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National and Regional Contests
India Contests
Chennai Mathematical Institute B.Sc. Entrance Exam
2024 CMI B.Sc. Entrance Exam
4
4
Part of
2024 CMI B.Sc. Entrance Exam
Problems
(1)
CMI just asks Schur directly
Source:
5/19/2024
(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
inequalities
algebra