4

Part of 2004 IMO Shortlist

Problems(2)

Source: Poland 2005, IMO Shortlist 2004, number theory problem 4

k

be a fixed integer greater than 1, and let

{m=4k^2-5}

. Show that there exist positive integers

a

b

such that the sequence

(x_n)

defined by x_0=a, x_1=b, x_{n+2}=x_{n+1}+x_n \text{for} n=0,1,2,\dots, has all of its terms relatively prime to

m

.
Proposed by Jaroslaw Wroblewski, Poland

quadraticsnumber theorySequencerelatively primeIMO Shortlist

sum of its entries not exceeding C in absolute value

Source: IMO ShortList 2004, combinatorics problem 4; KÃ¶mal

Consider a matrix of size

n\times n

whose entries are real numbers of absolute value not exceeding

1

. The sum of all entries of the matrix is

0

n

be an even positive integer. Determine the least number

C

such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding

C

in absolute value.
Proposed by Marcin Kuczma, Poland

linear algebramatrixalgebraprobabilityIMO Shortlistcombinatorics