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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2002 Vietnam National Olympiad
2002 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
2
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Vietnam NMO 2002_3
Let be given two positive integers
m
m
m
,
n
n
n
with
m
<
2001
m < 2001
m
<
2001
,
n
<
2002
n < 2002
n
<
2002
. Let distinct real numbers be written in the cells of a
2001
×
2002
2001 \times 2002
2001
×
2002
board (with
2001
2001
2001
rows and
2002
2002
2002
columns). A cell of the board is called bad if the corresponding number is smaller than at least
m
m
m
numbers in the same column and at least
n
n
n
numbers in the same row. Let
s
s
s
denote the total number of bad cells. Find the least possible value of
s
s
s
.
Vietnam NMO 2002_6
For a positive integer
n
n
n
, consider the equation \frac{1}{x\minus{}1}\plus{}\frac{1}{4x\minus{}1}\plus{}\cdots\plus{}\frac{1}{k^2x\minus{}1}\plus{}\cdots\plus{}\frac{1}{n^2x\minus{}1}\equal{}\frac{1}{2}. (a) Prove that, for every
n
n
n
, this equation has a unique root greater than
1
1
1
, which is denoted by
x
n
x_n
x
n
. (b) Prove that the limit of sequence
(
x
n
)
(x_n)
(
x
n
)
is
4
4
4
as
n
n
n
approaches infinity.
1
2
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Vietnam NMO 2002_1
Solve the equation \sqrt{4 \minus{} 3\sqrt{10 \minus{} 3x}} \equal{} x \minus{} 2.
Vietnam NMO 2002_4
Let
a
a
a
,
b
b
b
,
c
c
c
be real numbers for which the polynomial x^3 \plus{} ax^2 \plus{} bx \plus{} c has three real roots. Prove that 12ab \plus{} 27c \le 6a^3 \plus{} 10\left(a^2 \minus{} 2b\right)^{\frac {3}{2}} When does equality occur?
2
2
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Vietnam NMO 2002_2
An isosceles triangle
A
B
C
ABC
A
BC
with AB \equal{} AC is given on the plane. A variable circle
(
O
)
(O)
(
O
)
with center
O
O
O
on the line
B
C
BC
BC
passes through
A
A
A
and does not touch either of the lines
A
B
AB
A
B
and
A
C
AC
A
C
. Let
M
M
M
and
N
N
N
be the second points of intersection of
(
O
)
(O)
(
O
)
with lines
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. Find the locus of the orthocenter of triangle
A
M
N
AMN
A
MN
.
Again
Determine for which
n
n
n
positive integer the equation: a \plus{} b \plus{} c \plus{} d \equal{} n \sqrt {abcd} has positive integer solutions.