MathDB
Problems
Contests
National and Regional Contests
India Contests
India Pre-Regional Mathematical Olympiad
2013 India PRMO
2013 India PRMO
Part of
India Pre-Regional Mathematical Olympiad
Subcontests
(20)
17
1
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2013 preRMO p17, 3 circles, tangent, radius wanted
Let
S
S
S
be a circle with centre
O
O
O
. A chord
A
B
AB
A
B
, not a diameter, divides
S
S
S
into two regions
R
1
R_1
R
1
and
R
2
R_2
R
2
such that
O
O
O
belongs to
R
2
R_2
R
2
. Let
S
1
S_1
S
1
be a circle with centre in
R
1
R_1
R
1
, touching
A
B
AB
A
B
at
X
X
X
and
S
S
S
internally. Let
S
2
S_2
S
2
be a circle with centre in
R
2
R_2
R
2
, touching
A
B
AB
A
B
at
Y
Y
Y
, the circle
S
S
S
internally and passing through the centre of
S
S
S
. The point
X
X
X
lies on the diameter passing through the centre of
S
2
S_2
S
2
and
∠
Y
X
O
=
3
0
o
\angle YXO=30^o
∠
Y
XO
=
3
0
o
. If the radius of
S
2
S_2
S
2
is
100
100
100
then what is the radius of
S
1
S_1
S
1
?
19
1
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2013 preRMO p19, computational with right triangle and ratio of areas
In a triangle
A
B
C
ABC
A
BC
with
∠
B
C
A
=
9
0
o
\angle BC A = 90^o
∠
BC
A
=
9
0
o
, the perpendicular bisector of
A
B
AB
A
B
intersects segments
A
B
AB
A
B
and
A
C
AC
A
C
at
X
X
X
and
Y
Y
Y
, respectively. If the ratio of the area of quadrilateral
B
X
Y
C
BXYC
BX
Y
C
to the area of triangle
A
B
C
ABC
A
BC
is
13
:
18
13 : 18
13
:
18
and
B
C
=
12
BC = 12
BC
=
12
then what is the length of
A
C
AC
A
C
?
20
1
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2013 preRMO p20, <64 with three 1s in base 2 representation
What is the sum (in base
10
10
10
) of all the natural numbers less than
64
64
64
which have exactly three ones in their base
2
2
2
representation?
16
1
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2013 preRMO p16, common root, x^3 - 3x + b=0, x^2 + bx -3=0
Let
f
(
x
)
=
x
3
−
3
x
+
b
f(x) = x^3 - 3x + b
f
(
x
)
=
x
3
−
3
x
+
b
and
g
(
x
)
=
x
2
+
b
x
−
3
g(x) = x^2 + bx -3
g
(
x
)
=
x
2
+
b
x
−
3
, where
b
b
b
is a real number. What is the sum of all possible values of
b
b
b
for which the equations
f
(
x
)
f(x)
f
(
x
)
= 0 and
g
(
x
)
=
0
g(x) = 0
g
(
x
)
=
0
have a common root?
18
1
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2013 preRMO p18, 2013=sum of k consecutive positive integers, max k
What is the maximum possible value of
k
k
k
for which
2013
2013
2013
can be written as a sum of
k
k
k
consecutive positive integers?
15
1
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2013 preRMO p15, midpoints of segments of midpoints of ABCD, rectangle
Let
A
1
,
B
1
,
C
1
,
D
1
A_1,B_1,C_1,D_1
A
1
,
B
1
,
C
1
,
D
1
be the midpoints of the sides of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
and let
A
2
,
B
2
,
C
2
,
D
2
A_2, B_2, C_2, D_2
A
2
,
B
2
,
C
2
,
D
2
be the midpoints of the sides of the quadrilateral
A
1
B
1
C
1
D
1
A_1B_1C_1D_1
A
1
B
1
C
1
D
1
. If
A
2
B
2
C
2
D
2
A_2B_2C_2D_2
A
2
B
2
C
2
D
2
is a rectangle with sides
4
4
4
and
6
6
6
, then what is the product of the lengths of the diagonals of
A
B
C
D
ABCD
A
BC
D
?
14
1
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2013 preRMO p14, min odd m such 1+ 2 +...+ m square of an integer
Let
m
m
m
be the smallest odd positive integer for which
1
+
2
+
.
.
.
+
m
1+ 2 +...+ m
1
+
2
+
...
+
m
is a square of an integer and let
n
n
n
be the smallest even positive integer for which
1
+
2
+
.
.
.
+
n
1 + 2 + ... + n
1
+
2
+
...
+
n
is a square of an integer. What is the value of
m
+
n
m + n
m
+
n
?
13
1
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2013 preRMO p13, a,b \in {1,2,... ,1000}, 15/(a+b)
To each element of the set
S
=
{
1
,
2
,
.
.
.
,
1000
}
S = \{1,2,... ,1000\}
S
=
{
1
,
2
,
...
,
1000
}
a colour is assigned. Suppose that for any two elements
a
,
b
a, b
a
,
b
of
S
S
S
, if
15
15
15
divides
a
+
b
a + b
a
+
b
then they are both assigned the same colour. What is the maximum possible number of distinct colours used?
