MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
Taiwan National Olympiad
1996 Taiwan National Olympiad
1996 Taiwan National Olympiad
Part of
Taiwan National Olympiad
Subcontests
(6)
3
1
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Find the locus
Let be given points
A
,
B
A,B
A
,
B
on a circle and let
P
P
P
be a variable point on that circle. Let point
M
M
M
be determined by
P
P
P
as the point that is either on segment
P
A
PA
P
A
with
A
M
=
M
P
+
P
B
AM=MP+PB
A
M
=
MP
+
PB
or on segment
P
B
PB
PB
with
A
P
+
M
P
=
P
B
AP+MP=PB
A
P
+
MP
=
PB
. Find the locus of points
M
M
M
.
6
1
Hide problems
polynomial and integer sequence
Let
q
0
,
q
1
,
.
.
.
q_{0},q_{1},...
q
0
,
q
1
,
...
be a sequence of integers such that a) for any
m
>
n
m>n
m
>
n
we have
m
−
n
∣
q
m
−
q
n
m-n\mid q_{m}-q_{n}
m
−
n
∣
q
m
−
q
n
, and b)
∣
q
n
∣
≤
n
10
,
∀
n
≥
0
|q_{n}|\leq n^{10}, \ \forall n\geq 0
∣
q
n
∣
≤
n
10
,
∀
n
≥
0
.Prove there exists a polynomial
Q
Q
Q
such that
q
n
=
Q
(
n
)
,
∀
n
≥
0
q_{n}=Q(n), \ \forall n\geq 0
q
n
=
Q
(
n
)
,
∀
n
≥
0
.
5
1
Hide problems
|a_k-a_{k-1}|
Dertemine integers
a
1
,
a
2
,
.
.
.
,
a
99
=
a
0
a_{1},a_{2},...,a_{99}=a_{0}
a
1
,
a
2
,
...
,
a
99
=
a
0
satisfying
∣
a
k
−
a
k
−
1
∣
≥
1996
|a_{k}-a_{k-1}|\geq 1996
∣
a
k
−
a
k
−
1
∣
≥
1996
for all
k
=
1
,
2
,
.
.
.
,
99
k=1,2,...,99
k
=
1
,
2
,
...
,
99
, such that
m
=
max
1
≤
k
≤
99
∣
a
k
−
a
k
−
1
∣
m=\max_{1\leq k\leq 99} |a_{k}-a_{k-1}|
m
=
max
1
≤
k
≤
99
∣
a
k
−
a
k
−
1
∣
is minimum possible, and find the minimum value
m
∗
m^{*}
m
∗
of
m
m
m
.
4
1
Hide problems
not all the roots are real
Show that for any real numbers
a
3
,
a
4
,
.
.
.
,
a
85
a_{3},a_{4},...,a_{85}
a
3
,
a
4
,
...
,
a
85
, not all the roots of the equation
a
85
x
85
+
a
84
x
84
+
.
.
.
+
a
3
x
3
+
3
x
2
+
2
x
+
1
=
0
a_{85}x^{85}+a_{84}x^{84}+...+a_{3}x^{3}+3x^{2}+2x+1=0
a
85
x
85
+
a
84
x
84
+
...
+
a
3
x
3
+
3
x
2
+
2
x
+
1
=
0
are real.
2
1
Hide problems
a\leq a_i\leq\frac{1}{a_i}
Let
0
<
a
≤
1
0<a\leq 1
0
<
a
≤
1
be a real number and let
a
≤
a
i
≤
1
a
i
∀
i
=
1
,
1996
‾
a\leq a_{i}\leq\frac{1}{a_{i}}\forall i=\overline{1,1996}
a
≤
a
i
≤
a
i
1
∀
i
=
1
,
1996
are real numbers. Prove that for any nonnegative real numbers
k
i
(
i
=
1
,
2
,
.
.
.
,
1996
)
k_{i}(i=1,2,...,1996)
k
i
(
i
=
1
,
2
,
...
,
1996
)
such that
∑
i
=
1
1996
k
i
=
1
\sum_{i=1}^{1996}k_{i}=1
∑
i
=
1
1996
k
i
=
1
we have
(
∑
i
=
1
1996
k
i
a
i
)
(
∑
i
=
1
1996
k
i
a
i
)
≤
(
a
+
1
a
)
2
(\sum_{i=1}^{1996}k_{i}a_{i})(\sum_{i=1}^{1996}\frac{k_{i}}{a_{i}})\leq (a+\frac{1}{a})^{2}
(
∑
i
=
1
1996
k
i
a
i
)
(
∑
i
=
1
1996
a
i
k
i
)
≤
(
a
+
a
1
)
2
.
1
1
Hide problems
find x,y,z
Suppose that
a
,
b
,
c
a,b,c
a
,
b
,
c
are real numbers in
(
0
,
π
2
)
(0,\frac{\pi}{2})
(
0
,
2
π
)
such that
a
+
b
+
c
=
π
4
a+b+c=\frac{\pi}{4}
a
+
b
+
c
=
4
π
and
tan
a
=
1
x
,
tan
b
=
1
y
,
tan
c
=
1
z
\tan{a}=\frac{1}{x},\tan{b}=\frac{1}{y},\tan{c}=\frac{1}{z}
tan
a
=
x
1
,
tan
b
=
y
1
,
tan
c
=
z
1
, where
x
,
y
,
z
x,y,z
x
,
y
,
z
are positive integer numbers. Find
x
,
y
,
z
x,y,z
x
,
y
,
z
.