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Polish MO Finals
1959 Polish MO Finals
1
\frac{a+b}{2} \frac{a^2+b^2}{2} \frac{a^3+b^3}{2} \leq \frac{a^6+b^6}{2}.
\frac{a+b}{2} \frac{a^2+b^2}{2} \frac{a^3+b^3}{2} \leq \frac{a^6+b^6}{2}.
Source: Polish MO Finals 1959 p1
August 29, 2024
inequalities
algebra
Problem Statement
Prove that for any numbers
a
a
a
and
b
b
b
the inequality holds
a
+
b
2
⋅
a
2
+
b
2
2
⋅
a
3
+
b
3
2
≤
a
6
+
b
6
2
.
\frac{a+b}{2} \cdot \frac{a^2+b^2}{2} \cdot \frac{a^3+b^3}{2} \leq \frac{a^6+b^6}{2}.
2
a
+
b
⋅
2
a
2
+
b
2
⋅
2
a
3
+
b
3
≤
2
a
6
+
b
6
.
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