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Problems
Contests
National and Regional Contests
India Contests
Mathematics Talent Reward Programme (MTRP)
2024 Mathematical Talent Reward Programme
2024 Mathematical Talent Reward Programme
Part of
Mathematics Talent Reward Programme (MTRP)
Subcontests
(10)
10
1
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MTRP OBJECTIVE Q10
In MTRP district there are
10
10
10
cities. Bob the builder wants to make roads between cities in such a way so that one can go from one city to the other through exactly one unique path. The government has allotted him a budget of Rs.
20
20
20
and each road requires a positive integer amount (in Rs.) to build. In how many ways he can build such a network of roads? It is known that in the MTRP district, any positive integer amount of rupees can be used to construct a road, and that the full budget is used by Bob in the construction. Write the last two digits of your answer.
9
1
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MTRP OBJECTIVE Q9
Find the number of integer polynomials
P
P
P
such that
P
(
x
)
2
=
P
(
P
(
x
)
)
∀
x
P(x)^2 = P(P(x)) \forall x
P
(
x
)
2
=
P
(
P
(
x
))
∀
x
.
8
1
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MTRP OBJECTIVE Q8
Find the remainder when
202
4
202
3
202
2
2021..
.
3
2
+
202
5
202
1
201
7
2013..
.
5
1
2024^{2023^{2022^{2021...^{3^{2}}}}} + 2025^{2021^{2017^{2013...^{5^{1}}}}}
202
4
202
3
202
2
2021..
.
3
2
+
202
5
202
1
201
7
2013..
.
5
1
is divided by
19
19
19
.
7
1
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MTRP OBJECTIVE Q7
△
A
B
C
\bigtriangleup ABC
△
A
BC
triangle such that AB = AC, \angle BAC = 20 \textdegree.
P
P
P
is on
A
B
AB
A
B
such that
A
P
=
B
C
AP = BC
A
P
=
BC
, find
1
2
∠
A
P
C
\frac{1}{2}\angle APC
2
1
∠
A
PC
in degrees.
6
2
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MTRP OBJECTIVE Q6
Find the maximum possible length of a sequence consisting of non-zero integers, in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.
MTRP SUBJECTIVE Q6
Show that there exists an integer polynomial
P
P
P
such that
P
(
1
)
=
2024
P(1) = 2024
P
(
1
)
=
2024
and the set of prime divisors of {
P
(
2
k
)
P(2^k)
P
(
2
k
)
},
k
=
0
,
1
,
2
,
.
.
.
.
.
k=0,1,2,.....
k
=
0
,
1
,
2
,
.....
is an infinite set.
5
2
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MTRP OBJECTIVE Q5
How many positive integers
n
n
n
between
1
1
1
and
2024
2024
2024
(both included) are there such that
⌊
n
⌋
\lfloor{\sqrt{n}}\rfloor
⌊
n
⌋
divides
n
n
n
? (For
x
∈
R
,
⌊
n
⌋
x \in \mathbb{R}, \lfloor{n}\rfloor
x
∈
R
,
⌊
n
⌋
denotes the greatest integer less than or equal to
x
x
x
.)
(
A
)
44
(A) 44
(
A
)
44
(
B
)
132
(B) 132
(
B
)
132
(
C
)
1012
(C) 1012
(
C
)
1012
(
D
)
2024
(D) 2024
(
D
)
2024
MTRP SUBJECTIVE Q5
Let
f
:
N
⟶
N
f:\mathbb{N} \longrightarrow \mathbb{N}
f
:
N
⟶
N
such that
f
(
m
)
−
f
(
n
)
=
f
(
m
−
n
)
1
0
n
∀
m
>
n
∈
N
f(m) - f(n) = f(m-n)10^n \forall m>n \in \mathbb{N}
f
(
m
)
−
f
(
n
)
=
f
(
m
−
n
)
1
0
n
∀
m
>
n
∈
N
. Additionally, gcd
(
f
(
k
)
,
f
(
k
+
1
)
)
=
1
∀
k
∈
N
(f(k),f(k+1)) = 1 \forall k \in \mathbb{N}
(
f
(
k
)
,
f
(
k
+
1
))
=
1∀
k
∈
N
. Show that if
a
,
b
a,b
a
,
b
are coprime natural numbers, that is, gcd
(
a
,
b
)
=
1
(a,b) = 1
(
a
,
b
)
=
1
then
f
(
a
)
,
f
(
b
)
f(a),f(b)
f
(
a
)
,
f
(
b
)
are also coprime.
