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NATIONAL BOARD FOR HIGHER MATHEMATICS
AND
HOMI BHABHA CENTRE FOR SCIENCE EDUCATION
TATA INSTITUTE OF FUNDAMENTAL RESEARCH
Pre-REGIONAL MATHEMATICAL OLYMPIAD, 2012
Mumbai Region
October 7, 2012
QUESTION PAPER SET: A
ã There are 20 questions in this question paper. Each question

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NATIONAL BOARD FOR HIGHER MATHEMATICS
AND
HOMI BHABHA CENTRE FOR SCIENCE EDUCATIONTATA INSTITUTE OF FUNDAMENTAL RESEARCHPre-REGIONAL MATHEMATICAL OLYMPIAD, 2012
Mumbai RegionOctober 7, 2012QUESTION PAPER SET: A
ã
There are 20 questions in this question paper. Each question carries 5 marks.
ã
Answer all questions.
ã
Time allotted: 2 hours.
QUESTIONS
1. Rama was asked by her teacher to subtract 3 from a certain number and then divide theresult by 9. Instead, she subtracted 9 and then divided the result by 3. She got 43 as theanswer. What would have been her answer if she had solved the problem correctly?2. A triangle with perimeter 7 has integer side lengths. What is the maximum possible area of such a triangle?3. For how many pairs of positive integers (
x,y
) is
x
+ 3
y
= 100?4. The letters
R
,
M
, and
O
represent whole numbers. If
R
×
M
×
O
= 240,
R
×
O
+
M
= 46and
R
+
M
×
O
= 64, what is the value of
R
+
M
+
O
?5. Let
S
n
=
n
2
+ 20
n
+ 12,
n
a positive integer. What is the sum of all possible values of
n
forwhich
S
n
is a perfect square?6. A postman has to deliver ﬁve letters to ﬁve diﬀerent houses. Mischievously, he posts oneletter through each door without looking to see if it is the correct address. In how manydiﬀerent ways could he do this so that exactly two of the ﬁve houses receive the correctletters?7. In ∆
ABC
, we have
AC
=
BC
= 7 and
AB
= 2. Suppose that
D
is a point on line
AB
suchthat
B
lies between
A
and
D
and
CD
= 8. What is the length of the segment
BD
?8. In rectangle
ABCD
,
AB
= 5 and
BC
= 3. Points
F
and
G
are on line segment
CD
so that
DF
= 1 and
GC
= 2. Lines
AF
and
BG
intersect at
E
. What is the area of ∆
AEB
?9. Suppose that 4
X
1
= 5
,
5
X
2
= 6
,
6
X
3
= 7
,...,
126
X
123
= 127
,
127
X
124
= 128. What is thevalue of the product
X
1
X
2
...X
124
?10.
ABCD
is a square and
AB
= 1. Equilateral triangles
AYB
and
CXD
are drawn such that
X
and
Y
are inside the square. What is the length of
XY
?11. Let
P
(
n
) = (
n
+ 1)(
n
+ 3)(
n
+ 5)(
n
+ 7)(
n
+ 9). What is the largest integer that is a divisorof
P
(
n
) for all positive even integers
n
?12. If 1
2011 +
√
2011
2
−
1=
√
m
−√
n
, where
m
and
n
are positive integers, what is the valueof
m
+
n
?1
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13. If
a
=
b
−
c
,
b
=
c
−
d
,
c
=
d
−
a
and
abcd
= 0 then what is the value of
ab
+
bc
+
cd
+
da
?14.
O
and
I
are the circumcentre and incentre of ∆
ABC
respectively. Suppose
O
lies in theinterior of ∆
ABC
and
I
lies on the circle passing through
B
,
O
, and
C
. What is themagnitude of
BAC
in degrees?15. How many non-negative integral values of
x
satisfy the equation
x
5
=
x
7
?(Here [
x
] denotes the greatest integer less than or equal to
x
. For example [3
.
4] = 3 and[
−
2
.
3] =
−
3.)16. Let
N
be the set of natural numbers. Suppose
f
:
N
→
N
is a function satisfying thefollowing conditions.(a)
f
(
mn
) =
f
(
m
)
f
(
n
);(b)
f
(
m
)
< f
(
n
) if
m < n
;(c)
f
(2) = 2.What is the value of
20
k
=1
f
(
k
)?17. Let
x
1
,x
2
,x
3
be the roots of the equation
x
3
+3
x
+5 = 0. What is the value of the expression
x
1
+ 1
x
1
x
2
+ 1
x
2
x
3
+ 1
x
3
?18. What is the sum of the squares of the roots of the equation
x
2
−
7[
x
] + 5 = 0?(Here [
x
] denotes the greatest integer less than or equal to
x
. For example [3
.
4] = 3 and[
−
2
.
3] =
−
3.)19. How many integer pairs (
x,y
) satisfy
x
2
+ 4
y
2
−
2
xy
−
2
x
−
4
y
−
8 = 0?20.
PS
is a line segment of length 4 and
O
is the midpoint of
PS
. A semicircular arc is drawnwith
PS
as diameter. Let
X
be the midpoint of this arc.
Q
and
R
are points on the arc
PXS
such that
QR
is parallel to
PS
and the semicircular arc drawn with
QR
as diameteris tangent to
PS
. What is the area of the region
QXROQ
bounded by the two semicirculararcs?
END OF QUESTION PAPER
2
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Pre-RMO 2012 Answer Key for Set A
1. 152. 3
√
743. 334. 205. 166. 207. 38. 12.59. 3.510.
√
3
−
111. 1512. 201113. 0.514. 6015. 916. 21017.
−
29518. 9219. 620. 2
π
−
21
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