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Contests
National and Regional Contests
India Contests
Mathematics Talent Reward Programme (MTRP)
2019 Mathematical Talent Reward Programme
2019 Mathematical Talent Reward Programme
Part of
Mathematics Talent Reward Programme (MTRP)
Subcontests
(13)
MCQ: P 7
1
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Find the remainder
Let
n
n
n
be the number of isosceles triangles whose vertices are also the vertices of a regular 2019-gon. Then the remainder when
n
n
n
is divided by 100[*] 15 [*] 25 [*] 35 [*] 65
MCQ: P 6
1
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Find the limit
Find the limit
lim
n
→
∞
sin
n
!
\lim \limits_{n \to \infty} \sin{n!}
n
→
∞
lim
sin
n
!
[*] 1 [*] 0 [*]
π
4
\frac{\pi}{4}
4
π
[*] None of the above
MCQ: P 5
1
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Find the number ways you can choose two distinct integers
What is the number of ways you can choose two distinct integers
a
a
a
and
b
b
b
(unordered i.e. choosing
(
a
,
b
)
(a, b)
(
a
,
b
)
is same as choosing
(
b
,
a
)
(b, a)
(
b
,
a
)
from the set
{
1
,
2
,
⋯
,
100
}
\{1, 2, \cdots , 100\}
{
1
,
2
,
⋯
,
100
}
such that difference between them is atmost 10, i.e.
∣
a
−
b
∣
≤
10
|a-b|\leq 10
∣
a
−
b
∣
≤
10
[*]
(
100
2
)
−
(
90
2
)
{{100}\choose{2}} -{{90}\choose{2}}
(
2
100
)
−
(
2
90
)
[*]
(
100
2
)
−
90
{{100}\choose{2}} -90
(
2
100
)
−
90
[*]
(
100
2
)
−
(
90
2
)
−
100
{{100}\choose{2}} -{{90}\choose{2}}-100
(
2
100
)
−
(
2
90
)
−
100
[*] None of the above
MCQ: P 4
1
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Find the angle
Suppose
△
A
B
C
\triangle ABC
△
A
BC
is a triangle. From the vertex
A
A
A
draw the altitude
A
H
AH
A
H
, angle bisector (of
∠
B
A
C
\angle BAC
∠
B
A
C
)
A
P
AP
A
P
, median
A
D
AD
A
D
and these intersect the side
B
C
BC
BC
at the points (from left in order)
H
H
H
,
P
P
P
,
D
D
D
respectively. Let
∠
C
A
H
=
∠
H
A
P
=
∠
P
A
D
=
∠
D
A
B
\angle CAH = \angle HAP = \angle PAD = \angle DAB
∠
C
A
H
=
∠
H
A
P
=
∠
P
A
D
=
∠
D
A
B
. Then
∠
A
C
H
=
\angle ACH =
∠
A
C
H
=
[*]
22.
5
∘
22.5^{\circ}
22.
5
∘
[*]
4
5
∘
45^{\circ}
4
5
∘
[*]
67.
5
∘
67.5^{\circ}
67.
5
∘
[*] None of the above
MCQ: P 3
1
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Find the number of positive integral solutions
Find the number of positive integral solutions to the equation
∑
i
=
1
2019
1
0
a
i
=
∑
i
=
1
2019
1
0
b
i
\sum \limits_{i=1}^{2019} 10^{a_i}=\sum \limits_{i=1}^{2019} 10^{b_i}
i
=
1
∑
2019
1
0
a
i
=
i
=
1
∑
2019
1
0
b
i
, such that
a
1
<
a
2
<
⋯
<
a
2019
a_1<a_2<\cdots <a_{2019}
a
1
<
a
2
<
⋯
<
a
2019
,
b
1
<
b
2
<
⋯
<
b
2019
b_1<b_2<\cdots <b_{2019}
b
1
<
b
2
<
⋯
<
b
2019
and
a
2019
<
b
2019
a_{2019} < b_{2019}
a
2019
<
b
2019
[*] 1 [*] 2 [*] 2019 [*] None of the above
MCQ: P 2
1
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Find the number of integral solutions
What is the number of integral solutions of the equation
a
b
2
=
b
2
a
a^{b^2}=b^{2a}
a
b
2
=
b
2
a
, where a > 0 and
∣
b
∣
>
∣
a
∣
|b|>|a|
∣
b
∣
>
∣
a
∣
[*] 3 [*] 4 [*] 6 [*] 8
MCQ: P 1
1
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What can you say about f(x)
Let
f
:
(
0
,
∞
)
→
R
f : (0, \infty) \to \mathbb{R}
f
:
(
0
,
∞
)
→
R
is differentiable such that
lim
x
→
∞
f
(
x
)
=
2019
\lim \limits_{x \to \infty} f(x)=2019
x
→
∞
lim
f
(
x
)
=
2019
Then which of the following is correct?[*]
lim
x
→
∞
f
′
(
x
)
\lim \limits_{x \to \infty} f'(x)
x
→
∞
lim
f
′
(
x
)
always exists but not necessarily zero. [*]
lim
x
→
∞
f
′
(
x
)
\lim \limits_{x \to \infty} f'(x)
x
→
∞
lim
f
′
(
x
)
always exists and is equal to zero. [*]
lim
x
→
∞
f
′
(
x
)
\lim \limits_{x \to \infty} f'(x)
x
→
∞
lim
f
′
(
x
)
may not exist. [*]
lim
x
→
∞
f
′
(
x
)
\lim \limits_{x \to \infty} f'(x)
x
→
∞
lim
f
′
(
x
)
exists if
f
f
f
is twice differentiable.