4
2
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MTRP OBJECTIVE Q4
Two circles (centres
d
d
d
apart) have radii
15
,
95
15,95
15
,
95
. The external tangents to the circles cut at
60
60
60
degrees, find
d
d
d
.
(
A
)
40
(A) 40
(
A
)
40
(
B
)
80
(B) 80
(
B
)
80
(
C
)
120
(C) 120
(
C
)
120
(
D
)
160
(D) 160
(
D
)
160
MTRP SUBJECTIVE Q4
MTRPia in
2044
2044
2044
is highly advanced and a lot of the work is done by disc-shaped robots, each of radius
1
1
1
unit. In order to not collide with each other, there robots have a smaller
360
360
360
-degree camera mounted on top, as shown in the figure (robot
r
1
r_1
r
1
'sees' robot
r
2
r_2
r
2
). Each of there cameras themselves are smaller discs of radius
c
c
c
. Suppose there are three robots
r
1
,
r
2
,
r
3
r_1, r_2, r_3
r
1
,
r
2
,
r
3
placed 'consecutively' such that
r
2
r_2
r
2
is roughly in the middle. The angle between the lines joining the centres of
r
1
,
r
2
r_1, r_2
r
1
,
r
2
and
r
2
,
r
3
r_2, r_3
r
2
,
r
3
is given to be
θ
\theta
θ
. The distance between the centres of
r
1
,
r
2
=
r_1,r_2 =
r
1
,
r
2
=
distance between centres of
r
2
,
r
3
=
d
r_2,r_3 = d
r
2
,
r
3
=
d
. Show (with the aid of clear diagrams) that
r
1
r_1
r
1
and
r
3
r_3
r
3
can see each other iff
sin
θ
>
1
−
c
d
\sin{\theta} > \frac{1-c}{d}
sin
θ
>
d
1
−
c
. As a bonus, try to show that in a longer 'chain' of such robots (same
d
d
d
,
θ
\theta
θ
), if
sin
θ
>
1
−
c
d
\sin{\theta} > \frac{1-c}{d}
sin
θ
>
d
1
−
c
then all robots can see each other.
3
2
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MTRP OBJECTIVE Q3
The smallest positive integer which can be expressed as sum of positive perfect cubes (possibly with repetition and/or with a single element sum) in at least two different ways in
(
A
)
8
(A) 8
(
A
)
8
(
B
)
1729
(B) 1729
(
B
)
1729
(
C
)
2023
(C) 2023
(
C
)
2023
(
D
)
2024
(D) 2024
(
D
)
2024
MTRP SUBJECTIVE Q3
P
(
n
)
\mathcal{P}(\mathbb{n})
P
(
n
)
denotes the collection of all subsets of
N
\mathbb{N}
N
. Let
f
:
N
⟶
P
(
n
)
f:\mathbb{N} \longrightarrow \mathcal{P}(\mathbb{n})
f
:
N
⟶
P
(
n
)
be a function such that
f
(
n
)
=
⋃
d
∣
n
,
d
<
n
,
n
≥
2
f
(
d
)
f(n) = \bigcup_{d|n,d<n,n \geq 2} f(d)
f
(
n
)
=
d
∣
n
,
d
<
n
,
n
≥
2
⋃
f
(
d
)
Find the number of such functions
f
f
f
for which the range of
f
⊆
f \subseteq
f
⊆
{
1
,
2
,
3....2024
1,2,3....2024
1
,
2
,
3....2024
}.
