MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1975 Vietnam National Olympiad
1975 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(5)
5
1
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min and max of tan(3x)/tan^3x in (0, \pi /2)
Show that the sum of the (local) maximum and minimum values of the function
t
a
n
(
3
x
)
t
a
n
3
x
\frac{tan(3x)}{tan^3x}
t
a
n
3
x
t
an
(
3
x
)
on the interval
(
0
,
π
2
)
\big(0, \frac{\pi }{2}\big)
(
0
,
2
π
)
is rational.
4
1
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all terms of the arithmetic progression $-1, 18, 37, 56, ...$ whose only digit
Find all terms of the arithmetic progression
−
1
,
18
,
37
,
56
,
.
.
.
-1, 18, 37, 56, ...
−
1
,
18
,
37
,
56
,
...
whose only digit is
5
5
5
.
3
1
Hide problems
vol KOAC/vol KOBD = AC/BD iff 2AC·BD = AB^2 in a tetrahedron ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron with
B
A
⊥
A
C
,
D
B
⊥
(
B
A
C
)
BA \perp AC,DB \perp (BAC)
B
A
⊥
A
C
,
D
B
⊥
(
B
A
C
)
. Denote by
O
O
O
the midpoint of
A
B
AB
A
B
, and
K
K
K
the foot of the perpendicular from
O
O
O
to
D
C
DC
D
C
. Suppose that
A
C
=
B
D
AC = BD
A
C
=
B
D
. Prove that
V
K
O
A
C
V
K
O
B
D
=
A
C
B
D
\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}
V
K
OB
D
V
K
O
A
C
=
B
D
A
C
if and only if
2
A
C
⋅
B
D
=
A
B
2
2AC \cdot BD = AB^2
2
A
C
⋅
B
D
=
A
B
2
.
1
1
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Find a^8 + b^8 + c^8, if a, b, c are roots of x^3 - x + 1 = 0
The roots of the equation
x
3
−
x
+
1
=
0
x^3 - x + 1 = 0
x
3
−
x
+
1
=
0
are
a
,
b
,
c
a, b, c
a
,
b
,
c
. Find
a
8
+
b
8
+
c
8
a^8 + b^8 + c^8
a
8
+
b
8
+
c
8
.
2
1
Hide problems
VIETNAM MO 1975
Solve this equation
y
3
+
m
3
(
y
+
m
)
3
+
y
3
+
n
3
(
y
+
n
)
3
+
y
3
+
p
3
(
y
+
p
)
3
−
3
2
+
3
2
.
y
−
m
y
+
m
.
y
−
n
y
+
n
.
y
−
p
y
+
p
=
0
\frac{y^{3}+m^{3}}{\left ( y+m \right )^{3}}+\frac{y^{3}+n^{3}}{\left ( y+n \right )^{3}}+\frac{y^{3}+p^{3}}{\left ( y+p \right )^{3}}-\frac{3}{2}+\frac{3}{2}.\frac{y-m}{y+m}.\frac{y-n}{y+n}.\frac{y-p}{y+p}=0
(
y
+
m
)
3
y
3
+
m
3
+
(
y
+
n
)
3
y
3
+
n
3
+
(
y
+
p
)
3
y
3
+
p
3
−
2
3
+
2
3
.
y
+
m
y
−
m
.
y
+
n
y
−
n
.
y
+
p
y
−
p
=
0