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Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2007 India Regional Mathematical Olympiad
2007 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(6)
6
1
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Problem 6 of RMO 2007 (Regional Mathematical Olympiad-India)
Prove that: (a) 5<\sqrt {5}\plus{}\sqrt [3]{5}\plus{}\sqrt [4]{5} (b) 8>\sqrt {8}\plus{}\sqrt [3]{8}\plus{}\sqrt [4]{8} (c) n>\sqrt {n}\plus{}\sqrt [3]{n}\plus{}\sqrt [4]{n} for all integers
n
≥
9.
n\geq 9 .
n
≥
9.
[Weightage 16/100]
2
1
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Problem 2 of RMO 2007 (Regional Mathematical Olympiad-India)
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be three natural numbers such that
a
<
b
<
c
a < b < c
a
<
b
<
c
and gcd (c \minus{} a, c \minus{} b) \equal{} 1. Suppose there exists an integer
d
d
d
such that a \plus{} d, b \plus{} d, c \plus{} d form the sides of a right-angled triangle. Prove that there exist integers,
l
,
m
l,m
l
,
m
such that c \plus{} d \equal{} l^{2} \plus{} m^{2} . [Weightage 17/100]
3
1
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Problem 3 of RMO 2007 (Regional Mathematical Olympiad-India)
Find all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of real numbers such that whenever
α
\alpha
α
is a root of x^{2} \plus{} ax \plus{} b \equal{} 0, \alpha^{2} \minus{} 2 is also a root of the equation. [Weightage 17/100]
5
1
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Problem 5 of RMO 2007 (Regional Mathematical Olympiad-India)
A trapezium
A
B
C
D
ABCD
A
BC
D
, in which
A
B
AB
A
B
is parallel to
C
D
CD
C
D
, is inscribed in a circle with centre
O
O
O
. Suppose the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
of the trapezium intersect at
M
M
M
, and OM \equal{} 2. (a) If
∠
A
M
B
\angle AMB
∠
A
MB
is
6
0
∘
,
60^\circ ,
6
0
∘
,
find, with proof, the difference between the lengths of the parallel sides. (b) If
∠
A
M
D
\angle AMD
∠
A
M
D
is
6
0
∘
,
60^\circ ,
6
0
∘
,
find, with proof, the difference between the lengths of the parallel sides. [Weightage 17/100]
1
1
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Problem 1 of RMO 2007 (Regional Mathematical Olympiad-India)
Let
A
B
C
ABC
A
BC
be an acute-angled triangle;
A
D
AD
A
D
be the bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
with
D
D
D
on
B
C
BC
BC
; and
B
E
BE
BE
be the altitude from
B
B
B
on
A
C
AC
A
C
. Show that
∠
C
E
D
>
4
5
∘
.
\angle CED > 45^\circ .
∠
CE
D
>
4
5
∘
.
[weightage 17/100]
4
1
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Combinatorial
How many 6-digit numbers are there such that-: a)The digits of each number are all from the set
{
1
,
2
,
3
,
4
,
5
}
\{1,2,3,4,5\}
{
1
,
2
,
3
,
4
,
5
}
b)any digit that appears in the number appears at least twice ? (Example:
225252
225252
225252
is valid while
222133
222133
222133
is not) [weightage 17/100]