3

Part of 2006 Iran MO (2nd round)

Problems(2)

we can photograph k-1 pictures - Iran NMO 2006 - Problem3

Source:

In the night, stars in the sky are seen in different time intervals. Suppose for every

k

k>1

2

of them can be seen in one moment. Prove that we can photograph

k-1

pictures from the sky such that each of the mentioned stars is seen in at least one of the pictures. (The number of stars is finite. Define the moments that the

n^{th}

star is seen as

[a_n,b_n]

a_n<b_n

inductioncombinatorics proposedcombinatorics

The books become like the first - Iran NMO 2006 - Problem6

Source:

Some books are placed on each other. Someone first, reverses the upper book. Then he reverses the

2

upper books. Then he reverses the

3

upper books and continues like this. After he reversed all the books, he starts this operation from the first. Prove that after finite number of movements, the books become exactly like their initial configuration.

inductionleast common multiplecombinatorics proposedcombinatorics