MathDB

Let

n

be a fixed positive integer. Oriol has

n

cards, each of them with a

0

written on one side and

1

on the other. We place these cards in line, some face up and some face down (possibly all on the same side). We begin the following process consisting of

n

steps:
1) At the first step, Oriol flips the first card 2) At the second step, Oriol flips the first card and second card . . . n) At the last step Oriol flips all the cards
Let

s_0, s_1, s_2, \dots, s_n

be the sum of the numbers seen in the cards at the beggining, after the first step, after the second step,

\dots

after the last step, respectively.
a) Find the greatest integer

k

such that, no matter the initial card configuration, there exists at least

k

distinct numbers between

s_0, s_1, \dots, s_n

. b) Find all positive integers

m

such that, for each initial card configuration, there exists an index

r

such that

s_r = m

.
Proposed by Dorlir Ahmeti

Problem Statement