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National and Regional Contests
India Contests
Chennai Mathematical Institute B.Sc. Entrance Exam
2024 CMI B.Sc. Entrance Exam
2024 CMI B.Sc. Entrance Exam
Part of
Chennai Mathematical Institute B.Sc. Entrance Exam
Subcontests
(5)
5
1
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Standard Diophentine with a trap
Find all solutions for positive integers
(
x
,
y
,
k
,
m
)
(x,y,k,m)
(
x
,
y
,
k
,
m
)
such that
20
x
k
+
24
y
m
=
2024
20x^k+24y^m = 2024
20
x
k
+
24
y
m
=
2024
with
k
,
m
>
1
k, m > 1
k
,
m
>
1
4
1
Hide problems
CMI just asks Schur directly
(a) For non negetive
a
,
b
,
c
,
r
a,b,c, r
a
,
b
,
c
,
r
prove that
a
r
(
a
−
b
)
(
a
−
c
)
+
b
r
(
b
−
a
)
(
b
−
c
)
+
c
r
(
c
−
a
)
(
c
−
b
)
≥
0
a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r (c-a)(c-b) \geq 0
a
r
(
a
−
b
)
(
a
−
c
)
+
b
r
(
b
−
a
)
(
b
−
c
)
+
c
r
(
c
−
a
)
(
c
−
b
)
≥
0
(b) Find an inequality for non negative
a
,
b
,
c
a,b,c
a
,
b
,
c
with
a
4
+
b
4
+
c
4
+
a
b
c
(
a
+
b
+
c
)
a^4+b^4+c^4 + abc(a+b+c)
a
4
+
b
4
+
c
4
+
ab
c
(
a
+
b
+
c
)
on the greater side. (c) Prove that if
a
b
c
=
1
abc = 1
ab
c
=
1
for non negative
a
,
b
,
c
a,b,c
a
,
b
,
c
,
a
4
+
b
4
+
c
4
+
a
3
+
b
3
+
c
3
+
a
+
b
+
c
≥
a
2
+
b
2
c
+
b
2
+
c
2
a
+
c
2
+
a
2
b
+
3
a^4+b^4+c^4+a^3+b^3+c^3+a+b+c \geq \frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}+3
a
4
+
b
4
+
c
4
+
a
3
+
b
3
+
c
3
+
a
+
b
+
c
≥
c
a
2
+
b
2
+
a
b
2
+
c
2
+
b
c
2
+
a
2
+
3
3
1
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A lot of z
(a) FInd the number of complex roots of
Z
6
=
Z
+
Z
ˉ
Z^6 = Z + \bar{Z}
Z
6
=
Z
+
Z
ˉ
(b) Find the number of complex solutions of
Z
n
=
Z
+
Z
ˉ
Z^n = Z + \bar{Z}
Z
n
=
Z
+
Z
ˉ
for
n
∈
Z
+
n \in \mathbb{Z}^+
n
∈
Z
+
2
1
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CMI's Integral function
g
(
x
)
:
∫
10
x
log
10
(
log
10
(
t
2
−
1000
t
+
1
0
1000
)
)
d
t
g(x) \colon \int_{10}^{x} \log_{10}(\log_{10}(t^2-1000t+10^{1000})) dt
g
(
x
)
:
∫
10
x
lo
g
10
(
lo
g
10
(
t
2
−
1000
t
+
1
0
1000
))
d
t
(a) Find the domain of
g
(
x
)
g(x)
g
(
x
)
(b) Approximate the value of
g
(
1000
)
g(1000)
g
(
1000
)
(c) Find
x
∈
[
10
,
1000
]
x \in [10, 1000]
x
∈
[
10
,
1000
]
to maximize the slope of
g
(
x
)
g(x)
g
(
x
)
(d) Find
x
∈
[
10
,
1000
]
x \in [10, 1000]
x
∈
[
10
,
1000
]
to minimize the slope of
g
(
x
)
g(x)
g
(
x
)
(e) Determine, if it exists,
lim
x
→
∞
ln
(
x
)
g
(
x
)
\lim_{x \to \infty} \frac{\ln(x)}{g(x)}
lim
x
→
∞
g
(
x
)
l
n
(
x
)
1
1
Hide problems
ISI and CMI's newfound volume obsession
(a) Sketch qualitativly the region with maximum area such that it lies in the first quadrant and is bound by
y
=
x
2
−
x
3
y=x^2-x^3
y
=
x
2
−
x
3
and
y
=
k
x
y=kx
y
=
k
x
where
k
k
k
is a constent. The region must not have any other lines closing it. Note:
k
x
kx
k
x
lies above
x
2
−
x
3
x^2-x^3
x
2
−
x
3
(b) Find an expression for the volume of the solid obtained by spinning this region about the
y
y
y
axis.