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Contests
National and Regional Contests
India Contests
Mathematics Talent Reward Programme (MTRP)
2016 Mathematical Talent Reward Programme
2016 Mathematical Talent Reward Programme
Part of
Mathematics Talent Reward Programme (MTRP)
Subcontests
(21)
SAQ: P 6
1
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A problem on partitions from MTRP 2016
Consider the set
A
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
A=\{1,2,3,4,5,6,7,8,9\}
A
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
.A partition
Π
\Pi
Π
of
A
A
A
is collection of disjoint sets whose union is
A
A
A
. For example,
Π
1
=
{
{
1
,
2
}
,
{
3
,
4
,
5
}
,
{
6
,
7
,
8
,
9
}
}
\Pi_1=\{\{1,2\},\{3,4,5\},\{6,7,8,9\}\}
Π
1
=
{{
1
,
2
}
,
{
3
,
4
,
5
}
,
{
6
,
7
,
8
,
9
}}
and
Π
2
=
{
{
1
}
,
{
2
,
5
}
,
{
3
,
7
}
,
{
4
,
5
,
6
,
7
,
8
,
9
}
}
\Pi _2 =\{\{1\},\{2,5\},\{3,7\},\{4,5,6,7,8,9\}\}
Π
2
=
{{
1
}
,
{
2
,
5
}
,
{
3
,
7
}
,
{
4
,
5
,
6
,
7
,
8
,
9
}}
can be considered as partitions of
A
A
A
. For, each
Π
\Pi
Π
partition ,we consider the function
π
\pi
π
defined on the elements of
A
A
A
.
π
(
x
)
\pi (x)
π
(
x
)
denotes the cardinality of the subset in
Π
\Pi
Π
which contains
x
x
x
. For, example in case of
Π
1
\Pi_1
Π
1
,
π
1
(
1
)
=
π
1
(
2
)
=
2
\pi_1(1)=\pi_1(2)=2
π
1
(
1
)
=
π
1
(
2
)
=
2
,
π
1
(
3
)
=
π
1
(
4
)
=
π
1
(
5
)
=
3
\pi_1(3)=\pi_1(4)=\pi_1 (5)=3
π
1
(
3
)
=
π
1
(
4
)
=
π
1
(
5
)
=
3
, and
π
1
(
6
)
=
π
1
(
7
)
=
π
1
(
8
)
=
π
1
(
9
)
=
4
\pi_1(6)=\pi_1(7)=\pi_1(8)=\pi_1(9)=4
π
1
(
6
)
=
π
1
(
7
)
=
π
1
(
8
)
=
π
1
(
9
)
=
4
. For
Π
2
\Pi_2
Π
2
we have
π
2
(
1
)
=
1
\pi_2(1)=1
π
2
(
1
)
=
1
,
π
2
(
2
)
=
π
2
(
5
)
=
2
\pi_2(2)=\pi_2(5)=2
π
2
(
2
)
=
π
2
(
5
)
=
2
,
π
2
(
3
)
=
π
2
(
7
)
=
2
\pi_2(3)=\pi_2(7)=2
π
2
(
3
)
=
π
2
(
7
)
=
2
and
π
2
(
4
)
=
π
2
(
6
)
=
π
2
(
8
)
=
π
2
(
9
)
=
4
\pi_2(4)=\pi_2(6)=\pi_2(8)=\pi_2(9)=4
π
2
(
4
)
=
π
2
(
6
)
=
π
2
(
8
)
=
π
2
(
9
)
=
4
Given any two partitions
Π
\Pi
Π
and
Π
′
\Pi '
Π
′
, show that there are two numbers
x
x
x
and
y
y
y
in
A
A
A
, such that
π
(
x
)
=
π
′
(
x
)
\pi (x)= \pi '(x)
π
(
x
)
=
π
′
(
x
)
and
π
(
y
)
=
π
′
(
y
)
\pi (y)= \pi'(y)
π
(
y
)
=
π
′
(
y
)
.[Hint: Consider the case where there is a block of size greater than or equal to 4 in a partition and the alternative case]
SAQ: P 5
1
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Prove that the functions f and g are equal
Let
N
\mathbb{N}
N
be the set of all positive integers.
f
,
g
:
N
→
N
f,g:\mathbb{N} \to \mathbb{N}
f
,
g
:
N
→
N
be funtions such that
f
f
f
is onto and
g
g
g
is one-one and
f
(
n
)
≥
g
(
n
)
f(n)\geq g(n)
f
(
n
)
≥
g
(
n
)
for all positive integers
n
n
n
. Prove that
f
=
g
f=g
f
=
g
.
SAQ: P 4
1
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Show that we can always find a circle such that all points lie inside the circle
For any given
k
k
k
points in a plane, we define the diameter of the points as the maximum distance between any two points among the given points. Suppose
n
n
n
points are there in a plane with diameter
d
d
d
. Show that we can always find a circle with radius
3
2
d
\frac{\sqrt{3}}{2}d
2
3
d
such that all points lie inside the circle.
