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Combinatorics Difficulty 6.3 National olympiad Prove it Turkey

At the beginning Alice distributes $1000$ balls into $30$ boxes. After that Alice and Bob alternatively make moves, Alice begins. A person making move chooses a box and takes one ball from the chosen box. A person taking the very last ball from a box takes on that empty box. Find the maximal integer $k$ such that regardless of the strategy of Bob Alice can take at least $k$ boxes.

Solution

5. For any positive real $x$ we have $(x-1)^2(3x^2+4x+3) \ge 0$ . Therefore,
$3(x^4 + 1) \ge 2(x^3 + x^2 + x) \quad (1)$
By using (1) we get
$\frac{a^4+1}{b^3+b^2+b} \cdot \frac{b^4+1}{c^3+c^2+c} \cdot \frac{c^4+1}{a^3+a^2+a} \ge \frac{8}{27}$
Finally, by using of by AM-GM inequality
$\frac{a^4+1}{b^3+b^2+b} + \frac{b^4+1}{c^3+c^2+c} + \frac{c^4+1}{a^3+a^2+a} \ge 3 \cdot \sqrt[3]{\frac{8}{27}} = 2.$

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Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.