MathDB

1

Part of 2005 Iran MO (2nd round)

Problems(2)

There exists a finite configuratio- Iran NMO 2005 - Problem4

Source:

9/21/2010
We have a 2×n2\times n rectangle. We call each 1×11\times1 square a room and we show the room in the ithi^{th} row and jthj^{th} column as (i,j)(i,j). There are some coins in some rooms of the rectangle. If there exist more than 11 coin in each room, we can delete 22 coins from it and add 11 coin to its right adjacent room OR we can delete 22 coins from it and add 11 coin to its up adjacent room. Prove that there exists a finite configuration of allowable operations such that we can put a coin in the room (1,n)(1,n).
inductioncombinatorics proposedcombinatorics
Prove that 4p-3 is a square - Iran NMO 2005 - Problem1

Source:

9/21/2010
Let n,p>1n,p>1 be positive integers and pp be prime. We know that n∣p−1n|p-1 and p∣n3−1p|n^3-1. Prove that 4p−34p-3 is a perfect square.
number theoryLte