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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2000 Vietnam National Olympiad
2000 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
2
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Vietnam NMO 2000_3
Consider the polynomial P(x) \equal{} x^3 \plus{} 153x^2 \minus{} 111x \plus{} 38. (a) Prove that there are at least nine integers
a
a
a
in the interval
[
1
,
3
2000
]
[1, 3^{2000}]
[
1
,
3
2000
]
for which
P
(
a
)
P(a)
P
(
a
)
is divisible by
3
2000
3^{2000}
3
2000
. (b) Find the number of integers
a
a
a
in
[
1
,
3
2000
]
[1, 3^{2000}]
[
1
,
3
2000
]
with the property from (a).
Vietnam NMO 2000_6
Let
P
(
x
)
P(x)
P
(
x
)
be a nonzero polynomial such that, for all real numbers
x
x
x
, P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x). Determine the maximum possible number of real roots of
P
(
x
)
P(x)
P
(
x
)
.
1
2
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Vietnam NMO 2000_1
Given a real number
c
>
0
c > 0
c
>
0
, a sequence
(
x
n
)
(x_n)
(
x
n
)
of real numbers is defined by x_{n \plus{} 1} \equal{} \sqrt {c \minus{} \sqrt {c \plus{} x_n}} for
n
≥
0
n \ge 0
n
≥
0
. Find all values of
c
c
c
such that for each initial value
x
0
x_0
x
0
in
(
0
,
c
)
(0, c)
(
0
,
c
)
, the sequence
(
x
n
)
(x_n)
(
x
n
)
is defined for all
n
n
n
and has a finite limit
lim
x
n
\lim x_n
lim
x
n
when n\to \plus{} \infty.
Vietnam NMO 2000_4
For every integer
n
≥
3
n \ge 3
n
≥
3
and any given angle
α
\alpha
α
with
0
<
α
<
π
0 < \alpha < \pi
0
<
α
<
π
, let P_n(x) \equal{} x^n \sin\alpha \minus{} x \sin n\alpha \plus{} \sin(n \minus{} 1)\alpha. (a) Prove that there is a unique polynomial of the form f(x) \equal{} x^2 \plus{} ax \plus{} b which divides
P
n
(
x
)
P_n(x)
P
n
(
x
)
for every
n
≥
3
n \ge 3
n
≥
3
. (b) Prove that there is no polynomial g(x) \equal{} x \plus{} c which divides
P
n
(
x
)
P_n(x)
P
n
(
x
)
for every
n
≥
3
n \ge 3
n
≥
3
.
2
2
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Vietnam NMO 2000_2
Two circles
(
O
1
)
(O_1)
(
O
1
)
and
(
O
2
)
(O_2)
(
O
2
)
with respective centers
O
1
O_1
O
1
,
O
2
O_2
O
2
are given on a plane. Let
M
1
M_1
M
1
,
M
2
M_2
M
2
be points on
(
O
1
)
(O_1)
(
O
1
)
,
(
O
2
)
(O_2)
(
O
2
)
respectively, and let the lines
O
1
M
1
O_1M_1
O
1
M
1
and
O
2
M
2
O_2M_2
O
2
M
2
meet at
Q
Q
Q
. Starting simultaneously from these positions, the points
M
1
M_1
M
1
and
M
2
M_2
M
2
move clockwise on their own circles with the same angular velocity. (a) Determine the locus of the midpoint of
M
1
M
2
M_1M_2
M
1
M
2
. (b) Prove that the circumcircle of
△
M
1
Q
M
2
\triangle M_1QM_2
△
M
1
Q
M
2
passes through a fixed point.
Vietnam NMO 2000_5
Find all integers
n
≥
3
n \ge 3
n
≥
3
such that there are
n
n
n
points in space, with no three on a line and no four on a circle, such that all the circles pass through three points between them are congruent.