12
1
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2013 preRMO p12, computational with equilateral and rectangle
Let
A
B
C
ABC
A
BC
be an equilateral triangle. Let
P
P
P
and
S
S
S
be points on
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, and let
Q
Q
Q
and
R
R
R
be points on
B
C
BC
BC
such that
P
Q
R
S
PQRS
PQRS
is a rectangle. If
P
Q
=
3
P
S
PQ = \sqrt3 PS
PQ
=
3
PS
and the area of
P
Q
R
S
PQRS
PQRS
is
28
3
28\sqrt3
28
3
, what is the length of
P
C
PC
PC
?
11
1
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2013 preRMO p11 x^2+6y=-17, y^2+4z=1, z^2+2x=2, x^2 + y^2 + z^2 ?
Three real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
are such that
x
2
+
6
y
=
−
17
,
y
2
+
4
z
=
1
x^2 + 6y = -17, y^2 + 4z = 1
x
2
+
6
y
=
−
17
,
y
2
+
4
z
=
1
and
z
2
+
2
x
=
2
z^2 + 2x = 2
z
2
+
2
x
=
2
. What is the value of
x
2
+
y
2
+
z
2
x^2 + y^2 + z^2
x
2
+
y
2
+
z
2
?
2
1
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2013 preRMO p2, S_n=\sum_{k=0}^{n}\frac{1}{\sqrt{k+1}+\sqrt{k}}
Let
S
n
=
∑
k
=
0
n
1
k
+
1
+
k
S_n=\sum_{k=0}^{n}\frac{1}{\sqrt{k+1}+\sqrt{k}}
S
n
=
∑
k
=
0
n
k
+
1
+
k
1
. What is the value of
∑
n
=
1
99
1
S
n
+
S
n
−
1
\sum_{n=1}^{99}\frac{1}{S_n+S_{n-1}}
∑
n
=
1
99
S
n
+
S
n
−
1
1
?
10
1
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2013 preRMO p10, Carol multiplied largest with sum of other 2
Carol was given three numbers and was asked to add the largest of the three to the product of the other two. Instead, she multiplied the largest with the sum of the other two, but still got the right answer. What is the sum of the three numbers?
9
1
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2013 preRMO p9, orthocenter, incentre, circumcentre, 1 vetrex concyclic
In a triangle
A
B
C
ABC
A
BC
, let
H
,
I
H, I
H
,
I
and
O
O
O
be the orthocentre, incentre and circumcentre, respectively. If the points
B
,
H
,
I
,
C
B, H, I, C
B
,
H
,
I
,
C
lie on a circle, what is the magnitude of
∠
B
O
C
\angle BOC
∠
BOC
in degrees?
8
1
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2013 preRMO p8, computational geometry with a trapezium
Let
A
D
AD
A
D
and
B
C
BC
BC
be the parallel sides of a trapezium
A
B
C
D
ABCD
A
BC
D
. Let
P
P
P
and
Q
Q
Q
be the midpoints of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. If
A
D
=
16
AD = 16
A
D
=
16
and
B
C
=
20
BC = 20
BC
=
20
, what is the length of
P
Q
PQ
PQ
?
7
1
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2013 preRMO p7, Akbar and Birbal together have n marbles
Let Akbar and Birbal together have
n
n
n
marbles, where
n
>
0
n > 0
n
>
0
. Akbar says to Birbal, “ If I give you some marbles then you will have twice as many marbles as I will have.” Birbal says to Akbar, “ If I give you some marbles then you will have thrice as many marbles as I will have.” What is the minimum possible value of
n
n
n
for which the above statements are true?
6
1
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2013 preRMO p6, sum of digits S(5N + 2013)
Let
S
(
M
)
S(M)
S
(
M
)
denote the sum of the digits of a positive integer
M
M
M
written in base
10
10
10
. Let
N
N
N
be the smallest positive integer such that
S
(
N
)
=
2013
S(N) = 2013
S
(
N
)
=
2013
. What is the value of
S
(
5
N
+
2013
)
S(5N + 2013)
S
(
5
N
+
2013
)
?
5
1
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2013 preRMO p5, balls, n-1 red, n green, n+1 blue
There are
n
−
1
n -1
n
−
1
red balls,
n
n
n
green balls and
n
+
1
n + 1
n
+
1
blue balls in a bag. The number of ways of choosing two balls from the bag that have different colours is
299
299
299
. What is the value of
n
n
n
?
4
1
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2013 preRMO p4, X,Y,Z collinear XY = 10, XZ = 3, YZ=?
Three points
X
,
Y
,
Z
X, Y,Z
X
,
Y
,
Z
are on a striaght line such that
X
Y
=
10
XY = 10
X
Y
=
10
and
X
Z
=
3
XZ = 3
XZ
=
3
. What is the product of all possible values of
Y
Z
YZ
Y
Z
?
3
1
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2013 preRMO p3, x^2 + ax + 20 = 0 has integer roots
It is given that the equation
x
2
+
a
x
+
20
=
0
x^2 + ax + 20 = 0
x
2
+
a
x
+
20
=
0
has integer roots. What is the sum of all possible values of
a
a
a
?
1
1
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2013 preRMO p1 k(3^3 + 4^3 + 5^3) = a^n
What is the smallest positive integer
k
k
k
such that
k
(
3
3
+
4
3
+
5
3
)
=
a
n
k(3^3 + 4^3 + 5^3) = a^n
k
(
3
3
+
4
3
+
5
3
)
=
a
n
for some positive integers
a
a
a
and
n
n
n
, with
n
>
1
n > 1
n
>
1
?