Subcontests
(4)Problem 3, BMO 2020
Let k be a positive integer. Determine the least positive integer n, with n≥k+1, for which the game below can be played indefinitely:Consider n boxes, labelled b1,b2,...,bn. For each index i, box bi contains exactly i coins. At each step, the following three substeps are performed in order:
(1) Choose k+1 boxes;
(2) Of these k+1 boxes, choose k and remove at least half of the coins from each, and add to the remaining box, if labelled bi, a number of i coins.
(3) If one of the boxes is left empty, the game ends; otherwise, go to the next step.Proposed by Demetres Christofides, Cyprus Problem 2, BMO 2020
Denote Z>0={1,2,3,...} the set of all positive integers. Determine all functions f:Z>0→Z>0 such that, for each positive integer n,
i)∑k=1nf(k) is a perfect square, and
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ii)f(n) divides n3.Proposed by Dorlir Ahmeti, Albania