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National and Regional Contests
Latvia Contests
Latvia BW TST
2016 Latvia Baltic Way TST
5
5
Part of
2016 Latvia Baltic Way TST
Problems
(1)
a + b + d <= 3c if a^2 + ab + b^2 = 3c^2 and a^3 + a^2b + ab^2 + b^3 = 4d^3
Source: 2016 Latvia BW TST P5
12/17/2022
Given real positive numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
, for which the equalities
a
2
+
a
b
+
b
2
=
3
c
2
a^2 + ab + b^2 = 3c^2
a
2
+
ab
+
b
2
=
3
c
2
and
a
3
+
a
2
b
+
a
b
2
+
b
3
=
4
d
3
a^3 + a^2b + ab^2 + b^3 = 4d^3
a
3
+
a
2
b
+
a
b
2
+
b
3
=
4
d
3
are fulfilled. Prove that
a
+
b
+
d
≤
3
c
.
a + b + d \le 3c.
a
+
b
+
d
≤
3
c
.
algebra
inequalities