4

Part of 2018 Danube Mathematical Competition

Problems(2)

number of the subsets of M whose sum of elements equals n

Source: Danube 2018 junior p4

M

be the set of positive odd integers. For every positive integer

n

A(n)

the number of the subsets of

M

whose sum of elements equals

n

. For instance,

A(9) = 2

, because there are exactly two subsets of

M

with the sum of their elements equal to

9

\{9\}

\{1, 3, 5\}

. a) Prove that

A(n) \le A(n + 1)

for every integer

n \ge 2

. b) Find all the integers

n \ge 2

A(n) = A(n + 1)

combinatoricsSubsetsSets

Disconnected Chessboard Coloring

Source: China Mathematical Olympiad 2018 Q5

n \geq 3

be an odd number and suppose that each square in a

n \times n

chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares

a,b

are considered connected if there exists a sequence of squares

c_1,\ldots,c_k

c_1 = a, c_k = b

c_i, c_{i+1}

are adjacent for

i=1,2,\ldots,k-1

. \\ \\ Find the maximal number

M

such that there exists a coloring admitting

M

pairwise disconnected squares.