Team Selection Test for JBMO 2023

Question

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.

Accepted Answer

**5.** For any positive real x we have (x-1)^2(3x^2+4x+3) 0. Therefore, 3(x^4 + 1) 2(x^3 + x^2 + x) (1) By using (1) we get a^4+1b^3+b^2+b b^4+1c^3+c^2+c c^4+1a^3+a^2+a 827 Finally, by using of by AM-GM inequality a^4+1b^3+b^2+b + b^4+1c^3+c^2+c + c^4+1a^3+a^2+a 3 [3]827 = 2.