MathDB

1.1 Let

f(A)

denote the difference between the maximum value and the minimum value of a set

A

. Find the sum of

f(A)

A

ranges over the subsets of

\{1, 2, \dots, n\}

.
1.2 All cells of an

8 × 8

board are initially white. A move consists of flipping the color (white to black or vice versa) of cells in a

1\times 3

3\times 1

rectangle. Determine whether there is a finite sequence of moves resulting in the state where all

64

cells are black.
1.3 Prove that for all positive integers

m

, there exists a positive integer

n

such that the set

\{n, n + 1, n + 2, \dots , 3n\}

contains exactly

m

perfect squares.

Problem Statement