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Contests
National and Regional Contests
Russia Contests
Russian Team Selection Tests
Russian TST 2016
P1
Easy inequality
Easy inequality
Source: Russian TST 2016, Day 9 P1 (Groups A & B)
April 19, 2023
algebra
inequalities
Problem Statement
The positive numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
are such that
a
2
<
16
b
c
,
b
2
<
16
c
a
a^2<16bc, b^2<16ca
a
2
<
16
b
c
,
b
2
<
16
c
a
and
c
2
<
16
a
b
c^2<16ab
c
2
<
16
ab
. Prove that
a
2
+
b
2
+
c
2
<
2
(
a
b
+
b
c
+
c
a
)
.
a^2+b^2+c^2<2(ab+bc+ca).
a
2
+
b
2
+
c
2
<
2
(
ab
+
b
c
+
c
a
)
.
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