3

Part of 2002 Vietnam National Olympiad

Problems(2)

Vietnam NMO 2002_3

Source:

Let be given two positive integers

m

n

m < 2001

n < 2002

. Let distinct real numbers be written in the cells of a

2001 \times 2002

2001

2002

columns). A cell of the board is called bad if the corresponding number is smaller than at least

m

numbers in the same column and at least

n

numbers in the same row. Let

s

denote the total number of bad cells. Find the least possible value of

s

combinatorics unsolvedcombinatorics

Vietnam NMO 2002_6

Source:

For a positive integer

n

, consider the equation \frac{1}{x\minus{}1}\plus{}\frac{1}{4x\minus{}1}\plus{}\cdots\plus{}\frac{1}{k^2x\minus{}1}\plus{}\cdots\plus{}\frac{1}{n^2x\minus{}1}\equal{}\frac{1}{2}. (a) Prove that, for every

n

, this equation has a unique root greater than

1

, which is denoted by

x_n

. (b) Prove that the limit of sequence

(x_n)

4

n

approaches infinity.

algebrapolynomialalgebra unsolved