Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.3 National olympiad Prove it

106. Let $a, b, c$ be positive real numbers, prove: $\frac{a^{2}}{b}+\frac{b^{3}}{c^{2}}+\frac{c^{4}}{a^{3}} \geqslant -a + 2b + 2c$ . (2005 Ukrainian Mathematical Olympiad Problem)

Solution

106. By the AM-GM inequality, $\frac{a^{2}}{b}+a+b \geqslant 3 a, \frac{b^{3}}{c^{2}}+c+c \geqslant 3 b, \frac{c^{4}}{a^{3}}+a+a+a \geqslant 4 c$ , adding them up yields $\frac{a^{2}}{b}+\frac{b^{3}}{c^{2}}+\frac{c^{4}}{a^{3}} \geqslant -a+2 b+2 c$ .

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