MathDB
Problems
Contests
National and Regional Contests
India Contests
India Pre-Regional Mathematical Olympiad
2015 India PRMO
2015 India PRMO
Part of
India Pre-Regional Mathematical Olympiad
Subcontests
(20)
8
1
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PRMO 2015. p8. broken piece of a circular plate made of glass
8. The fi gure below shows a broken piece of a circular plate made of glass. https://cdn.artofproblemsolving.com/attachments/7/3/a49f60d803f802c54e2295932b34579514b4fe.png
C
C
C
is the midpoint of
A
B
AB
A
B
, and
D
D
D
is the midpoint of arc
A
B
AB
A
B
. Given that
A
B
=
24
AB = 24
A
B
=
24
cm and
C
D
=
6
CD = 6
C
D
=
6
cm, what is the radius of the plate in centimetres? (The fi gure is not drawn to scale.)
20
1
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PRMO 2015. p20.
20.
20.
20.
The circle
ω
\omega
ω
touches the circle
Ω
\Omega
Ω
internally at point
P
.
P.
P
.
The centre
O
O
O
of
Ω
\Omega
Ω
is outside
ω
.
\omega.
ω
.
Let
X
Y
XY
X
Y
be a diameter of
Ω
\Omega
Ω
which is also tangent to
ω
.
\omega.
ω
.
Assume
P
Y
>
P
X
.
PY>PX.
P
Y
>
PX
.
Let
P
Y
PY
P
Y
intersect
ω
\omega
ω
at
z
.
z.
z
.
If
Y
Z
=
2
P
Z
,
YZ=2PZ,
Y
Z
=
2
PZ
,
what is the magnitude of
∠
P
Y
X
\angle{PYX}
∠
P
Y
X
in degrees
?
?
?
19
1
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PRMO 2015. p19.
19.
19.
19.
The digits of a positive integer
n
n
n
are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when
n
n
n
is divided by
37
?
37 ?
37
?
18
1
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PRMO 2015. p18.
18.
18.
18.
A subset
B
B
B
of the set of first
100
100
100
positive integers has the property that no two elements of
B
B
B
sum to
125.
125.
125.
What is the maximum possible number of elements in
B
?
B ?
B
?
17
1
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PRMO 2015. p17.
17.
17.
17.
Let
a
,
a,
a
,
b
,
b,
b
,
and
c
.
c.
c
.
be such that
a
+
b
+
c
=
0
a+b+c=0
a
+
b
+
c
=
0
and
P
=
a
2
2
a
2
+
b
c
+
b
2
2
b
2
+
c
a
+
c
2
2
c
2
+
a
b
P=\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ca}+\frac{c^2}{2c^2+ab}
P
=
2
a
2
+
b
c
a
2
+
2
b
2
+
c
a
b
2
+
2
c
2
+
ab
c
2
is defined. What is the value of
P
?
P ?
P
?
16
1
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PRMO 2015. p16.
16.
16.
16.
In an acute angle triangle
A
B
C
,
ABC,
A
BC
,
let
D
D
D
be the foot of the altitude from
A
,
A,
A
,
and
E
E
E
be the midpoint of
B
C
.
BC.
BC
.
Let
F
F
F
be the midpoint of
A
C
.
AC.
A
C
.
Suppose
∠
B
A
E
=
4
0
o
.
\angle{BAE}=40^o.
∠
B
A
E
=
4
0
o
.
If
∠
D
A
E
=
∠
D
F
E
,
\angle{DAE}=\angle{DFE},
∠
D
A
E
=
∠
D
FE
,
What is the magnitude of
∠
A
D
F
\angle{ADF}
∠
A
D
F
in degrees
?
?
?
15
1
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PRMO 2015. p15.
15.
15.
15.
Let
n
n
n
be the largest integer that is the product of exactly
3
3
3
distinct prime numbers,
x
,
y
,
x,y,
x
,
y
,
and
10
x
+
y
,
10x+y,
10
x
+
y
,
where
x
x
x
and
y
y
y
are digits. What is the sum of digits of
n
?
n ?
n
?
14
1
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PRMO 2015. p14.
14.
14.
14.
If
3
x
+
2
y
=
985.
3^x+2^y=985.
3
x
+
2
y
=
985.
and
3
x
−
2
y
=
473.
3^x-2^y=473.
3
x
−
2
y
=
473.
What is the value of
x
y
?
xy ?
x
y
?
13
1
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PRMO 2015. p13.
13.
