MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
Taiwan National Olympiad
1997 Taiwan National Olympiad
1997 Taiwan National Olympiad
Part of
Taiwan National Olympiad
Subcontests
(9)
9
1
Hide problems
subset with at least $[\log_2{n}]+1$ elements
For
n
≥
k
≥
3
n\geq k\geq 3
n
≥
k
≥
3
, let
X
=
{
1
,
2
,
.
.
.
,
n
}
X=\{1,2,...,n\}
X
=
{
1
,
2
,
...
,
n
}
and let
F
k
F_{k}
F
k
a the family of
k
k
k
-element subsets of
X
X
X
, any two of which have at most
k
−
2
k-2
k
−
2
elements in common. Show that there exists a subset
M
k
M_{k}
M
k
of
X
X
X
with at least
[
log
2
n
]
+
1
[\log_{2}{n}]+1
[
lo
g
2
n
]
+
1
elements containing no subset in
F
k
F_{k}
F
k
.
8
1
Hide problems
$OD.OE.OF\geq 8R^3$
Let
O
O
O
be the circumcenter and
R
R
R
be the circumradius of an acute triangle
A
B
C
ABC
A
BC
. Let
A
O
AO
A
O
meet the circumcircle of
O
B
C
OBC
OBC
again at
D
D
D
,
B
O
BO
BO
meet the circumcircle of
O
C
A
OCA
OC
A
again at
E
E
E
, and
C
O
CO
CO
meet the circumcircle of
O
A
B
OAB
O
A
B
again at
F
F
F
. Show that
O
D
.
O
E
.
O
F
≥
8
R
3
OD.OE.OF\geq 8R^{3}
O
D
.
OE
.
OF
≥
8
R
3
.
2
1
Hide problems
find all points
Given a line segment
A
B
AB
A
B
in the plane, find all possible points
C
C
C
such that in the triangle
A
B
C
ABC
A
BC
, the altitude from
A
A
A
and the median from
B
B
B
have the same length.
1
1
Hide problems
$f$ is periodic
Let
a
a
a
be rational and
b
,
c
,
d
b,c,d
b
,
c
,
d
are real numbers, and let
f
:
R
→
[
−
1.1
]
f: \mathbb{R}\to [-1.1]
f
:
R
→
[
−
1.1
]
be a function satisfying
f
(
x
+
a
+
b
)
−
f
(
x
+
b
)
=
c
[
x
+
2
a
+
[
x
]
−
2
[
x
+
a
]
−
]
+
d
f(x+a+b)-f(x+b)=c[x+2a+[x]-2[x+a]-]+d
f
(
x
+
a
+
b
)
−
f
(
x
+
b
)
=
c
[
x
+
2
a
+
[
x
]
−
2
[
x
+
a
]
−
]
+
d
for all
x
x
x
. Show that
f
f
f
is periodic.
3
1
Hide problems
n of reals
Let
n
>
2
n>2
n
>
2
be an integer. Suppose that
a
1
,
a
2
,
.
.
.
,
a
n
a_{1},a_{2},...,a_{n}
a
1
,
a
2
,
...
,
a
n
are real numbers such that
k
i
=
a
i
−
1
+
a
i
+
1
a
i
k_{i}=\frac{a_{i-1}+a_{i+1}}{a_{i}}
k
i
=
a
i
a
i
−
1
+
a
i
+
1
is a positive integer for all
i
i
i
(Here
a
0
=
a
n
,
a
n
+
1
=
a
1
a_{0}=a_{n},a_{n+1}=a_{1}
a
0
=
a
n
,
a
n
+
1
=
a
1
). Prove that
2
n
≤
a
1
+
a
2
+
.
.
.
+
a
n
≤
3
n
2n\leq a_{1}+a_{2}+...+a_{n}\leq 3n
2
n
≤
a
1
+
a
2
+
...
+
a
n
≤
3
n
.
6
1
Hide problems
number of the form $2^p3^q$
Show that every number of the form
2
p
3
q
2^{p}3^{q}
2
p
3
q
, where
p
,
q
p,q
p
,
q
are nonnegative integers, divides some number of the form
a
2
k
1
0
2
k
+
a
2
k
−
2
1
0
2
k
−
2
+
.
.
.
+
a
2
1
0
2
+
a
0
a_{2k}10^{2k}+a_{2k-2}10^{2k-2}+...+a_{2}10^{2}+a_{0}
a
2
k
1
0
2
k
+
a
2
k
−
2
1
0
2
k
−
2
+
...
+
a
2
1
0
2
+
a
0
, where
a
2
i
∈
{
1
,
2
,
.
.
.
,
9
}
a_{2i}\in\{1,2,...,9\}
a
2
i
∈
{
1
,
2
,
...
,
9
}
5
1
Hide problems
tetrahedron
Let
A
B
C
D
ABCD
A
BC
D
is a tetrahedron. Show that a)If
A
B
=
C
D
,
A
C
=
D
B
,
A
D
=
B
C
AB=CD,AC=DB,AD=BC
A
B
=
C
D
,
A
C
=
D
B
,
A
D
=
BC
then triangles
A
B
C
,
A
B
D
,
A
C
D
,
B
C
D
ABC,ABD,ACD,BCD
A
BC
,
A
B
D
,
A
C
D
,
BC
D
are acute. b)If the triangles
A
B
C
,
A
B
D
,
A
C
D
,
B
C
D
ABC,ABD,ACD,BCD
A
BC
,
A
B
D
,
A
C
D
,
BC
D
have the same area , then
A
B
=
C
D
,
A
C
=
D
B
,
A
D
=
B
C
AB=CD,AC=DB,AD=BC
A
B
=
C
D
,
A
C
=
D
B
,
A
D
=
BC
.
7
1
Hide problems
Find all positive integers $k$
Find all positive integers
k
k
k
for which there exists a function
f
:
N
→
Z
f: \mathbb{N}\to\mathbb{Z}
f
:
N
→
Z
satisfying
f
(
1997
)
=
1998
f(1997)=1998
f
(
1997
)
=
1998
and
f
(
a
b
)
=
f
(
a
)
+
f
(
b
)
+
k
f
(
gcd
(
a
,
b
)
)
∀
a
,
b
f(ab)=f(a)+f(b)+kf(\gcd{(a,b)})\forall a,b
f
(
ab
)
=
f
(
a
)
+
f
(
b
)
+
k
f
(
g
cd
(
a
,
b
)
)
∀
a
,
b
.
4
1
Hide problems
fecma numbers
Let
k
=
2
2
n
+
1
k=2^{2^{n}}+1
k
=
2
2
n
+
1
for some
n
∈
N
n\in\mathbb{N}
n
∈
N
. Show that
k
k
k
is prime iff
k
∣
3
k
−
1
2
+
1
k|3^{\frac{k-1}{2}}+1
k
∣
3
2
k
−
1
+
1
.