MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1965 Vietnam National Olympiad
1965 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
2
1
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points in space with the same locus, circumcenter of a tetrahedron related
A
B
AB
A
B
and
C
D
CD
C
D
are two fixed parallel chords of the circle
S
S
S
.
M
M
M
is a variable point on the circle.
Q
Q
Q
is the intersection of the lines
M
D
MD
M
D
and
A
B
AB
A
B
.
X
X
X
is the circumcenter of the triangle
M
C
Q
MCQ
MCQ
. Find the locus of
X
X
X
. What happens to
X
X
X
as
M
M
M
tends to (1)
D
D
D
, (2)
C
C
C
? Find a point
E
E
E
outside the plane of
S
S
S
such that the circumcenter of the tetrahedron
M
C
Q
E
MCQE
MCQE
has the same locus as
X
X
X
.
1
1
Hide problems
minimum distance of 2 ships, a navy one and an enemy one
At a time
t
=
0
t = 0
t
=
0
, a navy ship is at a point
O
O
O
, while an enemy ship is at a point
A
A
A
cruising with speed
v
v
v
perpendicular to
O
A
=
a
OA = a
O
A
=
a
. The speed and direction of the enemy ship do not change. The strategy of the navy ship is to travel with constant speed
u
u
u
at a angle
0
<
ϕ
<
π
/
2
0 < \phi < \pi /2
0
<
ϕ
<
π
/2
to the line
O
A
OA
O
A
. 1) Let
ϕ
\phi
ϕ
be chosen. What is the minimum distance between the two ships? Under what conditions will the distance vanish? 2) If the distance does not vanish, what is the choice of
ϕ
\phi
ϕ
to minimize the distance? What are directions of the two ships when their distance is minimum?
3
1
Hide problems
min of Σx_i^m from i=1 to n, where Σx_i=k, x_i nonnegative real, m,n positive
1) Two nonnegative real numbers
x
,
y
x, y
x
,
y
have constant sum
a
a
a
. Find the minimum value of
x
m
+
y
m
x^m + y^m
x
m
+
y
m
, where m is a given positive integer. 2) Let
m
,
n
m, n
m
,
n
be positive integers and
k
k
k
a positive real number. Consider nonnegative real numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, . . . , x_n
x
1
,
x
2
,
...
,
x
n
having constant sum
k
k
k
. Prove that the minimum value of the quantity
x
1
m
+
.
.
.
+
x
n
m
x^m_1+ ... + x^m_n
x
1
m
+
...
+
x
n
m
occurs when
x
1
=
x
2
=
.
.
.
=
x
n
x_1 = x_2 = ... = x_n
x
1
=
x
2
=
...
=
x
n
.