MathDB

3

Part of 2010 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Functional equation f(x+y)=f(x)+f(y) on a restricted domain

Source: XIX Olimpíada Matemática Rioplatense (2010)

7/22/2011
Find all the functions f:N→Rf:\mathbb{N}\to\mathbb{R} that satisfy f(x+y)=f(x)+f(y) f(x+y)=f(x)+f(y) for all x,y∈Nx,y\in\mathbb{N} satisfying 106−1106<xy<106+110610^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}.
Note: N\mathbb{N} denotes the set of positive integers and R\mathbb{R} denotes the set of real numbers.
algebrafunctiondomainalgebra unsolved
Moving a pebble among 1, 2,..., n game

Source: XIX Olimpíada Matemática Rioplatense (2010)

7/22/2011
Alice and Bob play the following game. To start, Alice arranges the numbers 1,2,…,n1,2,\ldots,n in some order in a row and then Bob chooses one of the numbers and places a pebble on it. A player's turn consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the number kk at most kk times. The two players alternate taking turns beginning with Alice. The first player who cannot make a move loses. For each positive integer nn, determine who has a winning strategy.
combinatoricsgamegame strategyGame Theory