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Contests
National and Regional Contests
India Contests
Mathematics Talent Reward Programme (MTRP)
2017 Mathematical Talent Reward Programme
2017 Mathematical Talent Reward Programme
Part of
Mathematics Talent Reward Programme (MTRP)
Subcontests
(16)
SAQ: P 6
1
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Starting with A, B, C, D can you get a bigger square by some sequence of steps
Let us consider an infinite grid plane as shown below. We start with 4 points
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
, that form a square.We perform the following operation: We pick two points
X
X
X
and
Y
Y
Y
from the currant points.
X
X
X
is reflected about
Y
Y
Y
to get
X
′
X'
X
′
. We remove
X
X
X
and add
X
′
X'
X
′
to get a new set of 4 points and treat it as our currant points.For example in the figure suppose we choose
A
A
A
and
B
B
B
(we can choose any other pair too). Then reflect
A
A
A
about
B
B
B
to get
A
′
A'
A
′
. We remove
A
A
A
and add
A
′
A'
A
′
. Thus
A
′
A'
A
′
,
B
B
B
,
C
C
C
,
D
D
D
is our new 4 points. We may again choose
D
D
D
and
A
′
A'
A
′
from the currant points. Reflect
D
D
D
about
A
′
A'
A
′
to obtain
D
′
D'
D
′
and hence
A
′
A'
A
′
,
B
B
B
,
C
C
C
,
D
′
D'
D
′
are now new set of points. Then similar operation is performed on this new 4 points and so on.Starting with
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
can you get a bigger square by some sequence of such operations?
SAQ: P 5
1
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Show that there exists three numbers a, b, c in arithmatic progression
Let
N
\mathbb{N}
N
be the set of all natural numbers. Let
f
:
N
→
N
f:\mathbb{N} \to \mathbb{N}
f
:
N
→
N
be a bijective function. Show that there exists three numbers
a
a
a
,
b
b
b
,
c
c
c
in arithmatic progression such that
f
(
a
)
<
f
(
b
)
<
f
(
c
)
f(a)<f(b)<f(c)
f
(
a
)
<
f
(
b
)
<
f
(
c
)
SAQ: P 4
1
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Show that if Statement 1 is true then Statement 2 is also true
An irreducible polynomial is a not-constant polynomial that cannot be factored into product of two non-constant polynomials. Consider the following statements :- Statement 1 :
p
(
x
)
p(x)
p
(
x
)
be any monic irreducible polynomial with integer coefficients and degree
≥
4
\geq 4
≥
4
. Then
p
(
n
)
p(n)
p
(
n
)
is a prime for at least one natural number
n
n
n
Statement 2 :
n
2
+
1
n^2+1
n
2
+
1
is prime for infinitely many values of natural number
n
n
n
Show that if Statement 1 is true then Statement 2 is also true
SAQ: P 3
1
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Find out if there exists such an function
Let
f
:
[
0
,
1
]
→
[
0
,
1
]
f:[0,1]\to [0,1]
f
:
[
0
,
1
]
→
[
0
,
1
]
be a continuous function. We say
f
≡
0
f\equiv 0
f
≡
0
if
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
for all
x
∈
[
0
,
1
]
x\in [0,1]
x
∈
[
0
,
1
]
and similarly
f
≢
0
f\not\equiv 0
f
≡
0
if there exists at least one
x
∈
[
0
,
1
]
x\in [0,1]
x
∈
[
0
,
1
]
such that
f
(
x
)
≠
0
f(x)\neq 0
f
(
x
)
=
0
. Suppose
f
≢
0
f\not\equiv 0
f
≡
0
,
f
∘
f
≢
0
f \circ f \not\equiv 0
f
∘
f
≡
0
but
f
∘
f
∘
f
≡
0
f \circ f \circ f \equiv 0
f
∘
f
∘
f
≡
0
. Do there exists such an
f
f
f
? If yes construct such an function, if no prove it
SAQ: P 2
1
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Prove this inequality from MTRP 2017
Let
a
a
a
,
b
b
b
,
c
c
c
be positive reals such that
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
. Show that
a
b
+
c
+
b
c
+
a
+
c
a
+
b
≤
6
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
\sqrt{\frac{a}{b+c}} + \sqrt{\frac{b}{c+a}} + \sqrt{\frac{c}{a+b}} \leq \frac{6}{\sqrt(a+b)(b+c)(c+a)}
b
+
c
a
+
c
+
a
b
+
a
+
b
c
≤
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
6
SAQ: P 1
1
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Prove that P(x+1)=Q(x-1) has a real solution if P(x)=Q(x) has no real solution
A monic polynomial is a polynomial whose highest degree coefficient is 1. Let
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
be monic polynomial with real coefficients and
d
e
g
P
(
x
)
=
d
e
g
Q
(
x
)
=
10
degP(x)=degQ(x)=10
d
e
g
P
(
x
)
=
d
e
g
Q
(
x
)
=
10
. Prove that if the equation
P
(
x
)
=
Q
(
x
)
P(x)=Q(x)
P
(
x
)
=
Q
(
x
)
has no real solutions then
P
(
x
+
1
)
=
Q
(
x
−
1
)
P(x+1)=Q(x-1)
P
(
x
+
1
)
=
Q
(
x
−
1
)
has a real solution
MCQ: P 10
1
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Find out about the property of the function
Let
f
:
R
→
R
f:\mathbb{R}\to \mathbb{R}
f
:
R
→
R
be a differentiable function such that
lim
x
→
∞
f
′
(
x
)
=
1
\lim \limits_{x\to \infty}f'(x)=1
x
→
∞
lim
f
′
(
x
)
=
1
, then [*]
f
f
f
is increasing [*]
f
f
f
is unbounded [*]
f
′
f'
f
′
is bounded [*] All of these
MCQ: P 9
1
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Find the length of AB
From a point
P
P
P
outside of a circle with centre
O
O
O
, tangent segments
P
A
PA
P
A
and
P
B
PB
PB
are drawn.
