1

Part of 2001 Cuba MO

Problems(2)

numbers in 3 x3 board

Source: 2001 Cuba MO 1.1

In each square of a

3 \times 3

board a real number is written. The element of the

i

-th row and the

j

-th column is equal to abso;uteof the difference of the sum of the elements of column

j

and the sum of the elements of row

i

. Prove that every element of the board is equal to the sum or difference of two other elements on the board.

f(x) f(k - x) > 0

Source: 2001 Cuba MO 2.1

f

be a linear function such that

f(0) = -5

f(f(0)) = -15

. Find the values of

k \in R

for which the solutions of the inequality

f(x) \cdot f(k - x) > 0

, lie in an interval of length

2

inequalitiesalgebra