MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
Taiwan National Olympiad
1994 Taiwan National Olympiad
1994 Taiwan National Olympiad
Part of
Taiwan National Olympiad
Subcontests
(6)
4
1
Hide problems
mean and the standard deviation
Prove that there are infinitely many positive integers
n
n
n
with the following property: For any
n
n
n
integers
a
1
,
a
2
,
.
.
.
,
a
n
a_{1},a_{2},...,a_{n}
a
1
,
a
2
,
...
,
a
n
which form in arithmetic progression, both the mean and the standard deviation of the set
{
a
1
,
a
2
,
.
.
.
,
a
n
}
\{a_{1},a_{2},...,a_{n}\}
{
a
1
,
a
2
,
...
,
a
n
}
are integers. Remark. The mean and standard deviation of the set
{
x
1
,
x
2
,
.
.
.
,
x
n
}
\{x_{1},x_{2},...,x_{n}\}
{
x
1
,
x
2
,
...
,
x
n
}
are defined by
x
‾
=
x
1
+
x
2
+
.
.
.
+
x
n
n
\overline{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n}
x
=
n
x
1
+
x
2
+
...
+
x
n
and
∑
(
x
i
−
x
‾
)
2
n
\sqrt{\frac{\sum (x_{i}-\overline{x})^{2}}{n}}
n
∑
(
x
i
−
x
)
2
, respectively.
5
1
Hide problems
addition table on $X$
Given
X
=
{
0
,
a
,
b
,
c
}
X=\{0,a,b,c\}
X
=
{
0
,
a
,
b
,
c
}
, let
M
(
X
)
=
{
f
∣
f
:
X
→
X
}
M(X)=\{f|f: X\to X\}
M
(
X
)
=
{
f
∣
f
:
X
→
X
}
denote the set of all functions from
X
X
X
into itself. An addition table on
X
X
X
is given us follows:
+
+
+
0
0
0
a
a
a
b
b
b
c
c
c
0
0
0
0
0
0
a
a
a
b
b
b
c
c
c
a
a
a
a
a
a
0
0
0
c
c
c
b
b
b
b
b
b
b
b
b
c
c
c
0
0
0
a
a
a
c
c
c
c
c
c
b
b
b
a
a
a
0
0
0
a)If
S
=
{
f
∈
M
(
X
)
∣
f
(
x
+
y
+
x
)
=
f
(
x
)
+
f
(
y
)
+
f
(
x
)
∀
x
,
y
∈
X
}
S=\{f\in M(X)|f(x+y+x)=f(x)+f(y)+f(x)\forall x,y\in X\}
S
=
{
f
∈
M
(
X
)
∣
f
(
x
+
y
+
x
)
=
f
(
x
)
+
f
(
y
)
+
f
(
x
)
∀
x
,
y
∈
X
}
, find
∣
S
∣
|S|
∣
S
∣
. b)If
I
=
{
f
∈
M
(
X
)
∣
f
(
x
+
x
)
=
f
(
x
)
+
f
(
x
)
∀
x
∈
X
}
I=\{f\in M(X)|f(x+x)=f(x)+f(x)\forall x\in X\}
I
=
{
f
∈
M
(
X
)
∣
f
(
x
+
x
)
=
f
(
x
)
+
f
(
x
)
∀
x
∈
X
}
, find
∣
I
∣
|I|
∣
I
∣
.
2
1
Hide problems
|f(\alpha)-g(\alpha)|
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers and
α
\alpha
α
be any real number. Denote
f
(
α
)
=
a
b
c
(
a
α
+
b
α
+
c
α
)
,
g
(
α
)
=
a
2
+
α
(
b
+
c
−
a
)
+
b
2
+
α
(
−
b
+
c
+
a
)
+
c
2
+
α
(
b
−
c
+
a
)
f(\alpha)=abc(a^{\alpha}+b^{\alpha}+c^{\alpha}), g(\alpha)=a^{2+\alpha}(b+c-a)+b^{2+\alpha}(-b+c+a)+c^{2+\alpha}(b-c+a)
f
(
α
)
=
ab
c
(
a
α
+
b
α
+
c
α
)
,
g
(
α
)
=
a
2
+
α
(
b
+
c
−
a
)
+
b
2
+
α
(
−
b
+
c
+
a
)
+
c
2
+
α
(
b
−
c
+
a
)
. Determine
min
∣
f
(
α
)
−
g
(
α
)
∣
\min{|f(\alpha)-g(\alpha)|}
min
∣
f
(
α
)
−
g
(
α
)
∣
and
max
∣
f
(
α
)
−
g
(
α
)
∣
\max{|f(\alpha)-g(\alpha)|}
max
∣
f
(
α
)
−
g
(
α
)
∣
, if they are exists.
1
1
Hide problems
quadrilateral with $AD=BC$
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with
A
D
=
B
C
AD=BC
A
D
=
BC
and
A
^
+
B
^
=
12
0
0
\widehat{A}+\widehat{B}=120^{0}
A
+
B
=
12
0
0
. Let us draw equilateral
A
C
P
,
D
C
Q
,
D
B
R
ACP,DCQ,DBR
A
CP
,
D
CQ
,
D
BR
away from
A
B
AB
A
B
. Prove that the points
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
are collinear.
6
1
Hide problems
$T_n(x)=\frac{1}{2^n}[(x+\sqrt{1-x^2})^n+(x-\sqrt{1-x^2})^n]
For
−
1
≤
x
≤
1
-1\leq x\leq 1
−
1
≤
x
≤
1
and
n
∈
N
n\in\mathbb N
n
∈
N
define
T
n
(
x
)
=
1
2
n
[
(
x
+
1
−
x
2
)
n
+
(
x
−
1
−
x
2
)
n
]
T_{n}(x)=\frac{1}{2^{n}}[(x+\sqrt{1-x^{2}})^{n}+(x-\sqrt{1-x^{2}})^{n}]
T
n
(
x
)
=
2
n
1
[(
x
+
1
−
x
2
)
n
+
(
x
−
1
−
x
2
)
n
]
. a)Prove that
T
n
T_{n}
T
n
is a monic polynomial of degree
n
n
n
in
x
x
x
and that the maximum value of
∣
T
n
(
x
)
∣
|T_{n}(x)|
∣
T
n
(
x
)
∣
is
1
2
n
−
1
\frac{1}{2^{n-1}}
2
n
−
1
1
. b)Suppose that
p
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
1
x
+
a
0
∈
R
[
x
]
p(x)=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\in\mathbb{R}[x]
p
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
1
x
+
a
0
∈
R
[
x
]
is a monic polynomial of degree
n
n
n
such that
p
(
x
)
>
−
1
2
n
−
1
p(x)>-\frac{1}{2^{n-1}}
p
(
x
)
>
−
2
n
−
1
1
forall
x
x
x
,
−
1
≤
x
≤
1
-1\leq x\leq 1
−
1
≤
x
≤
1
. Prove that there exists
x
0
x_{0}
x
0
,
−
1
≤
x
0
≤
1
-1\leq x_{0}\leq 1
−
1
≤
x
0
≤
1
such that
p
(
x
0
)
≥
1
2
n
−
1
p(x_{0})\geq\frac{1}{2^{n-1}}
p
(
x
0
)
≥
2
n
−
1
1
.
3
1
Hide problems
$a$ in base $5$ contains at least $1994$ nonzero digits
Let
a
a
a
be a positive integer such that
5
1994
−
1
∣
a
5^{1994}-1\mid a
5
1994
−
1
∣
a
. Prove that the expression of
a
a
a
in base
5
5
5
contains at least
1994
1994
1994
nonzero digits.