1
O
A
2
+
1
P
A
2
=
1
16
\frac{1}{OA^2}+\frac{1}{PA^2}=\frac{1}{16}
O
A
2
1
+
P
A
2
1
=
16
1
then
A
B
=
AB=
A
B
=
[*] 4 [*] 6 [*] 8 [*] 10
MCQ: P 8
1
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Find the number of finite sequences
How many finite sequances
x
1
,
x
2
,
⋯
,
x
m
x_1,x_2,\cdots,x_m
x
1
,
x
2
,
⋯
,
x
m
are there such that
x
i
=
1
x_i=1
x
i
=
1
or 2 and
∑
i
=
1
m
x
i
=
10
\sum \limits_{i=1}^mx_i=10
i
=
1
∑
m
x
i
=
10
?[*] 89 [*] 73 [*] 107 [*] 119
MCQ: P 7
1
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Find the length of the diagonal BD
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with sides
A
B
=
2
AB=2
A
B
=
2
,
B
C
=
C
D
=
4
BC=CD=4
BC
=
C
D
=
4
and
D
A
=
5
DA=5
D
A
=
5
. The opposite angles
A
A
A
and
C
C
C
are equal. The length of diagonal
B
D
BD
B
D
equals[*]
2
6
2\sqrt{6}
2
6
[*]
3
3
3\sqrt{3}
3
3
[*]
3
6
3\sqrt{6}
3
6
[*]
2
3
2\sqrt{3}
2
3
MCQ: P 6
1
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Find the value of p(5)
Let
p
(
x
)
p(x)
p
(
x
)
be a polynomial of degree 4 with leading coefficients 1. Suppose
p
(
1
)
=
1
p(1)=1
p
(
1
)
=
1
,
p
(
2
)
=
2
p(2)=2
p
(
2
)
=
2
,
p
(
3
)
=
3
p(3)=3
p
(
3
)
=
3
,
p
(
4
)
=
4
p(4)=4
p
(
4
)
=
4
. Then
p
(
5
)
=
p(5)=
p
(
5
)
=
[*] 5 [*]
25
6
\frac{25}{6}
6
25
[*] 29 [*] 35
MCQ: P 5
1
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Find the number of ordered quadruples
Compute the number of ordered quadruples of positive integers
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
such that
a
!
⋅
b
!
⋅
c
!
⋅
d
!
=
24
!
a!\cdot b!\cdot c!\cdot d!=24!
a
!
⋅
b
!
⋅
c
!
⋅
d
!
=
24
!
[*] 4 [*] 4! [*]
4
4
4^4
4
4
[*] None of these
MCQ: P 4
1
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What can you say about the sum of fibonacci numbers till 2017
Let
F
1
=
F
2
=
1
F_1=F_2=1
F
1
=
F
2
=
1
. We define inductively
F
n
+
1
=
F
n
+
F
n
−
1
F_{n+1}=F_n+F_{n-1}
F
n
+
1
=
F
n
+
F
n
−
1
for all
n
≥
2
n\geq 2
n
≥
2
. Then the sum
F
1
+
F
2
+
⋯
+
F
2017
F_1+F_2+\cdots+F_{2017}
F
1
+
F
2
+
⋯
+
F
2017
is[*] Even but not divisible by 3 [*] Odd but divisible by 3 [*] Odd and leaves remainder 1 when divisible by 3 [*] None of these
MCQ: P 3
1
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Find a+b
Let
p
(
x
)
=
x
4
−
4
x
3
+
2
x
2
+
a
x
+
b
p(x)=x^4-4x^3+2x^2+ax+b
p
(
x
)
=
x
4
−
4
x
3
+
2
x
2
+
a
x
+
b
. Suppose that for every root
λ
\lambda
λ
of
p
p
p
,
1
λ
\frac{1}{\lambda}
λ
1
is also a root of
p
p
p
. Then
a
+
b
=
a+b=
a
+
b
=
[*] -3 [*] -6 [*] -4 [*] -8
MCQ: P 2
1
Hide problems
Find the limit
lim
x
→
∞
(
sin
x
x
)
1
x
2
=
\lim \limits_{x\to \infty} \left(\frac{\sin x}{x}\right)^{\frac{1}{x^2}}=
x
→
∞
lim
(
x
s
i
n
x
)
x
2
1
=
[*]
e
\sqrt{e}
e
[*]
∞
\infty
∞
[*] Does not exists [*] None of these
MCQ: P 1
1
Hide problems
Find the number of real solutions of the equation
The number of real solutions of the equation
(
9
10
)
x
=
−
3
+
x
−
x
2
\left(\frac{9}{10}\right)^x=-3+x-x^2
(
10
9
)
x
=
−
3
+
x
−
x
2
is[*] 2 [*] 0 [*] 1 [*] None of these