MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1964 Vietnam National Olympiad
1964 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(4)
4
1
Hide problems
f_n = (a^{n+1} - b^{n+1})/\sqrt5, where a, b are real such a+b=1, ab=-1, a>b
Define the sequence of positive integers
f
n
f_n
f
n
by
f
0
=
1
,
f
1
=
1
,
f
n
+
2
=
f
n
+
1
+
f
n
f_0 = 1, f_1 = 1, f_{n+2} = f_{n+1} + f_n
f
0
=
1
,
f
1
=
1
,
f
n
+
2
=
f
n
+
1
+
f
n
. Show that
f
n
=
(
a
n
+
1
−
b
n
+
1
)
5
f_n =\frac{ (a^{n+1} - b^{n+1})}{\sqrt5}
f
n
=
5
(
a
n
+
1
−
b
n
+
1
)
, where
a
,
b
a, b
a
,
b
are real numbers such that
a
+
b
=
1
,
a
b
=
−
1
a + b = 1, ab = -1
a
+
b
=
1
,
ab
=
−
1
and
a
>
b
a > b
a
>
b
.
3
1
Hide problems
old vietanmese locus in space
Let
P
P
P
be a plane and two points
A
∈
(
P
)
,
O
∉
(
P
)
A \in (P),O \notin (P)
A
∈
(
P
)
,
O
∈
/
(
P
)
. For each line in
(
P
)
(P)
(
P
)
through
A
A
A
, let
H
H
H
be the foot of the perpendicular from
O
O
O
to the line. Find the locus
(
c
)
(c)
(
c
)
of
H
H
H
. Denote by
(
C
)
(C)
(
C
)
the oblique cone with peak
O
O
O
and base
(
c
)
(c)
(
c
)
. Prove that all planes, either parallel to
(
P
)
(P)
(
P
)
or perpendicular to
O
A
OA
O
A
, intersect
(
C
)
(C)
(
C
)
by circles. Consider the two symmetric faces of
(
C
)
(C)
(
C
)
that intersect
(
C
)
(C)
(
C
)
by the angles
α
\alpha
α
and
β
\beta
β
respectively. Find a relation between
α
\alpha
α
and
β
\beta
β
.
2
1
Hide problems
find number of roots x + | x^2 - 1 | = k, k real
Draw the graph of the functions
y
=
∣
x
2
−
1
∣
y = | x^2 - 1 |
y
=
∣
x
2
−
1∣
and
y
=
x
+
∣
x
2
−
1
∣
y = x + | x^2 -1 |
y
=
x
+
∣
x
2
−
1∣
. Find the number of roots of the equation
x
+
∣
x
2
−
1
∣
=
k
x + | x^2 - 1 | = k
x
+
∣
x
2
−
1∣
=
k
, where
k
k
k
is a real constant.
1
1
Hide problems
cos a + cos (a +2\pi /3 ) + cos (a+4\pi /3)
Given an arbitrary angle
α
\alpha
α
, compute
c
o
s
α
+
c
o
s
(
α
+
2
π
3
)
+
c
o
s
(
α
+
4
π
3
)
cos \alpha + cos \big( \alpha +\frac{2\pi }{3 }\big) + cos \big( \alpha +\frac{4\pi }{3 }\big)
cos
α
+
cos
(
α
+
3
2
π
)
+
cos
(
α
+
3
4
π
)
and
s
i
n
α
+
s
i
n
(
α
+
2
π
3
)
+
s
i
n
(
α
+
4
π
3
)
sin \alpha + sin \big( \alpha +\frac{2\pi }{3 } \big) + sin \big( \alpha +\frac{4\pi }{3 } \big)
s
in
α
+
s
in
(
α
+
3
2
π
)
+
s
in
(
α
+
3
4
π
)
. Generalize this result and justify your answer.