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Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1968 Vietnam National Olympiad
1968 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(2)
1
1
Hide problems
2 roots of f_n(x) =x^2 - b^nx- a^n in (-1,1), for every n
Let
a
a
a
and
b
b
b
satisfy
a
≥
b
>
0
,
a
+
b
=
1
a \ge b >0, a + b = 1
a
≥
b
>
0
,
a
+
b
=
1
. i) Prove that if
m
m
m
and
n
n
n
are positive integers with
m
<
n
m < n
m
<
n
, then
a
m
−
a
n
≥
b
m
−
b
n
>
0
a^m - a^n \ge b^m- b^n > 0
a
m
−
a
n
≥
b
m
−
b
n
>
0
. ii) For each positive integer
n
n
n
, consider a quadratic function
f
n
(
x
)
=
x
2
−
b
n
x
−
a
n
f_n(x) = x^2 - b^nx- a^n
f
n
(
x
)
=
x
2
−
b
n
x
−
a
n
. Show that
f
(
x
)
f(x)
f
(
x
)
has two roots that are in between
−
1
-1
−
1
and
1
1
1
.
2
1
Hide problems
VietNam MO 1968-Pr.2
L
L
L
and
M
M
M
are two parallel lines a distance
d
d
d
apart. Given
r
r
r
and
x
x
x
, construct a triangle
A
B
C
ABC
A
BC
, with
A
A
A
on
L
L
L
, and
B
B
B
and
C
C
C
on
M
M
M
, such that the inradius is
r
r
r
, and angle
A
=
x
A = x
A
=
x
. Calculate angles
B
B
B
and
C
C
C
in terms of
d
d
d
,
r
r
r
and
x
x
x
. If the incircle touches the side
B
C
BC
BC
at
D
D
D
, find a relation between
B
D
BD
B
D
and
D
C
DC
D
C