MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1979 Vietnam National Olympiad
1979 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(6)
5
1
Hide problems
x^2 - 2x[x] +x - k= 0 , find real k, so that it has >=2 non-negative roots
Find all real numbers
k
k
k
such that
x
2
−
2
x
[
x
]
+
x
−
k
=
0
x^2 - 2 x [x] + x - k = 0
x
2
−
2
x
[
x
]
+
x
−
k
=
0
has at least two non-negative roots.
6
1
Hide problems
find 2 points in plane so that t4 points form a regular tetrahedron
A
B
C
D
ABCD
A
BC
D
is a rectangle with
B
C
/
A
B
=
2
BC / AB = \sqrt2
BC
/
A
B
=
2
.
A
B
E
F
ABEF
A
BEF
is a congruent rectangle in a different plane. Find the angle
D
A
F
DAF
D
A
F
such that the lines
C
A
CA
C
A
and
B
F
BF
BF
are perpendicular. In this configuration, find two points on the line
C
A
CA
C
A
and two points on the line
B
F
BF
BF
so that the four points form a regular tetrahedron.
4
1
Hide problems
there is a polynomial p(x) such that p(2 cos x) = 2 cos nx., for every n>0
For each integer
n
>
0
n > 0
n
>
0
show that there is a polynomial
p
(
x
)
p(x)
p
(
x
)
such that
p
(
2
c
o
s
x
)
=
2
c
o
s
n
x
p(2 cos x) = 2 cos nx
p
(
2
cos
x
)
=
2
cos
n
x
.
3
1
Hide problems
find a point so that ratio of areas = ratio of perimeters
A
B
C
ABC
A
BC
is a triangle. Find a point
X
X
X
on
B
C
BC
BC
such that : area
A
B
X
ABX
A
BX
/ area
A
C
X
ACX
A
CX
= perimeter
A
B
X
ABX
A
BX
/ perimeter
A
C
X
ACX
A
CX
.
2
1
Hide problems
x^3 + ax^2 + bx + c , x^3 +α^3 x^2 + β^3 x + γ^3, 3 real roots each, related
Find all real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
such that
x
3
+
a
x
2
+
b
x
+
c
x^3 + ax^2 + bx + c
x
3
+
a
x
2
+
b
x
+
c
has three real roots
α
,
β
,
γ
\alpha, \beta,\gamma
α
,
β
,
γ
(not necessarily all distinct) and the equation
x
3
+
α
3
x
2
+
β
3
x
+
γ
3
x^3 + \alpha^3 x^2 + \beta^3 x + \gamma^3
x
3
+
α
3
x
2
+
β
3
x
+
γ
3
has roots
α
3
,
β
3
,
γ
3
\alpha^3, \beta^3,\gamma^3
α
3
,
β
3
,
γ
3
.
1
1
Hide problems
triangle with sides x^4+ x^3+ 2x^2+ x+ 1, 2x^3+ x^2+ 2x+ 1, x^4 - 1 if x>1
Show that for all
x
>
1
x > 1
x
>
1
there is a triangle with sides,
x
4
+
x
3
+
2
x
2
+
x
+
1
,
2
x
3
+
x
2
+
2
x
+
1
,
x
4
−
1.
x^4 + x^3 + 2x^2 + x + 1, 2x^3 + x^2 + 2x + 1, x^4 - 1.
x
4
+
x
3
+
2
x
2
+
x
+
1
,
2
x
3
+
x
2
+
2
x
+
1
,
x
4
−
1.