MathDB
Problems
Contests
National and Regional Contests
India Contests
India Pre-Regional Mathematical Olympiad
2014 India PRMO
2014 India PRMO
Part of
India Pre-Regional Mathematical Olympiad
Subcontests
(20)
20
1
Hide problems
2014 preRMO p20, A,B subsets of {1,2,..., 5}
What is the number of ordered pairs
(
A
,
B
)
(A,B)
(
A
,
B
)
where
A
A
A
and
B
B
B
are subsets of
{
1
,
2
,
.
.
.
,
5
}
\{1,2,..., 5\}
{
1
,
2
,
...
,
5
}
such that neither
A
⊆
B
A \subseteq B
A
⊆
B
nor
B
⊆
A
B \subseteq A
B
⊆
A
?
19
1
Hide problems
2014 preRMO p19, sum x_i=1, sum x_i/(1-x_i)=1, sum x^2_i/(1-x_i)=?
Let
x
1
,
x
2
,
.
.
.
,
x
2014
x_1,x_2,... ,x_{2014}
x
1
,
x
2
,
...
,
x
2014
be real numbers different from
1
1
1
, such that
x
1
+
x
2
+
.
.
.
+
x
2014
=
1
x_1 + x_2 +...+x_{2014} = 1
x
1
+
x
2
+
...
+
x
2014
=
1
and
x
1
1
−
x
1
+
x
2
1
−
x
2
+
.
.
.
+
x
2014
1
−
x
2014
=
1
\frac{x_1}{1-x_1}+\frac{x_2}{1-x_2}+...+\frac{x_{2014}}{1-x_{2014}}=1
1
−
x
1
x
1
+
1
−
x
2
x
2
+
...
+
1
−
x
2014
x
2014
=
1
What is the value of
x
1
2
1
−
x
1
+
x
2
2
1
−
x
2
+
.
.
.
+
x
2014
2
1
−
x
2014
\frac{x^2_1}{1-x_1}+\frac{x^2_2}{1-x_2}+...+\frac{x^2_{2014}}{1-x_{2014}}
1
−
x
1
x
1
2
+
1
−
x
2
x
2
2
+
...
+
1
−
x
2014
x
2014
2
?
18
1
Hide problems
2014 preRMO p18, f(mn) = f(m)f(n), f 1-1, N-> N, f (999) ?
Let
f
f
f
be a one-to-one function from the set of natural numbers to itself such that
f
(
m
n
)
=
f
(
m
)
f
(
n
)
f(mn) = f(m)f(n)
f
(
mn
)
=
f
(
m
)
f
(
n
)
for all natural numbers
m
m
m
and
n
n
n
. What is the least possible value of
f
(
999
)
f (999)
f
(
999
)
?
17
1
Hide problems
2014 preRMO p17, x^2+ax+b = 0 has integer roots , N(b) = 20
For a natural number
b
b
b
, let
N
(
b
)
N(b)
N
(
b
)
denote the number of natural numbers
a
a
a
for which the equation
x
2
+
a
x
+
b
=
0
x^2 + ax + b = 0
x
2
+
a
x
+
b
=
0
has integer roots. What is the smallest value of
b
b
b
for which
N
(
b
)
=
20
N(b) = 20
N
(
b
)
=
20
?
16
1
Hide problems
2014 preRMO p16, angle chasing candidate, incenter, incircle related
In a triangle
A
B
C
ABC
A
BC
, let
I
I
I
denote the incenter. Let the lines
A
I
,
B
I
AI,BI
A
I
,
B
I
and
C
I
CI
C
I
intersect the incircle at
P
,
Q
P,Q
P
,
Q
and
R
R
R
, respectively. If
∠
B
A
C
=
4
0
o
\angle BAC = 40^o
∠
B
A
C
=
4
0
o
, what is the value of
∠
Q
P
R
\angle QPR
∠
QPR
in degrees ?
15
1
Hide problems
2014 preRMO p15, computational with right triangle and midpoints
Let
X
O
Y
XOY
XO
Y
be a triangle with
∠
X
O
Y
=
9
0
o
\angle XOY = 90^o
∠
XO
Y
=
9
0
o
. Let
M
M
M
and
N
N
N
be the midpoints of legs
O
X
OX
OX
and
O
Y
OY
O
Y
, respectively. Suppose that
X
N
=
19
XN = 19
XN
=
19
and
Y
M
=
22
YM =22
Y
M
=
22
. What is
X
Y
XY
X
Y
?
14
1
Hide problems
2014 preRMO p14, 8-ounce mixture of coffee and milk for Manjul’s family
One morning, each member of Manjul’s family drank an
8
8
8
-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank
1
/
7
1/7
1/7
-th of the total amount of milk and
2
/
17
2/17
2/17
-th of the total amount of coffee. How many people are there in Manjul’s family?
13
1
Hide problems
2014 preRMO p13, 8n/(9999-n) an integer, if 1<=n<= 2014
For how many natural numbers
n
n
n
between
1
1
1
and
2014
2014
2014
(both inclusive) is
8
n
9999
−
n
\frac{8n}{9999-n}
9999
−
n
8
n
an integer?
12
1
Hide problems
2014 preRMO p12, sum of radii of triangle incircles wanted
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
∠
D
A
B
=
∠
B
D
C
=
9
0
o
\angle DAB =\angle B DC = 90^o
∠
D
A
B
=
∠
B
D
C
=
9
0
o
. Let the incircles of triangles
A
B
D
ABD
A
B
D
and
B
C
D
BCD
BC
D
touch
B
D
BD
B
D
at
P
P
P
and
Q
Q
Q
, respectively, with
P
P
P
lying in between
B
B
B
and
Q
Q
Q
. If
A
D
=
999
AD = 999
A
D
=
999
and
P
Q
=
200
PQ = 200
PQ
=
200
then what is the sum of the radii of the incircles of triangles
A
B
D
ABD
A
B
D
and
B
D
C
BDC
B
D
C
?
