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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2008 Vietnam National Olympiad
2008 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(7)
6
1
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Vietnam MO 2008 Inequality
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be distinct non-negative real numbers. Prove that \frac{1}{(x\minus{}y)^2} \plus{} \frac{1}{(y\minus{}z)^2} \plus{} \frac{1}{(z\minus{}x)^2} \geq \frac{4}{xy \plus{} yz \plus{} zx}. When does the equality hold?
5
1
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Number of digits of which does not exceed 2008
What is the total number of natural numbes divisible by 9 the number of digits of which does not exceed 2008 and at least two of the digits are 9s?
4
1
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x_(n+2) = 2^(-x_n) + 0.5
he sequence of real number
(
x
n
)
(x_n)
(
x
n
)
is defined by x_1 \equal{} 0, x_2 \equal{} 2 and x_{n\plus{}2} \equal{} 2^{\minus{}x_n} \plus{} \frac{1}{2} \forall n \equal{} 1,2,3 \ldots Prove that the sequence has a limit as
n
n
n
approaches \plus{}\infty. Determine the limit.
3
1
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n(2n + 1)(5n + 2) divides m
Let m \equal{} 2007^{2008}, how many natural numbers n are there such that
n
<
m
n < m
n
<
m
and n(2n \plus{} 1)(5n \plus{} 2) is divisible by
m
m
m
(which means that m \mid n(2n \plus{} 1)(5n \plus{} 2)) ?
2
1
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Opposite ray of EC such that angle BME equals angle ECA
Given a triangle with acute angle
∠
B
E
C
,
\angle BEC,
∠
BEC
,
let
E
E
E
be the midpoint of
A
B
.
AB.
A
B
.
Point
M
M
M
is chosen on the opposite ray of
E
C
EC
EC
such that \angle BME \equal{} \angle ECA. Denote by
θ
\theta
θ
the measure of angle
∠
B
E
C
.
\angle BEC.
∠
BEC
.
Evaluate
M
C
A
B
\frac{MC}{AB}
A
B
MC
in terms of
θ
.
\theta.
θ
.
1
1
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x^2 + y^3 = 29 and log_3(x) * log_2(y) = 1
Determine the number of solutions of the simultaneous equations x^2 \plus{} y^3 \equal{} 29 and \log_3 x \cdot \log_2 y \equal{} 1.
7
1
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A nice problem. Olympic VietNam 2007-2008 (New).
Let
A
D
AD
A
D
is centroid of
A
B
C
ABC
A
BC
triangle. Let
(
d
)
(d)
(
d
)
is the perpendicular line with
A
D
AD
A
D
. Let
M
M
M
is a point on
(
d
)
(d)
(
d
)
. Let
E
,
F
E, F
E
,
F
are midpoints of
M
B
,
M
C
MB, MC
MB
,
MC
respectively. The line through point
E
E
E
and perpendicular with
(
d
)
(d)
(
d
)
meet
A
B
AB
A
B
at
P
P
P
. The line through point
F
F
F
and perpendicular with
(
d
)
(d)
(
d
)
meet
A
C
AC
A
C
at
Q
Q
Q
. Let
(
d
′
)
(d')
(
d
′
)
is a line through point
M
M
M
and perpendicular with
P
Q
PQ
PQ
. Prove
(
d
′
)
(d')
(
d
′
)
always pass a fixed point.