SAQ: P 3
1
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Prove that there are n consecutive composite numbers all less than 4^{n+2}
Prove that for any positive integer
n
n
n
there are
n
n
n
consecutive composite numbers all less than
4
n
+
2
4^{n+2}
4
n
+
2
.
SAQ: P 2
1
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Determine the minimum possible height of the whole structure.
5 blocks of volume 1 cm
3
^3
3
, 1 cm
3
^3
3
, 1 cm
3
^3
3
, 1 cm
3
^3
3
and 4 cm
3
^3
3
are placed one above another to form a structure as shown in the figure. Suppose sum of surface areas of upper face of each is 48 cm
2
^2
2
. Determine the minimum possible height of the whole structure.
SAQ: P 1
1
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Show that there exist a polynomial P(x)
Show that there exist a polynomial
P
(
x
)
P(x)
P
(
x
)
whose one cofficient is
1
2016
\frac{1}{2016}
2016
1
and remaining cofficients are rational numbers, such that
P
(
x
)
P(x)
P
(
x
)
is an integer for any integer
x
x
x
.
MCQ: P 15
1
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Find number of solutions for x when x takes values 1,2,...,100
Suppose
50
x
50x
50
x
is divisible by 100 and
k
x
kx
k
x
is not divisible by 100 for all
k
=
1
,
2
,
⋯
,
49
k=1,2,\cdots, 49
k
=
1
,
2
,
⋯
,
49
Find number of solutions for
x
x
x
when
x
x
x
takes values
1
,
2
,
⋯
100
1,2,\cdots 100
1
,
2
,
⋯
100
.[*] 20 [*] 25 [*] 15 [*] 50
MCQ: P 14
1
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Find the value of x so that x[x[x[x]]]=88 where [x] is greatest integer function
Let
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denotes the greatest integer less than or equal to
x
x
x
. Find
x
x
x
such that
x
⌊
x
⌊
x
⌊
x
⌋
⌋
⌋
=
88
x\lfloor x\lfloor x\lfloor x\rfloor\rfloor \rfloor = 88
x
⌊
x
⌊
x
⌊
x
⌋⌋⌋
=
88
[*]
π
\pi
π
[*] 3.14[*]
22
7
\frac{22}{7}
7
22
[*] All of these
MCQ: P 13
1
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Find the value of P(0)
Let
P
(
x
)
=
x
2
+
b
x
+
c
P(x)=x^2+bx+c
P
(
x
)
=
x
2
+
b
x
+
c
. Suppose
P
(
P
(
1
)
)
=
P
(
P
(
−
2
)
)
=
0
P(P(1))=P(P(-2))=0
P
(
P
(
1
))
=
P
(
P
(
−
2
))
=
0
and
P
(
1
)
≠
P
(
−
2
)
P(1)\neq P(-2)
P
(
1
)
=
P
(
−
2
)
. Then
P
(
0
)
=
P(0)=
P
(
0
)
=
[*]
−
5
2
-\frac{5}{2}
−
2
5
[*]
−
3
2
-\frac{3}{2}
−
2
3
[*]
−
7
4
-\frac{7}{4}
−
4
7
[*]
6
7
\frac{6}{7}
7
6
MCQ: P 12
1
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Find the number of points of discontinuity
Let
f
(
x
)
=
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
f(x)=(x-1)(x-2)(x-3)
f
(
x
)
=
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
. Consider
g
(
x
)
=
m
i
n
{
f
(
x
)
,
f
′
(
x
)
}
g(x)=min\{f(x),f'(x)\}
g
(
x
)
=
min
{
f
(
x
)
,
f
′
(
x
)}
. Then the number of points of discontinuity are [*] 0 [*] 1 [*] 2 [*] More than 2
MCQ: P 11
1
Hide problems
What is the perimeter triangle BDP
In rectangle
A
B
C
D
ABCD
A
BC
D
,
A
D
=
1
AD=1
A
D
=
1
,
P
P
P
is on
A
B
AB
A
B
and
D
B
DB
D
B
and
D
P
DP
D
P
trisect
∠
A
D
C
\angle ADC
∠
A
D
C
. What is the perimeter
△
B
D
P
\triangle BDP
△
B
D
P
[*]
3
+
3
3
3+\frac{\sqrt{3}}{3}
3
+
3
3
[*]
2
+
4
3
3
2+\frac{4\sqrt{3}}{3}
2
+
3
4
3
[*]
2
+
2
2
2+2\sqrt{2}
2
+
2
2
[*]
3
+
3
5
2
\frac{3+3\sqrt{5}}{2}
2
3
+
3
5
MCQ: P 10
1
Hide problems
