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Part of 2010 Bundeswettbewerb Mathematik

Problems(2)

3 number game from 1 to 9999, 3 left to construct triangle, winning strategy

Source: Germany Federal - Bundeswettbewerb Mathematik 2010, round 1, p2

4/9/2020
There are 99999999 rods with lengths 1,2,...,9998,99991, 2, ..., 9998, 9999. The players Anja and Bernd alternately remove one of the sticks, with Anja starting. The game ends when there are only three bars left. If from those three bars, a not degenerate triangle can be constructed then Anja wins, otherwise Bernd. Who has a winning strategy?
combinatoricsgamegame strategywinning strategy
a_1 = 1, a_{n + 1} = [\sqrt{a_1+a_2+...+a_n} ]

Source: Germany Federal - Bundeswettbewerb Mathematik 2010, round 2, p2

4/14/2020
The sequence of numbers a1,a2,a3,...a_1, a_2, a_3, ... is defined recursively by a1=1,an+1=⌊a1+a2+...+an⌋a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor for n≥1n \ge 1. Find all numbers that appear more than twice at this sequence.
floor functionrecurrence relationSequenceradicalalgebra