11
1
Hide problems
2014 preRMO p11, gcd, xy = x + y + (x, y)
For natural numbers
x
x
x
and
y
y
y
, let
(
x
,
y
)
(x,y)
(
x
,
y
)
denote the greatest common divisor of
x
x
x
and
y
y
y
. How many pairs of natural numbers
x
x
x
and
y
y
y
with
x
≤
y
x \le y
x
≤
y
satisfy the equation
x
y
=
x
+
y
+
(
x
,
y
)
xy = x + y + (x, y)
x
y
=
x
+
y
+
(
x
,
y
)
?
10
1
Hide problems
2014 preRMO p10, computational with ratios and areas
In a triangle
A
B
C
,
X
ABC, X
A
BC
,
X
and
Y
Y
Y
are points on the segments
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
A
X
:
X
B
=
1
:
2
AX : XB = 1 : 2
A
X
:
XB
=
1
:
2
and
A
Y
:
Y
C
=
2
:
1
AY :YC = 2:1
A
Y
:
Y
C
=
2
:
1
. If the area of triangle
A
X
Y
AXY
A
X
Y
is
10
10
10
, then what is the area of triangle
A
B
C
ABC
A
BC
?
9
1
Hide problems
2014 preRMO p9, a,b roots of x^2-kx+l=0, a+1/b,b+1/a roots of x^2-px+q=0
Natural numbers
k
,
l
,
p
k, l,p
k
,
l
,
p
and
q
q
q
are such that if
a
a
a
and
b
b
b
are roots of
x
2
−
k
x
+
l
=
0
x^2 - kx + l = 0
x
2
−
k
x
+
l
=
0
then
a
+
1
b
a +\frac1b
a
+
b
1
and
b
+
1
a
b + \frac1a
b
+
a
1
are the roots of
x
2
−
p
x
+
q
=
0
x^2 -px + q = 0
x
2
−
p
x
+
q
=
0
. What is the sum of all possible values of
q
q
q
?
8
1
Hide problems
2014 preRMO p8, S, SU{15}, SU{15,1} have means M, M+2,M+1
Let
S
S
S
be a set of real numbers with mean
M
M
M
. If the means of the sets
S
∪
{
15
}
S\cup \{15\}
S
∪
{
15
}
and
S
∪
{
15
,
1
}
S\cup \{15,1\}
S
∪
{
15
,
1
}
are
M
+
2
M + 2
M
+
2
and
M
+
1
M + 1
M
+
1
, respectively, then how many elements does
S
S
S
have?
7
1
Hide problems
2014 preRMO p7, if x^{x^4}=4, then x^{x^2}+x^{x^8} =?
If
x
x
4
=
4
x^{x^4}=4
x
x
4
=
4
what is the value of
x
x
2
+
x
x
8
x^{x^2}+x^{x^8}
x
x
2
+
x
x
8
?
6
1
Hide problems
2014 preRMO p6, x^2 -nx + 2014 = 0 integer roots
What is the smallest possible natural number
n
n
n
for which the equation
x
2
−
n
x
+
2014
=
0
x^2 -nx + 2014 = 0
x
2
−
n
x
+
2014
=
0
has integer roots?
5
1
Hide problems
2014 preRMO p5, a+1=b+2=c+3=d+4=e+5=a+b+c+d+e+3
If real numbers
a
,
b
,
c
,
d
,
e
a, b, c, d, e
a
,
b
,
c
,
d
,
e
satisfy
a
+
1
=
b
+
2
=
c
+
3
=
d
+
4
=
e
+
5
=
a
+
b
+
c
+
d
+
e
+
3
a + 1 = b + 2 = c + 3 = d + 4 = e + 5 = a + b + c + d + e + 3
a
+
1
=
b
+
2
=
c
+
3
=
d
+
4
=
e
+
5
=
a
+
b
+
c
+
d
+
e
+
3
, what is the value of
a
2
+
b
2
+
c
2
+
d
2
+
e
2
a^2 + b^2 + c^2 + d^2 + e^2
a
2
+
b
2
+
c
2
+
d
2
+
e
2
?
4
1
Hide problems
2014 preRMO p4, integer sidelenths, max perimeter
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is
17
17
17
. What is the greatest possible perimeter of the triangle?
3
1
Hide problems
2014 preRMO p3, ABCD with perp. diagonals, AB=20, BC=70, CD=90, DA=?
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with perpendicular diagonals. If
A
B
=
20
,
B
C
=
70
AB = 20, BC = 70
A
B
=
20
,
BC
=
70
and
C
D
=
90
CD = 90
C
D
=
90
, then what is the value of
D
A
DA
D
A
?
2
1
Hide problems
2014 preRMO p2, sequence, each term is sum of cubes of digits of last term
The first term of a sequence is
2014
2014
2014
. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the
2014
2014
2014
th term of the sequence?
1
1
Hide problems
2014 preRMO p1, largest prime factor of k if k^2 < 2014 < (k +1)^2,
A natural number
k
k
k
is such that
k
2
<
2014
<
(
k
+
1
)
2
k^2 < 2014 < (k +1)^2
k
2
<
2014
<
(
k
+
1
)
2
. What is the largest prime factor of
k
k
k
?