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Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2011 India Regional Mathematical Olympiad
2011 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(6)
3
2
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Indian RMO 2011 P3
A natural number
n
n
n
is chosen strictly between two consecutive perfect squares. The smaller of these two squares is obtained by subtracting
k
k
k
from
n
n
n
and the larger by adding
l
l
l
to
n
.
n.
n
.
Prove that
n
−
k
l
n-kl
n
−
k
l
is a perfect square.
a,b,c in HP; (a^2+b^2) etc in GP [RMO2-2011, India]
Let
a
,
b
,
c
>
0.
a,b,c>0.
a
,
b
,
c
>
0.
If
1
a
,
1
b
,
1
c
\frac 1a,\frac 1b,\frac 1c
a
1
,
b
1
,
c
1
are in arithmetic progression, and if
a
2
+
b
2
,
b
2
+
c
2
,
c
2
+
a
2
a^2+b^2,b^2+c^2,c^2+a^2
a
2
+
b
2
,
b
2
+
c
2
,
c
2
+
a
2
are in geometric progression, show that
a
=
b
=
c
.
a=b=c.
a
=
b
=
c
.
1
2
Hide problems
Indian RMO 2011: Question 1
Let
A
B
C
ABC
A
BC
be a triangle. Let
D
,
E
,
F
D, E, F
D
,
E
,
F
be points respectively on the segments
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
such that
A
D
,
B
E
,
C
F
AD, BE, CF
A
D
,
BE
,
CF
concur at the point
K
K
K
. Suppose
B
D
D
C
=
B
F
F
A
\frac{BD}{DC} = \frac {BF}{FA}
D
C
B
D
=
F
A
BF
and
∠
A
D
B
=
∠
A
F
C
\angle ADB = \angle AFC
∠
A
D
B
=
∠
A
FC
. Prove that
∠
A
B
E
=
∠
C
A
D
\angle ABE = \angle CAD
∠
A
BE
=
∠
C
A
D
.
AO and HM meet on odot(ABC) - [RMO2, India]
Let
A
B
C
ABC
A
BC
be an acute angled scalene triangle with circumcentre
O
O
O
and orthocentre
H
.
H.
H
.
If
M
M
M
is the midpoint of
B
C
,
BC,
BC
,
then show that
A
O
AO
A
O
and
H
M
HM
H
M
intersect on the circumcircle of
A
B
C
.
ABC.
A
BC
.
6
2
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Indian RMO 2011: Question 6
Find all pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of real numbers such that
1
6
x
2
+
y
+
1
6
x
+
y
2
=
1
16^{x^{2}+y} + 16^{x+y^{2}} = 1
1
6
x
2
+
y
+
1
6
x
+
y
2
=
1
Largest real constant lambda(a,b,c>0)
Find the largest real constant
λ
\lambda
λ
such that
λ
a
b
c
a
+
b
+
c
≤
(
a
+
b
)
2
+
(
a
+
b
+
4
c
)
2
\frac{\lambda abc}{a+b+c}\leq (a+b)^2+(a+b+4c)^2
a
+
b
+
c
λab
c
≤
(
a
+
b
)
2
+
(
a
+
b
+
4
c
)
2
For all positive real numbers
a
,
b
,
c
.
a,b,c.
a
,
b
,
c
.
5
2
Hide problems
Indian RMO 2011: Question 5
Let
A
B
C
ABC
A
BC
be a triangle and let
B
B
1
,
C
C
1
BB_1,CC_1
B
B
1
,
C
C
1
be respectively the bisectors of
∠
B
,
∠
C
\angle{B},\angle{C}
∠
B
,
∠
C
with
B
1
B_1
B
1
on
A
C
AC
A
C
and
C
1
C_1
C
1
on
A
B
AB
A
B
, Let
E
,
F
E,F
E
,
F
be the feet of perpendiculars drawn from
A
A
A
onto
B
B
1
,
C
C
1
BB_1,CC_1
B
B
1
,
C
C
1
respectively. Suppose
D
D
D
is the point at which the incircle of
A
B
C
ABC
A
BC
touches
A
B
AB
A
B
. Prove that
A
D
=
E
F
AD=EF
A
D
=
EF
Convex quadrilateral and midpoints [RMO2-2011, India]
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. Let
E
,
F
,
G
,
H
E,F,G,H
E
,
F
,
G
,
H
be the midpoints of
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
respectively. If
A
C
,
B
D
,
E
G
,
F
H
AC,BD,EG,FH
A
C
,
B
D
,
EG
,
F
H
concur at a point
O
,
O,
O
,
prove that
A
B
C
D
ABCD
A
BC
D
is a parallelogram.
2
2
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Indian RMO 2011: Question 2
Let
(
a
1
,
a
2
,
a
3
,
.
.
.
,
a
2011
)
(a_1,a_2,a_3,...,a_{2011})
(
a
1
,
a
2
,
a
3
,
...
,
a
2011
)
be a permutation of the numbers
1
,
2
,
3
,
.
.
.
,
2011
1,2,3,...,2011
1
,
2
,
3
,
...
,
2011
. Show that there exist two numbers
j
,
k
j,k
j
,
k
such that
1
≤
j
<
k
≤
2011
1\leq{j}<k\leq2011
1
≤
j
<
k
≤
2011
and
∣
a
j
−
j
∣
=
∣
a
k
−
k
∣
|a_j-j|=|a_k-k|
∣
a
j
−
j
∣
=
∣
a
k
−
k
∣
2n+1, 3n+1 perfect squares [RMO2-2011, India]
Let
n
n
n
be a positive integer such that
2
n
+
1
2n+1
2
n
+
1
and
3
n
+
1
3n+1
3
n
+
1
are both perfect squares. Show that
5
n
+
3
5n+3
5
n
+
3
is a composite number.
4
2
Hide problems
Indian RMO 2011: Question 4
Consider a
20
20
20
-sided convex polygon
K
K
K
, with vertices
A
1
,
A
2
,
.
.
.
,
A
20
A_1, A_2,...,A_{20}
A
1
,
A
2
,
...
,
A
20
in that order. Find the number of ways in which three sides of
K
K
K
can be chosen so that every pair among them has at least two sides of
K
K
K
between them. (For example
(
A
1
A
2
,
A
4
A
5
,
A
11
A
12
)
(A_1A_2, A_4A_5, A_{11}A_{12})
(
A
1
A
2
,
A
4
A
5
,
A
11
A
12
)
is an admissible triple while
(
A
1
A
2
,
A
4
A
5
,
A
19
A
20
)
(A_1A_2, A_4A_5, A_{19}A_{20})
(
A
1
A
2
,
A
4
A
5
,
A
19
A
20
)
is not.
4 digit numbers from {0,1,...,5} [RMO2-2011, India]
Find the number of 4-digit numbers with distinct digits chosen from the set
{
0
,
1
,
2
,
3
,
4
,
5
}
\{0,1,2,3,4,5\}
{
0
,
1
,
2
,
3
,
4
,
5
}
in which no two adjacent digits are even.