IMO 2017 Shortlist N2
Let p ě 2 be a prime number. Eduardo and Fernando play the following game making moves alternately: in ea h move, the ur...
Category: Number Theory
Problem
Let p ě 2 be a prime number. Eduardo and Fernando play the following game making
moves alternately: in ea h move, the urrent player hooses an index i in the set t0,1,...,p´1u
that was not hosen before by either of the two players and then hooses an element ai of the
set t0,1,2,3,4,5,6,7,8,9u. Eduardo has the rst move. The game ends after all the indi es
i P t0,1,...,p ´ 1u have been hosen. Then the following number is omputed:
M “ a0 10 ¨ a1 ¨¨¨ ` 10p´1
¨ ap´1 “
p´1 ÿ
j“0
aj ¨ 10j
.
The goal of Eduardo is to make the number M divisible by p, and the goal of Fernando is to
prevent this.
Prove that Eduardo has a winning strategy.
(Moro o)