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Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2008 India National Olympiad
2008 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
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Existence of Polynomials!
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with integer coefficients. Prove that there exist two polynomials
Q
(
x
)
Q(x)
Q
(
x
)
and
R
(
x
)
R(x)
R
(
x
)
, again with integer coefficients, such that (i)
P
(
x
)
⋅
Q
(
x
)
P(x) \cdot Q(x)
P
(
x
)
⋅
Q
(
x
)
is a polynomial in
x
2
x^2
x
2
, and (ii)
P
(
x
)
⋅
R
(
x
)
P(x) \cdot R(x)
P
(
x
)
⋅
R
(
x
)
is a polynomial in
x
3
x^3
x
3
.
3
1
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Prove the existence of an integer M
Let
A
A
A
be a set of real numbers such that
A
A
A
has at least four elements. Suppose
A
A
A
has the property that a^2 \plus{} bc is a rational number for all distinct numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
in
A
A
A
. Prove that there exists a positive integer
M
M
M
such that
a
M
a\sqrt{M}
a
M
is a rational number for every
a
a
a
in
A
A
A
.
5
1
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problem 5 of Indian Mathematical Olympiad 2008
Let
A
B
C
ABC
A
BC
be a triangle;
Γ
A
,
Γ
B
,
Γ
C
\Gamma_A,\Gamma_B,\Gamma_C
Γ
A
,
Γ
B
,
Γ
C
be three equal, disjoint circles inside
A
B
C
ABC
A
BC
such that
Γ
A
\Gamma_A
Γ
A
touches
A
B
AB
A
B
and
A
C
AC
A
C
;
Γ
B
\Gamma_B
Γ
B
touches
A
B
AB
A
B
and
B
C
BC
BC
; and
Γ
C
\Gamma_C
Γ
C
touches
B
C
BC
BC
and
C
A
CA
C
A
. Let
Γ
\Gamma
Γ
be a circle touching circles
Γ
A
,
Γ
B
,
Γ
C
\Gamma_A, \Gamma_B, \Gamma_C
Γ
A
,
Γ
B
,
Γ
C
externally. Prove that the line joining the circum-centre
O
O
O
and the in-centre
I
I
I
of triangle
A
B
C
ABC
A
BC
passes through the centre of
Γ
\Gamma
Γ
.
2
1
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A nice Indian MO problem
Find all triples
(
p
,
x
,
y
)
\left(p,x,y\right)
(
p
,
x
,
y
)
such that p^x\equal{}y^4\plus{}4, where
p
p
p
is a prime and
x
x
x
and
y
y
y
are natural numbers.
1
1
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Concyclic Points - INMO 2008
Let
A
B
C
ABC
A
BC
be triangle,
I
I
I
its in-center;
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
be the reflections of
I
I
I
in
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
respectively. Suppose the circum-circle of triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
passes through
A
A
A
. Prove that
B
1
,
C
1
,
I
,
I
1
B_1,C_1,I,I_1
B
1
,
C
1
,
I
,
I
1
are concylic, where
I
1
I_1
I
1
is the in-center of triangle
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
.
4
1
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Plane Coloring - INMO 2008
All the points with integer coordinates in the
x
y
xy
x
y
-Plane are coloured using three colours, red, blue and green, each colour being used at least once. It is known that the point
(
0
,
0
)
(0,0)
(
0
,
0
)
is red and the point
(
0
,
1
)
(0,1)
(
0
,
1
)
is blue. Prove that there exist three points with integer coordinates of distinct colours which form the vertices of a right-angled triangle.