MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2009 India National Olympiad
2009 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
2
1
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binary type sequence
Define a a sequence {<{a_n}>}^{\infty}_{n\equal{}1} as follows a_n\equal{}0, if number of positive divisors of
n
n
n
is odd a_n\equal{}1, if number of positive divisors of
n
n
n
is even (The positive divisors of
n
n
n
include
1
1
1
as well as
n
n
n
.)Let x\equal{}0.a_1a_2a_3........ be the real number whose decimal expansion contains
a
n
a_n
a
n
in the
n
n
n
-th place,
n
≥
1
n\geq1
n
≥
1
.Determine,with proof,whether
x
x
x
is rational or irrational.
6
1
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a^3 + b^3 = c^3
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that a^3 \plus{} b^3 \equal{} c^3.Prove that: a^2 \plus{} b^2 \minus{} c^2 > 6(c \minus{} a)(c \minus{} b).
4
1
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isosceles triangle or angles in GP
All the points in the plane are colored using three colors.Prove that there exists a triangle with vertices having the same color such that either it is isosceles or its angles are in geometric progression.
3
1
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greatest integer function
Find all real numbers
x
x
x
such that: [x^2\plus{}2x]\equal{}{[x]}^2\plus{}2[x] (Here
[
x
]
[x]
[
x
]
denotes the largest integer not exceeding
x
x
x
.)
5
1
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inequality of altitude
Let
A
B
C
ABC
A
BC
be an acute angled triangle and let
H
H
H
be its ortho centre. Let
h
m
a
x
h_{max}
h
ma
x
denote the largest altitude of the triangle
A
B
C
ABC
A
BC
. Prove that:AH \plus{} BH \plus{} CH\leq2h_{max}
1
1
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collinearity
Let
A
B
C
ABC
A
BC
be a tringle and let
P
P
P
be an interior point such that \angle BPC \equal{} 90 ,\angle BAP \equal{} \angle BCP.Let
M
,
N
M,N
M
,
N
be the mid points of
A
C
,
B
C
AC,BC
A
C
,
BC
respectively.Suppose BP \equal{} 2PM.Prove that
A
,
P
,
N
A,P,N
A
,
P
,
N
are collinear.