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National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2019 Hanoi Open Mathematics Competitions
14
14
Part of
2019 Hanoi Open Mathematics Competitions
Problems
(1)
max of M =3(ab+bc+ca)-2abc if a+b+c=3, a,b,c>=0 (HOMC 2019 JI-14)
Source:
11/7/2020
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be nonnegative real numbers satisfying
a
+
b
+
c
=
3
a + b + c =3
a
+
b
+
c
=
3
. a) If
c
>
3
2
c > \frac32
c
>
2
3
, prove that
3
(
a
b
+
b
c
+
c
a
)
−
2
a
b
c
<
7
3(ab + bc + ca) - 2abc < 7
3
(
ab
+
b
c
+
c
a
)
−
2
ab
c
<
7
. b) Find the greatest possible value of
M
=
3
(
a
b
+
b
c
+
c
a
)
−
2
a
b
c
M =3(ab + bc + ca) - 2abc
M
=
3
(
ab
+
b
c
+
c
a
)
−
2
ab
c
.
algebra
inequalities
max