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Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1962 Vietnam National Olympiad
1962 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(5)
5
1
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Trigonometric equation
Solve the equation \sin^6x \plus{} \cos^6x \equal{} \frac{1}{4}.
4
1
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Find the volume of the tetrahedron
Let be given a tetrahedron
A
B
C
D
ABCD
A
BC
D
such that triangle
B
C
D
BCD
BC
D
equilateral and AB \equal{} AC \equal{} AD. The height is
h
h
h
and the angle between two planes
A
B
C
ABC
A
BC
and
B
C
D
BCD
BC
D
is
α
\alpha
α
. The point
X
X
X
is taken on
A
B
AB
A
B
such that the plane
X
C
D
XCD
XC
D
is perpendicular to
A
B
AB
A
B
. Find the volume of the tetrahedron
X
B
C
D
XBCD
XBC
D
.
3
1
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Tetrahedron
Let
A
B
C
D
ABCD
A
BC
D
is a tetrahedron. Denote by
A
′
A'
A
′
,
B
′
B'
B
′
the feet of the perpendiculars from
A
A
A
and
B
B
B
, respectively to the opposite faces. Show that
A
A
′
AA'
A
A
′
and
B
B
′
BB'
B
B
′
intersect if and only if
A
B
AB
A
B
is perpendicular to
C
D
CD
C
D
. Do they intersect if AC \equal{} AD \equal{} BC \equal{} BD?
2
1
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Determine derivative
Let f(x) \equal{} (1 \plus{} x)\cdot\sqrt{(2 \plus{} x^2)}\cdot\sqrt[3]{(3 \plus{} x^3)}. Determine
f
′
(
1
)
f'(1)
f
′
(
1
)
.
1
1
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Inequality for positive real numbers
Prove that for positive real numbers
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
, we have \frac{1}{\frac{1}{a}\plus{}\frac{1}{b}}\plus{}\frac{1}{\frac{1}{c}\plus{}\frac{1}{d}}\le\frac{1}{\frac{1}{a\plus{}c}\plus{}\frac{1}{b\plus{}d}}