MathDB
Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2010 India Regional Mathematical Olympiad
2010 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(6)
6
1
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Number theory 2
For each integer
n
≥
1
n \ge 1
n
≥
1
define
a
n
=
[
n
[
n
]
]
a_n = \left[\frac{n}{\left[\sqrt{n}\right]}\right]
a
n
=
[
[
n
]
n
]
(where
[
x
]
[x]
[
x
]
denoted the largest integer not exceeding
x
x
x
, for any real number
x
x
x
). Find the number of all
n
n
n
in the set
{
1
,
2
,
3
,
⋯
,
2010
}
\{1, 2, 3, \cdots , 2010\}
{
1
,
2
,
3
,
⋯
,
2010
}
for which
a
n
>
a
n
+
1
a_n > a_{n+1}
a
n
>
a
n
+
1
5
1
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Reflection
Let
A
B
C
ABC
A
BC
be a triangle in which
∠
A
=
6
0
∘
\angle A = 60^\circ
∠
A
=
6
0
∘
. Let
B
E
BE
BE
and
C
F
CF
CF
be the bisectors of
∠
B
\angle B
∠
B
and
∠
C
\angle C
∠
C
with
E
E
E
on
A
C
AC
A
C
and
F
F
F
on
A
B
AB
A
B
. Let
M
M
M
be the reflection of
A
A
A
in line
E
F
EF
EF
. Prove that
M
M
M
lies on
B
C
BC
BC
.
4
1
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Number theory
Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.
3
1
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Divsibility (combinatorics)
Find the number of
4
4
4
-digit numbers (in base
10
10
10
) having non-zero digits and which are divisible by
4
4
4
but not by
8
8
8
.
2
1
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Quadratic Polynomial
Let
P
1
(
x
)
=
a
x
2
−
b
x
−
c
P_1(x) = ax^2 - bx - c
P
1
(
x
)
=
a
x
2
−
b
x
−
c
,
P
2
(
x
)
=
b
x
2
−
c
x
−
a
P_2(x) = bx^2 - cx - a
P
2
(
x
)
=
b
x
2
−
c
x
−
a
,
P
3
(
x
)
=
c
x
2
−
a
x
−
b
P_3(x) = cx^2 - ax - b
P
3
(
x
)
=
c
x
2
−
a
x
−
b
be three quadratic polynomials. Suppose there exists a real number
α
\alpha
α
such that
P
1
(
α
)
=
P
2
(
α
)
=
P
3
(
α
)
P_1(\alpha) = P_2(\alpha) = P_3(\alpha)
P
1
(
α
)
=
P
2
(
α
)
=
P
3
(
α
)
. Prove that
a
=
b
=
c
a = b = c
a
=
b
=
c
.
1
1
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Hexagon
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon in which diagonals
A
D
,
B
E
,
C
F
AD, BE, CF
A
D
,
BE
,
CF
are concurrent at
O
O
O
. Suppose
[
O
A
F
]
[OAF]
[
O
A
F
]
is geometric mean of
[
O
A
B
]
[OAB]
[
O
A
B
]
and
[
O
E
F
]
[OEF]
[
OEF
]
and
[
O
B
C
]
[OBC]
[
OBC
]
is geometric mean of
[
O
A
B
]
[OAB]
[
O
A
B
]
and
[
O
C
D
]
[OCD]
[
OC
D
]
. Prove that
[
O
E
D
]
[OED]
[
OE
D
]
is the geometric mean of
[
O
C
D
]
[OCD]
[
OC
D
]
and
[
O
E
F
]
[OEF]
[
OEF
]
. (Here
[
X
Y
Z
]
[XYZ]
[
X
Y
Z
]
denotes are of
△
X
Y
Z
\triangle XYZ
△
X
Y
Z
)