SAQ: P 6
1
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Prove that there cannot be 5 collinear points
Consider a finite set of points,
Φ
\Phi
Φ
, in the
R
2
\mathbb{R}^2
R
2
, such that they follow the following properties :[*]
Φ
\Phi
Φ
doesn't contain the origin
{
(
0
,
0
)
}
\{(0,0)\}
{(
0
,
0
)}
and not all points are collinear. [*] If
α
∈
Φ
\alpha \in \Phi
α
∈
Φ
, then
−
α
∈
Φ
-\alpha \in \Phi
−
α
∈
Φ
,
c
α
∉
Φ
c\alpha \notin \Phi
c
α
∈
/
Φ
for
c
≠
1
c\neq 1
c
=
1
or
−
1
-1
−
1
[*] If
α
,
β
\alpha, \ \beta
α
,
β
are in
Φ
\Phi
Φ
, then the reflection of
β
\beta
β
in the line passing through the origin and perpendicular to the line containing origin and
α
\alpha
α
is in
Φ
\Phi
Φ
[*] If
α
=
(
a
,
b
)
,
β
=
(
c
,
d
)
\alpha = (a,b) , \ \beta = (c,d)
α
=
(
a
,
b
)
,
β
=
(
c
,
d
)
, (both
α
,
β
∈
Φ
\alpha, \ \beta \in \Phi
α
,
β
∈
Φ
) then
2
(
a
c
+
b
d
)
c
2
+
d
2
∈
Z
\frac{2(ac+bd)}{c^2+d^2} \in \mathbb{Z}
c
2
+
d
2
2
(
a
c
+
b
d
)
∈
Z
Prove that there cannot be 5 collinear points in
Φ
\Phi
Φ
SAQ: P 5
1
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Find the number of MTRP-numbers
Let a fi xed natural number m be given. Call a positive integer n to be an MTRP-number iff[*]
n
≡
1
(
m
o
d
m
)
n \equiv 1\ (mod\ m)
n
≡
1
(
m
o
d
m
)
[*] Sum of digits in decimal representation of
n
2
n^2
n
2
is greater than equal to sum of digits in decimal representation of
n
n
n
How many MTRP-numbers are there ?
SAQ: P 4
1
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Find such numbers
Are there infinitely many natural numbers
n
n
n
such that the sum of 2019th powers of the digits of
n
n
n
is equal to
n
n
n
? You don't need to find any such
n
n
n
. Just provide mathematical justi fication if you think there are in finitely many or finitely many such natural numbers
SAQ: P 3
1
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Prove this inequality
Suppose
a
a
a
,
b
b
b
,
c
c
c
are three positive real numbers with
a
+
b
+
c
=
3
a + b + c = 3
a
+
b
+
c
=
3
. Prove that
a
b
2
+
c
+
b
c
2
+
a
+
c
a
2
+
b
≥
3
2
\frac{a}{b^2 + c}+\frac{b}{c^2 + a}+\frac{c}{a^2 + b}\geq \frac{3}{2}
b
2
+
c
a
+
c
2
+
a
b
+
a
2
+
b
c
≥
2
3
SAQ: P 2
1
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Find All Matrices
How many
n
×
n
n\times n
n
×
n
matrices
A
A
A
, with all entries from the set
{
0
,
1
,
2
}
\{0, 1, 2\}
{
0
,
1
,
2
}
, are there, such that for all
i
=
1
,
2
,
⋯
,
n
i=1,2,\cdots,n
i
=
1
,
2
,
⋯
,
n
A
i
i
>
∑
j
=
1
j
≠
i
n
A
i
j
A_{ii} > \displaystyle{\sum \limits_{j=1 j\neq i}^n} A_{ij}
A
ii
>
j
=
1
j
=
i
∑
n
A
ij
[Where
A
i
j
A_{ij}
A
ij
is the
(
i
,
j
)
(i,j)
(
i
,
j
)
th element of the matrix
A
A
A
]
SAQ: P 1
1
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Find all functions
Find all functions
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
x
)
≥
0
∀
x
∈
R
f(x)\geq 0\ \forall \ x\in \mathbb{R}
f
(
x
)
≥
0
∀
x
∈
R
,
f
′
(
x
)
f'(x)
f
′
(
x
)
exists
∀
x
∈
R
\forall \ x\in \mathbb{R}
∀
x
∈
R
and
f
′
(
x
)
≥
0
∀
x
∈
R
f'(x)\geq 0\ \forall \ x\in \mathbb{R}
f
′
(
x
)
≥
0
∀
x
∈
R
and
f
(
n
)
=
0
∀
n
∈
Z
f(n)=0\ \forall \ n\in \mathbb{Z}
f
(
n
)
=
0
∀
n
∈
Z