MathDB

2

Part of 2009 Iran MO (2nd Round)

Problems(2)

Neighbors in the garden - Iran NMO 2009 - Problem 2

Source:

9/20/2010
In some of the 1×1 1\times1 squares of a square garden 50×50 50\times50 we've grown apple, pomegranate and peach trees (At most one tree in each square). We call a 1×1 1\times1 square a room and call two rooms neighbor if they have one common side. We know that a pomegranate tree has at least one apple neighbor room and a peach tree has at least one apple neighbor room and one pomegranate neighbor room. We also know that an empty room (a room in which there’s no trees) has at least one apple neighbor room and one pomegranate neighbor room and one peach neighbor room. Prove that the number of empty rooms is not greater than 1000. 1000.
combinatorics proposedcombinatorics
Prove the inequality - Iran NMO 2009 - Problem 5

Source:

9/20/2010
Let a1<a2<⋯<an a_1<a_2<\cdots<a_n be positive integers such that for every distinct 1≤i,j≤n1\leq{i,j}\leq{n} we have aj−ai a_j-a_i divides ai a_i . Prove that iaj≤jai for 1≤i<j≤n ia_j\leq{ja_i} \qquad \text{ for } 1\leq{i}<j\leq{n}
inequalitiesinductionnumber theory proposednumber theory