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Rioplatense Mathematical Olympiad, Level 3
1999 Rioplatense Mathematical Olympiad, Level 3
4
sum 1/ \sqrt[3]{1^2}+\sqrt[3]{1 x 2}+\sqrt[3]{2^2} } + ...>9/2
sum 1/ \sqrt[3]{1^2}+\sqrt[3]{1 x 2}+\sqrt[3]{2^2} } + ...>9/2
Source: Rioplatense 1999 L3 P4
September 19, 2022
inequalities
algebra
Problem Statement
Prove the following inequality:
1
1
2
3
+
1
⋅
2
3
+
2
2
3
+
1
3
2
3
+
3
⋅
4
3
+
4
2
3
+
.
.
.
+
1
99
9
2
3
+
999
⋅
1000
3
+
100
0
2
3
>
9
2
\frac{1}{\sqrt[3]{1^2}+\sqrt[3]{1 \cdot 2}+\sqrt[3]{2^2} }+\frac{1}{\sqrt[3]{3^2}+\sqrt[3]{3 \cdot 4}+\sqrt[3]{4^2} }+...+ \frac{1}{\sqrt[3]{999^2}+\sqrt[3]{999 \cdot 1000}+\sqrt[3]{1000^2} }> \frac{9}{2}
3
1
2
+
3
1
⋅
2
+
3
2
2
1
+
3
3
2
+
3
3
⋅
4
+
3
4
2
1
+
...
+
3
99
9
2
+
3
999
⋅
1000
+
3
100
0
2
1
>
2
9
(The member on the left has 500 fractions.)
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