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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1986 Vietnam National Olympiad
1986 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
2
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Compute M(n + 2)
Suppose
M
(
y
)
M(y)
M
(
y
)
is a polynomial of degree
n
n
n
such that M(y) \equal{} 2^y for y \equal{} 1, 2, \ldots, n \plus{} 1. Compute M(n \plus{} 2).
Find the n-th term of the sequence
A sequence of positive integers is constructed as follows: the first term is
1
1
1
, the following two terms are
2
2
2
,
4
4
4
, the following three terms are
5
5
5
,
7
7
7
,
9
9
9
, the following four terms are
10
10
10
,
12
12
12
,
14
14
14
,
16
16
16
, etc. Find the
n
n
n
-th term of the sequence.
2
2
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Geometry Inequality
Let
R
R
R
,
r
r
r
be respectively the circumradius and inradius of a regular
1986
1986
1986
-gonal pyramid. Prove that \frac{R}{r}\ge 1\plus{}\frac{1}{\cos\frac{\pi}{1986}} and find the total area of the surface of the pyramid when the equality occurs.
Find n s.t the inequality holds for all real numbers x_i
Find all
n
>
1
n > 1
n
>
1
such that the inequality \sum_{i\equal{}1}^nx_i^2\ge x_n\sum_{i\equal{}1}^{n\minus{}1}x_i holds for all real numbers
x
1
x_1
x
1
,
x
2
x_2
x
2
,
…
\ldots
…
,
x
n
x_n
x
n
.
1
2
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Inequality
Let
1
2
≤
a
1
,
a
2
,
…
,
a
n
≤
5
\frac{1}{2}\le a_1, a_2, \ldots, a_n \le 5
2
1
≤
a
1
,
a
2
,
…
,
a
n
≤
5
be given real numbers and let
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots, x_n
x
1
,
x
2
,
…
,
x
n
be real numbers satisfying 4x_i^2\minus{} 4a_ix_i \plus{} \left(a_i \minus{} 1\right)^2 \le 0. Prove that \sqrt{\sum_{i\equal{}1}^n\frac{x_i^2}{n}}\le\sum_{i\equal{}1}^n\frac{x_i}{n}\plus{}1
Find the locus of points
Let
A
B
C
D
ABCD
A
BC
D
be a square of side
2
a
2a
2
a
. An equilateral triangle
A
M
B
AMB
A
MB
is constructed in the plane through
A
B
AB
A
B
perpendicular to the plane of the square. A point
S
S
S
moves on
A
B
AB
A
B
such that SB\equal{}x. Let
P
P
P
be the projection of
M
M
M
on
S
C
SC
SC
and
E
E
E
,
O
O
O
be the midpoints of
A
B
AB
A
B
and
C
M
CM
CM
respectively. (a) Find the locus of
P
P
P
as
S
S
S
moves on
A
B
AB
A
B
. (b) Find the maximum and minimum lengths of
S
O
SO
SO
.