Find the maximum possible cardinality of S
Let
A
=
{
1
,
2
,
⋯
,
100
}
A=\{1,2,\cdots ,100\}
A
=
{
1
,
2
,
⋯
,
100
}
. Let
S
S
S
be a subset of power set of
A
A
A
such that any two elements of
S
S
S
has nonzero intersection (Note that elements of
S
S
S
are actually some subsets of
A
A
A
). Then the maximum possible cardinality of
S
S
S
is[*]
2
99
2^{99}
2
99
[*]
2
99
+
1
2^{99}+1
2
99
+
1
[*]
2
99
+
2
98
2^{99}+2^{98}
2
99
+
2
98
[*] None of these
MCQ: P 9
1
Hide problems
Find the value of f(7)
f
f
f
be a function satisfying
2
f
(
x
)
+
3
f
(
−
x
)
=
x
2
+
5
x
2f(x)+3f(-x)=x^2+5x
2
f
(
x
)
+
3
f
(
−
x
)
=
x
2
+
5
x
. Find
f
(
7
)
f(7)
f
(
7
)
[*]
−
105
4
-\frac{105}{4}
−
4
105
[*]
−
126
5
-\frac{126}{5}
−
5
126
[*]
−
120
7
-\frac{120}{7}
−
7
120
[*]
−
132
7
-\frac{132}{7}
−
7
132
MCQ: P 8
1
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Find a possible choice of p such that 16p+1 is a perfect cube
Let
p
p
p
be a prime such that
16
p
+
1
16p+1
16
p
+
1
is a perfect cube. A possible choice for
p
p
p
is [*] 283 [*] 307 [*] 593 [*] 691
MCQ: P 7
1
Hide problems
Find the limit from MTRP 2016 MCQ 7
Let
{
x
}
\{x\}
{
x
}
denote the fractional part of
x
x
x
. Then
lim
n
→
∞
{
(
1
+
2
)
2
n
}
\lim \limits_{n\to \infty} \left\{ \left(1+\sqrt{2}\right)^{2n}\right\}
n
→
∞
lim
{
(
1
+
2
)
2
n
}
equals[*] 0 [*] 0.5 [*] 1 [*] Does not exists
MCQ: P 6
1
Hide problems
Find the number of solutions of the equation 3^x+4^x=8^x in reals
Number of solutions of the equation
3
x
+
4
x
=
8
x
3^x+4^x=8^x
3
x
+
4
x
=
8
x
in reals is[*] 0 [*] 1 [*] 2 [*]
∞
\infty
∞
MCQ: P 5
1
Hide problems
Find about the quadrilateral ABCD whose vertices satisfy z^4+z^3+z^2+z+1=0
A
B
C
D
ABCD
A
BC
D
is a quadrilateral on complex plane whose four vertices satisfy
z
4
+
z
3
+
z
2
+
z
+
1
=
0
z^4+z^3+z^2+z+1=0
z
4
+
z
3
+
z
2
+
z
+
1
=
0
. Then
A
B
C
D
ABCD
A
BC
D
is a[*] Rectangle [*] Rhombus [*] Isosceles Trapezium [*] Square
MCQ: P 4
1
Hide problems
Find the sum of primes below 1000
There are 168 primes below 1000. Then sum of all primes below 1000 is [*] 11555 [*] 76127 [*] 57298 [*] 81722
MCQ: P 3
1
Hide problems
Find the where the complex number z/1-z^2 lies on
z
z
z
is a complex number and
∣
z
∣
=
1
|z|=1
∣
z
∣
=
1
and
z
2
≠
1
z^2\neq 1
z
2
=
1
. Then
z
1
−
z
2
\frac{z}{1-z^2}
1
−
z
2
z
lies on [*] A line not passing through origin [*]
∣
z
∣
=
2
|z|=2
∣
z
∣
=
2
[*]
x
x
x
-axis [*]
y
y
y
-axis
MCQ: P 2
1
Hide problems
Find the limit of the infinite sum of this function
Let
f
f
f
be a function satisfying
f
(
x
+
y
+
z
)
=
f
(
x
)
+
f
(
y
)
+
f
(
z
)
f(x+y+z)=f(x)+f(y)+f(z)
f
(
x
+
y
+
z
)
=
f
(
x
)
+
f
(
y
)
+
f
(
z
)
for all integers
x
x
x
,
y
y
y
,
z
z
z
. Suppose
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
,
f
(
2
)
=
2
f(2)=2
f
(
2
)
=
2
. Then
lim
n
→
∞
1
n
3
∑
r
=
1
n
4
r
f
(
3
r
)
\lim \limits_{n\to \infty} \frac{1}{n^3} \sum \limits_{r=1}^n 4rf(3r)
n
→
∞
lim
n
3
1
r
=
1
∑
n
4
r
f
(
3
r
)
equals[*] 4 [*] 6 [*] 12 [*] 24
MCQ: P 1
1
Hide problems
Find the sum of roots of the equation sinx tanx=x^2
Sum of the roots in the range
(
−
π
2
,
π
2
)
\left(-\frac{\pi}{2},\frac{\pi}{2} \right)
(
−
2
π
,
2
π
)
of the equation
sin
x
tan
x
=
x
2
\sin x\tan x=x^2
sin
x
tan
x
=
x
2
is[*]
π
2
\frac{\pi}{2}
2
π
[*] 0 [*] 1 [*] None of these