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Contests
National and Regional Contests
India Contests
India National Olympiad
1989 India National Olympiad
1989 India National Olympiad
Part of
India National Olympiad
Subcontests
(5)
1
1
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problem 1 of Indian Mathematical Olympiad 1989
Prove that the Polynomial f(x) \equal{} x^{4} \plus{} 26x^{3} \plus{} 56x^{2} \plus{} 78x \plus{} 1989 can't be expressed as a product f(x) \equal{} p(x)q(x) , where
p
(
x
)
p(x)
p
(
x
)
and
q
(
x
)
q(x)
q
(
x
)
are both polynomial with integral coefficients and with degree at least
1
1
1
.
5
1
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problem 5 of Indian Mathematical Olympiad 1989
For positive integers
n
n
n
, define
A
(
n
)
A(n)
A
(
n
)
to be
(
2
n
)
!
(
n
!
)
2
\frac {(2n)!}{(n!)^{2}}
(
n
!
)
2
(
2
n
)!
. Determine the sets of positive integers
n
n
n
for which (a)
A
(
n
)
A(n)
A
(
n
)
is an even number, (b)
A
(
n
)
A(n)
A
(
n
)
is a multiple of
4
4
4
.
3
1
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problem 3 of Indian Mathematical Olympiad 1989
Let
A
A
A
denote a subset of the set
{
1
,
11
,
21
,
31
,
…
,
541
,
551
}
\{ 1,11,21,31, \dots ,541,551 \}
{
1
,
11
,
21
,
31
,
…
,
541
,
551
}
having the property that no two elements of
A
A
A
add up to
552
552
552
. Prove that
A
A
A
can't have more than
28
28
28
elements.
2
1
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problem 2 of Indian Mathematical Olympiad 1989
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
and
d
d
d
be any four real numbers, not all equal to zero. Prove that the roots of the polynomial f(x) \equal{} x^{6} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d can't all be real.
6
1
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problem 6 of Indian Mathematical Olympiad 1989
Triangle
A
B
C
ABC
A
BC
has incentre
I
I
I
and the incircle touches
B
C
,
C
A
BC, CA
BC
,
C
A
at
D
,
E
D, E
D
,
E
respectively. Let
B
I
BI
B
I
meet
D
E
DE
D
E
at
G
G
G
. Show that
A
G
AG
A
G
is perpendicular to
B
G
BG
BG
.