1

Part of 2018 Bundeswettbewerb Mathematik

Problems(2)

Classical game with a monotonicity condition

Source: Bundeswettbewerb Mathematik 2018, Round 2, Problem 1

Anja and Bernd take turns in removing stones from a heap, initially consisting of

n

n \ge 2

). Anja begins, removing at least one but not all the stones. Afterwards, in each turn the player has to remove at least one stone and at most as many stones as removed in the preceding move. The player removing the last stone wins.
Depending on the value of

n

, which player can ensure a win?

gamecombinatoricscombinatorics proposed

Number with a "convex" sequence of digits

Source: Bundeswettbewerb Mathematik 2018, Round 1 - #1

Find the largest positive integer with the property that each digit apart from the first and the last one is smaller than the arithmetic mean of her neighbours.

Digitsnumber theoryarithmetic