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Chennai Mathematical Institute B.Sc. Entrance Exam
2023 CMI B.Sc. Entrance Exam
2023 CMI B.Sc. Entrance Exam
Part of
Chennai Mathematical Institute B.Sc. Entrance Exam
Subcontests
(6)
6
1
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Finding finite trajectories of a particular function
Consider a positive integer
a
>
1
a > 1
a
>
1
. If
a
a
a
is not a perfect square then at the next move we add
3
3
3
to it and if it is a perfect square we take the square root of it. Define the trajectory of a number
a
a
a
as the set obtained by performing this operation on
a
a
a
. For example the cardinality of
3
3
3
is
{
3
,
6
,
9
}
\{3, 6, 9\}
{
3
,
6
,
9
}
. Find all
n
n
n
such that the cardinality of
n
n
n
is finite. The following part problems may attract partial credit.
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Show that the cardinality of the trajectory of a number cannot be
1
1
1
or
2
2
2
.
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(
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Show that
{
3
,
6
,
9
}
\{3, 6, 9\}
{
3
,
6
,
9
}
is the only trajectory with cardinality
3
3
3
.
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<span class='latex-bold'>(c)</span>
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Show that there for all
k
≥
3
k \geq 3
k
≥
3
, there exists a number such that the cardinality of its trajectory is
k
k
k
.
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d
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d
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Give an example of a number with cardinality of trajectory as infinity.
5
1
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Results on composition of functions
In whatever follows
f
f
f
denotes a differentiable function from
R
\mathbb{R}
R
to
R
\mathbb{R}
R
.
f
∘
f
f \circ f
f
∘
f
denotes the composition of
f
(
x
)
f(x)
f
(
x
)
.
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(
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<span class='latex-bold'>(a)</span>
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If
f
∘
f
(
x
)
=
f
(
x
)
∀
x
∈
R
f\circ f(x) = f(x) \forall x \in \mathbb{R}
f
∘
f
(
x
)
=
f
(
x
)
∀
x
∈
R
then for all
x
x
x
,
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
or
f
′
(
f
(
x
)
)
=
f'(f(x)) =
f
′
(
f
(
x
))
=
. Fill in the blank and justify.
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Assume that the range of
f
f
f
is of the form
(
−
∞
,
+
∞
)
,
[
a
,
∞
)
,
(
−
∞
,
b
]
,
[
a
,
b
]
\left(-\infty , +\infty \right), [a, \infty ),(- \infty , b], [a, b]
(
−
∞
,
+
∞
)
,
[
a
,
∞
)
,
(
−
∞
,
b
]
,
[
a
,
b
]
. Show that if
f
∘
f
=
f
f \circ f = f
f
∘
f
=
f
, then the range of
f
f
f
is
R
\mathbb{R}
R
. (Hint: Consider a maximal element in the range of f)
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<span class='latex-bold'>(c)</span>
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(
c
)
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If
g
g
g
satisfies
g
∘
g
∘
g
=
g
g \circ g \circ g = g
g
∘
g
∘
g
=
g
, then
g
g
g
is onto. Prove that
g
g
g
is either strictly increasing or strictly decreasing. Furthermore show that if
g
g
g
is strictly increasing, then
g
g
g
is unique.
4
1
Hide problems
Results on n students with distinct heights
In a class there are n students with unequal heights.
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<span class='latex-bold'>(a)</span>
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−
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>
(
a
)
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Find the number of orderings of the students such that the shortest person is not at the front and the tallest person is not at the end.
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<span class='latex-bold'>(b)</span>
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Define the badness of an ordering as the maximum number
k
k
k
such that there are
k
k
k
many people with height greater than in front of a person. For example: the sequence
66
,
61
,
65
,
64
,
62
,
70
66, 61, 65, 64, 62, 70
66
,
61
,
65
,
64
,
62
,
70
has badness
3
3
3
since there are
3
3
3
numbers greater than
62
62
62
in front of it. Let
f
k
(
n
)
f_k(n)
f
k
(
n
)
denote the number of orderings of
n
n
n
with badness
k
k
k
. Find
f
k
(
n
)
f_k(n)
f
k
(
n
)
. (Hint: Consider
g
k
(
n
)
g_k(n)
g
k
(
n
)
as the number of orderings of n with badness less than or equal to
k
k
k
)
1
1
Hide problems
n|2023^n-1
We will consider odd natural numbers
n
n
n
such that
n
∣
202
3
n
−
1
n|2023^n-1
n
∣202
3
n
−
1
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a
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<span class='latex-bold'>a.</span>
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Find the smallest two such numbers.
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b
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<span class='latex-bold'>b.</span>
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b
.
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Prove that there exists infinitely many such
n
n
n
3
1
Hide problems
If r is a double root of a degree 4 polynomial, then results on its coefficients
Consider the polynomial
p
(
x
)
=
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
p(x) = x^4 + ax^3 + bx^2 + cx + d
p
(
x
)
=
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
. It is given that
p
(
x
)
p(x)
p
(
x
)
has its only root at
x
=
r
x = r
x
=
r
i.e
p
(
r
)
=
0
p(r) = 0
p
(
r
)
=
0
.
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<span class='latex-bold'>(a)</span>
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(
a
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Show that if
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are rational then
r
r
r
is rational.
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b
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<span class='latex-bold'>(b)</span>
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(
b
)
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>
Show that if
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are integers then
r
r
r
is an integer. (Hint: Consider the roots of
p
′
(
x
)
p'(x)
p
′
(
x
)
)
2
1
Hide problems
g(m + n) = g(m) + mn(m + n) + g(n)
Solve for
g
:
Z
+
→
Z
+
g : \mathbb{Z}^+ \to \mathbb{Z}^+
g
:
Z
+
→
Z
+
such that
g
(
m
+
n
)
=
g
(
m
)
+
m
n
(
m
+
n
)
+
g
(
n
)
g(m + n) = g(m) + mn(m + n) + g(n)
g
(
m
+
n
)
=
g
(
m
)
+
mn
(
m
+
n
)
+
g
(
n
)
Show that
g
(
n
)
g(n)
g
(
n
)
is of the form
∑
i
=
0
d
c
i
n
i
\sum_{i=0}^{d} {c_i n^i}
∑
i
=
0
d
c
i
n
i
\\ and find necessary and sufficient conditions on
d
d
d
and
c
0
,
c
1
,
⋯
,
c
d
c_0, c_1, \cdots , c_d
c
0
,
c
1
,
⋯
,
c
d