MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1993 Vietnam National Olympiad
1993 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
2
2
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locus of center parallelogram
A
B
C
D
ABCD
A
BC
D
is a quadrilateral such that
A
B
AB
A
B
is not parallel to
C
D
CD
C
D
, and
B
C
BC
BC
is not parallel to
A
D
AD
A
D
. Variable points
P
,
Q
,
R
,
S
P, Q, R, S
P
,
Q
,
R
,
S
are taken on
A
B
,
B
C
,
C
D
,
D
A
AB, BC, CD, DA
A
B
,
BC
,
C
D
,
D
A
respectively so that
P
Q
R
S
PQRS
PQRS
is a parallelogram. Find the locus of its center.
$1993$ points are arranged in a circle
1993
1993
1993
points are arranged in a circle. At time
0
0
0
each point is arbitrarily labeled
+
1
+1
+
1
or
−
1
-1
−
1
. At times
n
=
1
,
2
,
3
,
.
.
.
n = 1, 2, 3, ...
n
=
1
,
2
,
3
,
...
the vertices are relabeled. At time
n
n
n
a vertex is given the label
+
1
+1
+
1
if its two neighbours had the same label at time
n
−
1
n-1
n
−
1
, and it is given the label
−
1
-1
−
1
if its two neighbours had different labels at time
n
−
1
n-1
n
−
1
. Show that for some time
n
>
1
n > 1
n
>
1
the labeling will be the same as at time
1.
1.
1.
3
2
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find a function
Find a function
f
(
n
)
f(n)
f
(
n
)
on the positive integers with positive integer values such that
f
(
f
(
n
)
)
=
1993
n
1945
f( f(n) ) = 1993 n^{1945}
f
(
f
(
n
))
=
1993
n
1945
for all
n
n
n
.
two sequence
Define the sequences
a
0
,
a
1
,
a
2
,
.
.
.
a_{0}, a_{1}, a_{2}, ...
a
0
,
a
1
,
a
2
,
...
and
b
0
,
b
1
,
b
2
,
.
.
.
b_{0}, b_{1}, b_{2}, ...
b
0
,
b
1
,
b
2
,
...
by
a
0
=
2
,
b
0
=
1
,
a
n
+
1
=
2
a
n
b
n
/
(
a
n
+
b
n
)
,
b
n
+
1
=
a
n
+
1
b
n
a_{0}= 2, b_{0}= 1, a_{n+1}= 2a_{n}b_{n}/(a_{n}+b_{n}), b_{n+1}= \sqrt{a_{n+1}b_{n}}
a
0
=
2
,
b
0
=
1
,
a
n
+
1
=
2
a
n
b
n
/
(
a
n
+
b
n
)
,
b
n
+
1
=
a
n
+
1
b
n
. Show that the two sequences converge to the same limit, and find the limit.
1
2
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min and max
f
:
[
−
1995
,
1995
]
→
R
f : [-\sqrt{1995},\sqrt{1995}] \to\mathbb{R}
f
:
[
−
1995
,
1995
]
→
R
is defined by
f
(
x
)
=
x
(
1993
+
1995
−
x
2
)
f(x) = x(1993+\sqrt{1995-x^{2}})
f
(
x
)
=
x
(
1993
+
1995
−
x
2
)
. Find its maximum and minimum values.
right tetrahedron at A
The tetrahedron
A
B
C
D
ABCD
A
BC
D
has its vertices on the fixed sphere
S
S
S
. Prove that
A
B
2
+
A
C
2
+
A
D
2
−
B
C
2
−
B
D
2
−
C
D
2
AB^{2}+AC^{2}+AD^{2}-BC^{2}-BD^{2}-CD^{2}
A
B
2
+
A
C
2
+
A
D
2
−
B
C
2
−
B
D
2
−
C
D
2
is minimum iff
A
B
⊥
A
C
,
A
C
⊥
A
D
,
A
D
⊥
A
B
AB\perp AC,AC\perp AD,AD\perp AB
A
B
⊥
A
C
,
A
C
⊥
A
D
,
A
D
⊥
A
B
.