MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
Taiwan National Olympiad
1995 Taiwan National Olympiad
1995 Taiwan National Olympiad
Part of
Taiwan National Olympiad
Subcontests
(6)
6
1
Hide problems
there are exactly $k$ pairs $(x_1,x_2)$...
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are integers such that
(
a
,
b
)
=
(
c
,
d
)
=
1
(a,b)=(c,d)=1
(
a
,
b
)
=
(
c
,
d
)
=
1
and
a
d
−
b
c
=
k
>
0
ad-bc=k>0
a
d
−
b
c
=
k
>
0
. Prove that there are exactly
k
k
k
pairs
(
x
1
,
x
2
)
(x_{1},x_{2})
(
x
1
,
x
2
)
of rational numbers with
0
≤
x
1
,
x
2
<
1
0\leq x_{1},x_{2}<1
0
≤
x
1
,
x
2
<
1
for which both
a
x
1
+
b
x
2
,
c
x
1
+
d
x
2
ax_{1}+bx_{2},cx_{1}+dx_{2}
a
x
1
+
b
x
2
,
c
x
1
+
d
x
2
are integers.
5
1
Hide problems
Euler's line
Let
P
P
P
be a point on the circumcircle of a triangle
A
1
A
2
A
3
A_{1}A_{2}A_{3}
A
1
A
2
A
3
, and let
H
H
H
be the orthocenter of the triangle. The feet
B
1
,
B
2
,
B
3
B_{1},B_{2},B_{3}
B
1
,
B
2
,
B
3
of the perpendiculars from
P
P
P
to
A
2
A
3
,
A
3
A
1
,
A
1
A
2
A_{2}A_{3},A_{3}A_{1},A_{1}A_{2}
A
2
A
3
,
A
3
A
1
,
A
1
A
2
lie on a line. Prove that this line bisects the segment
P
H
PH
P
H
.
4
1
Hide problems
exists plynomial with two conditions
Let
m
1
,
m
2
,
.
.
.
,
m
n
m_{1},m_{2},...,m_{n}
m
1
,
m
2
,
...
,
m
n
be mutually distinct integers. Prove that there exists a
f
(
x
)
∈
Z
[
x
]
f(x)\in\mathbb{Z}[x]
f
(
x
)
∈
Z
[
x
]
of degree
n
n
n
satisfying the following two conditions: a)
f
(
m
i
)
=
−
1
∀
i
=
1
,
2
,
.
.
.
,
n
f(m_{i})=-1\forall i=1,2,...,n
f
(
m
i
)
=
−
1∀
i
=
1
,
2
,
...
,
n
. b)
f
(
x
)
f(x)
f
(
x
)
is irreducible.
3
1
Hide problems
$n$ persons meet in a meeting
Suppose that
n
n
n
persons meet in a meeting, and that each of the persons is acquainted to exactly
8
8
8
others. Any two acquainted persons have exactly
4
4
4
common acquaintances, and any two non-acquainted persons have exactly
2
2
2
common acquaintances. Find all possible values of
n
n
n
.
2
1
Hide problems
single operation
Given a sequence of eight integers
x
1
,
x
2
,
.
.
.
,
x
8
x_{1},x_{2},...,x_{8}
x
1
,
x
2
,
...
,
x
8
in a single operation one replaces these numbers with
∣
x
1
−
x
2
∣
,
∣
x
2
−
x
3
∣
,
.
.
.
,
∣
x
8
−
x
1
∣
|x_{1}-x_{2}|,|x_{2}-x_{3}|,...,|x_{8}-x_{1}|
∣
x
1
−
x
2
∣
,
∣
x
2
−
x
3
∣
,
...
,
∣
x
8
−
x
1
∣
. Find all the eight-term sequences of integers which reduce to a sequence with all the terms equal after finitely many single operations.
1
1
Hide problems
$P(x)=a_0+a_1x+...+a_nx^n\in\mathbb{C}[x]$
Let
P
(
x
)
=
a
0
+
a
1
x
+
.
.
.
+
a
n
x
n
∈
C
[
x
]
P(x)=a_{0}+a_{1}x+...+a_{n}x^{n}\in\mathbb{C}[x]
P
(
x
)
=
a
0
+
a
1
x
+
...
+
a
n
x
n
∈
C
[
x
]
, where
a
n
=
1
a_{n}=1
a
n
=
1
. The roots of
P
(
x
)
P(x)
P
(
x
)
are
b
1
,
b
2
,
.
.
.
,
b
n
b_{1},b_{2},...,b_{n}
b
1
,
b
2
,
...
,
b
n
, where
∣
b
1
∣
,
∣
b
2
∣
,
.
.
.
,
∣
b
j
∣
>
1
|b_{1}|,|b_{2}|,...,|b_{j}|>1
∣
b
1
∣
,
∣
b
2
∣
,
...
,
∣
b
j
∣
>
1
and
∣
b
j
+
1
∣
,
.
.
.
,
∣
b
n
∣
≤
1
|b_{j+1}|,...,|b_{n}|\leq 1
∣
b
j
+
1
∣
,
...
,
∣
b
n
∣
≤
1
. Prove that
∏
i
=
1
j
∣
b
i
∣
≤
∣
a
0
∣
2
+
∣
a
1
∣
2
+
.
.
.
+
∣
a
n
∣
2
\prod_{i=1}^{j}|b_{i}|\leq\sqrt{|a_{0}|^{2}+|a_{1}|^{2}+...+|a_{n}|^{2}}
∏
i
=
1
j
∣
b
i
∣
≤
∣
a
0
∣
2
+
∣
a
1
∣
2
+
...
+
∣
a
n
∣
2
.