3

Part of 1997 Iran MO (2nd round)

Problems(2)

Swapping -1 with +1 - Iran NMO 1997 (Second Round)- Problem3

Source:

n\times n

table and we’ve written numbers

0,+1 \ or \ -1

1\times1

square such that in every row or column, there is only one

+1

-1

. Prove that by swapping the rows with each other and the columns with each other finitely, we can swap

+1

-1

combinatorics proposedcombinatorics

Maximum of p - Iran NMO 1997 (Second Round) - Problem6

Source:

a,b

be positive integers and

p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}

be a prime number. Find the maximum value of

p

and justify your answer.

number theory proposednumber theory