13.
13.
At a party, each man danced with exactly four women and each woman danced with exactly three men. Nine men attended the party. How many women attended the party
?
?
?
12
1
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PRMO 2015. p12.
12.
12.
12.
In a rectangle
A
B
C
D
ABCD
A
BC
D
A
B
=
8
AB=8
A
B
=
8
and
B
C
=
20.
BC=20.
BC
=
20.
Let
P
P
P
be a point on
A
D
AD
A
D
such that
∠
B
P
C
=
9
0
o
.
\angle{BPC}=90^o.
∠
BPC
=
9
0
o
.
If
r
1
,
r
2
,
r
3
.
r_1,r_2,r_3.
r
1
,
r
2
,
r
3
.
are the radii of the incircles of triangles
A
P
B
,
APB,
A
PB
,
B
P
C
,
BPC,
BPC
,
and
C
P
D
.
CPD.
CP
D
.
what is the value of
r
1
+
r
2
+
r
3
?
r_1+r_2+r_3 ?
r
1
+
r
2
+
r
3
?
11
1
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PRMO 2015. p11.
11.
11.
11.
Let
a
,
a,
a
,
b
,
b,
b
,
and
c
c
c
be real numbers such that
a
−
7
b
+
8
c
=
4.
a-7b+8c=4.
a
−
7
b
+
8
c
=
4.
and
8
a
+
4
b
−
c
=
7.
8a+4b-c=7.
8
a
+
4
b
−
c
=
7.
What is the value of
a
2
−
b
2
+
c
2
?
a^2-b^2+c^2 ?
a
2
−
b
2
+
c
2
?
10
1
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PRMO 2015. p10.
10.
10.
10.
A
2
×
3
2\times 3
2
×
3
rectangle and a
3
×
4
3 \times 4
3
×
4
rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square
?
?
?
9
1
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PRMO 2015. p9.
9.
9.
9.
What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length
12
?
12 ?
12
?
7
1
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PRMO 2015. p7.
7.
7.
7.
Let
E
(
n
)
E(n)
E
(
n
)
denote the sum of even digits of
n
.
n.
n
.
For example,
E
(
1243
)
=
2
+
4
=
6.
E(1243)=2+4=6.
E
(
1243
)
=
2
+
4
=
6.
What is the value of
E
(
1
)
+
E
(
2
)
+
E
(
3
)
+
.
.
.
+
E
(
100
)
?
E(1)+E(2)+E(3)+...+E(100) ?
E
(
1
)
+
E
(
2
)
+
E
(
3
)
+
...
+
E
(
100
)?
6
1
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PRMO 2015. p6.
6.
6.
6.
How many two digit positive integers
N
N
N
have the property that the sum of
N
N
N
and the number obtained by reversing the order of the digits of
N
N
N
is a perfect square
?
?
?
5
1
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PRMO 2015. p5.
5.
5.
5.
Let
P
(
x
)
P(x)
P
(
x
)
be a non - zero polynomial with integer coefficients. If
P
(
n
)
P(n)
P
(
n
)
is divisible by
n
n
n
for each integer polynomial
n
.
n.
n
.
What is the value of
P
(
0
)
?
P(0) ?
P
(
0
)?
4
1
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PRMO 2015. p4.
4.
4.
4.
How many line segments have both their endpoints located at the vertices of a given cube
?
?
?
3
1
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PRMO 2015. p3.
3.
3.
3.
Positive integers
a
a
a
and
b
b
b
are such that
a
+
b
=
a
b
+
b
a
.
a+b=\frac{a}{b}+\frac{b}{a}.
a
+
b
=
b
a
+
a
b
.
What is the value of
a
2
+
b
2
?
a^2+b^2 ?
a
2
+
b
2
?
2
1
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PRMO 2015. p2.
2.
2.
2.
The equations
x
2
−
4
x
+
k
=
0
x^2-4x+k=0
x
2
−
4
x
+
k
=
0
and
x
2
+
k
x
−
4
=
0
,
x^2+kx-4=0,
x
2
+
k
x
−
4
=
0
,
where
k
k
k
is a real number, have exactly one common root. What is the value of
k
?
k ?
k
?
1
1
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PRMO 2015. p1.
1.
1.
1.
A man walks a certain distance and rides back in
3
3
4
;
3\frac{3}{4};
3
4
3
;
he could ride both ways in
2
1
2
2\frac{1}{2}
2
2
1
hours. How many hours would it take him to walk both ways
?
?
?