∈
P
∗
−
S
such that
q
+
1
q+1
q
+
1
divides
(
p
1
+
1
)
(
p
2
+
1
)
.
.
.
(
p
k
+
1
)
(p_{1}+1)(p_{2}+1)...(p_{k}+1)
(
p
1
+
1
)
(
p
2
+
1
)
...
(
p
k
+
1
)
.
3
1
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$1999$ people participating in an exhibition
There are
1999
1999
1999
people participating in an exhibition. Among any
50
50
50
people there are two who don't know each other. Prove that there are
41
41
41
people, each of whom knows at most
1958
1958
1958
people.
2
1
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$a_i+a_j\leq a_{i+j}\leq a_i+a_j+1$
Let
a
1
,
a
2
,
.
.
.
,
a
1999
a_{1},a_{2},...,a_{1999}
a
1
,
a
2
,
...
,
a
1999
be a sequence of nonnegative integers such that for any
i
,
j
i,j
i
,
j
with
i
+
j
≤
1999
i+j\leq 1999
i
+
j
≤
1999
,
a
i
+
a
j
≤
a
i
+
j
≤
a
i
+
a
j
+
1
a_{i}+a_{j}\leq a_{i+j}\leq a_{i}+a_{j}+1
a
i
+
a
j
≤
a
i
+
j
≤
a
i
+
a
j
+
1
. Prove that there exists a real number
x
x
x
such that
a
n
=
[
n
x
]
∀
n
a_{n}=[nx]\forall n
a
n
=
[
n
x
]
∀
n
.
1
1
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integer equation
Find all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of positive integers such that
(
x
+
1
)
y
+
1
+
1
=
(
x
+
2
)
z
+
1
(x+1)^{y+1}+1=(x+2)^{z+1}
(
x
+
1
)
y
+
1
+
1
=
(
x
+
2
)
z
+
1
.