2
2
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MTRP OBJECTIVE Q2
How many triangles are in this figure? [asy] import olympiad; pair A = (0,0); pair B = (0,1); pair C = (0,2); pair D = (0,3); pair E = (0,4); pair F = (1,0); pair G = (2,0); pair H = (3,0); pair I = (4,0); pair J = (1,4); pair K = (2,4); pair L = (3,4); pair M = (4,4); pair N = (4,3); pair O = (4,2); pair P = (4,1); draw(A--E--I--A); draw(M--E--I--M); draw(B--F); draw(C--G); draw(D--H); draw(L--N); draw(O--K); draw(P--J); draw(B--F); draw(B--F); draw(H--P); draw(G--O); draw(F--N); draw(B--L); draw(C--K); draw(D--J); draw(A--M); [/asy]
(
A
)
56
(A) 56
(
A
)
56
(
B
)
60
(B) 60
(
B
)
60
(
C
)
64
(C) 64
(
C
)
64
(
D
)
68
(D) 68
(
D
)
68
MTRP SUBJECTIVE Q2
Find positive reals
a
,
b
,
c
a,b,c
a
,
b
,
c
such that:
a
b
+
c
+
b
c
+
a
+
c
a
+
b
=
2
\sqrt{\frac{a}{b+c}} + \sqrt{\frac{b}{c+a}} + \sqrt{\frac{c}{a+b}} = 2
b
+
c
a
+
c
+
a
b
+
a
+
b
c
=
2
1
2
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MTRP OBJECTIVE Q1
Hari the milkman delivers milk to his customers everyday by travelling on his cycle. Each litre of milk costs him Rs.
20
20
20
, and he sells it at Rs.
24
24
24
. One day while riding his cycle with
20
20
20
L, Hari trips and loses
5
5
5
L of it, and he decides to mix some water with the rest of the milk. His customers can detect if the milk is more than
10
10
10
% impure (
1
1
1
L water in
10
10
10
L misture). Given that he doesn't wish to make his customers angry, what is his maximum profit for the day?
(
A
)
(A)
(
A
)
Rs
12
12
12
profit
(
B
)
(B)
(
B
)
Rs
24
24
24
profit
(
C
)
(C)
(
C
)
No profit
(
D
)
(D)
(
D
)
Rs
12
12
12
loss
MTRP SUBJECTIVE Q1
The Integration Premier League has
n
n
n
teams competing. The tournament follows a round-robin system, that is, where every pair of teams play each other exactly once. So every team plays exactly
n
−
1
n-1
n
−
1
matches. The top
m
≤
n
m \leq n
m
≤
n
temas at the end of the tournament qualify for the playoffs. Assume there are no tied matches.Let
A
(
m
,
n
)
A(m,n)
A
(
m
,
n
)
be the minimum number of matches a team has to win to gurantee selection for the playoffs, regardless of what their run rate is. For example,
A
(
n
,
n
)
=
0
A(n,n) = 0
A
(
n
,
n
)
=
0
(everyone qualifies anyway so no need to win!) and
A
(
1
,
n
)
=
n
−
1
A(1,n) = n-1
A
(
1
,
n
)
=
n
−
1
(even if you lose to just one other team, they might defeat everyone and qualify instead of you). Answer the following:
(
A
)
(A)
(
A
)
FInd the value of
A
(
2
,
4
)
,
A
(
2
,
6
)
A(2,4),A(2,6)
A
(
2
,
4
)
,
A
(
2
,
6
)
and
A
(
4
,
10
)
A(4,10)
A
(
4
,
10
)
with proof (explain why a smaller value can still lead to the team not qualifying, and show that the respective values themselves are enough).
(
B
)
(B)
(
B
)
Show that
A
(
n
−
1
,
n
)
=
n
2
A(n-1,n) = \frac{n}{2}
A
(
n
−
1
,
n
)
=
2
n
when
n
n
n
even and
=
n
+
1
2
= \frac{n+1}{2}
=
2
n
+
1
when
n
n
n
odd.
(
C
)
(C)
(
C
)
For bonus marks, try to find a general pattern for
A
(
m
,
n
)
A(m,n)
A
(
m
,
n
)
.