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National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2009 Grand Duchy of Lithuania
2009 Grand Duchy of Lithuania
Part of
Grand Duchy of Lithuania
Subcontests
(4)
5
1
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a table having all entries divisible by n
Consider a table whose entries are integers. Adding a same integer to all entries on a same row, or on a same column, is called an operation. It is given that, for infinitely many positive integers
n
n
n
, one can obtain, through a finite number of operations, a table having all entries divisible by
n
n
n
. Prove that, through a finite number of operations, one can obtain the table whose all entries are zeroes.
3
1
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x^2+ 2 = 4\sqrt{x^3+1}
Solve the equation
x
2
+
2
=
4
x
3
+
1
x^2+ 2 = 4\sqrt{x^3+1}
x
2
+
2
=
4
x
3
+
1
2
1
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2b^3+ 9a^2 d - 7abc <= 0 for f(x) = ax^3 + bx^2 + cx + d, f(0) < 0
Let
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
f(x) = ax^3 + bx^2 + cx + d
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
be a polynomial with real coefficients. Given that
f
(
x
)
f(x)
f
(
x
)
has three real positive roots and that
f
(
0
)
<
0
f(0) < 0
f
(
0
)
<
0
, prove that
2
b
3
+
9
a
2
d
−
7
a
b
c
≤
0
2b^3+ 9a^2 d - 7abc \le 0
2
b
3
+
9
a
2
d
−
7
ab
c
≤
0
.
1
1
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N is a multiple of 2009 and the sum of its digits equals 2009
The natural number
N
N
N
is a multiple of
2009
2009
2009
and the sum of its (decimal) digits equals
2009
2009
2009
. (a) Find one such number. (b) Find the smallest such number.