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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1963 Vietnam National Olympiad
1963 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(5)
5
1
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Find the side length and the area
The triangle
A
B
C
ABC
A
BC
has semiperimeter
p
p
p
. Find the side length
B
C
BC
BC
and the area
S
S
S
in terms of
∠
A
\angle A
∠
A
,
∠
B
\angle B
∠
B
and
p
p
p
. In particular, find
S
S
S
if
p
≈
23.6
p \approx 23.6
p
≈
23.6
,
∠
A
≈
5
2
∘
4
2
′
\angle A \approx 52^{\circ}42'
∠
A
≈
5
2
∘
4
2
′
,
∠
B
≈
4
6
∘
1
6
′
\angle B \approx 46^{\circ}16'
∠
B
≈
4
6
∘
1
6
′
.
4
1
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Volume of the tetrahedron
The tetrahedron
S
.
A
B
C
S.ABC
S
.
A
BC
has the faces
S
B
C
SBC
SBC
and
A
B
C
ABC
A
BC
perpendicular. The three angles at
S
S
S
are all
6
0
∘
60^{\circ}
6
0
∘
and SB \equal{} SC \equal{} 1. Find the volume of the tetrahedron.
3
1
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Another trigonometric equation
Solve the equation \sin^3x \cos 3x \plus{} \cos^3x \sin 3x \equal{} \frac{3}{8}.
2
1
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Roots of an equation
For what values of
m
m
m
does the equation x^2 \plus{} (2m \plus{} 6)x \plus{} 4m \plus{} 12 \equal{} 0 has two real roots, both of them greater than \minus{}1.
1
1
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Men and women in a conference
A conference has
47
47
47
people attending. One woman knows
16
16
16
of the men who are attending, another knows
17
17
17
, and so on up to the
m
m
m
-th woman who knows all
n
n
n
of the men who are attending. Find
m
m
m
and
n
n
n
.