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National and Regional Contests
Azerbaijan Contests
JBMO TST - Azerbaijan
2022 Azerbaijan JBMO TST
A2
A2
Part of
2022 Azerbaijan JBMO TST
Problems
(1)
An inequality where cyc \frac{1}{a} \ge \frac{3}{abc}
Source: Azerbaijan 2022 JBMO TST
7/3/2022
For positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
,
1
a
+
1
b
+
1
c
≥
3
a
b
c
\frac{1}{a}+\frac{1}{b} + \frac{1}{c} \ge \frac{3}{abc}
a
1
+
b
1
+
c
1
≥
ab
c
3
is true. Prove that:
a
2
+
b
2
a
2
+
b
2
+
1
+
b
2
+
c
2
b
2
+
c
2
+
1
+
c
2
+
a
2
c
2
+
a
2
+
1
≥
2
\frac{a^2+b^2}{a^2+b^2+1}+\frac{b^2+c^2}{b^2+c^2+1}+\frac{c^2+a^2}{c^2+a^2+1} \ge 2
a
2
+
b
2
+
1
a
2
+
b
2
+
b
2
+
c
2
+
1
b
2
+
c
2
+
c
2
+
a
2
+
1
c
2
+
a
2
≥
2
inequalities
Algebra - Inequality
algebra
JBMO TST
Azerbaijan
Junior