4

Part of 2017 Auckland Mathematical Olympiad

Problems(2)

average mean, integers 1 to n related (2017 Auckland MO J4)

Source:

The positive integers from

1

n

inclusive are written on a whiteboard. After one number is erased, the average (arithmetic mean) of the remaining

n - 1

22

n

is odd, determine

n

and the number that was erased. Explain your reasoning.

11 empty boxes and a pile of stones (2017 Auckland MO S4)

Source:

11

empty boxes and a pile of stones. Two players play the following game by alternating moves: In one move a player takes

10

stones from the pile and places them into boxes, taking care to place no more than one stone in any box. The winner is the player after whose move there appear

21

stones in one of the boxes for the first time. If a player wants to guarantee that they win the game, should they go first or second? Explain your reasoning.

combinatoricsgamewinning strategy