3

Part of 1999 Iran MO (2nd round)

Problems(2)

100*100 garden - Iran NMO 1999 (Second Round) Problem3

Source:

100\times100

garden and we’ve plant

10000

1\times1

squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree.

combinatorics proposedcombinatorics

A_1,...,A_n points - Iran NMO 1999 (Second Round) Problem6

Source:

A_1,A_2,\cdots,A_n

n

distinct points on the plane (

n>1

). We consider all the segments

A_iA_j

i<j\leq{n}

and color the midpoints of them. What's the minimum number of colored points? (In fact, if

k

colored points coincide, we count them

1

combinatorics proposedcombinatorics