3

Part of 2001 Turkey MO (2nd round)

Problems(2)

Turkey NMO 2001 Problem 3

Source: Turkey NMO 2001 Problem 3

One wants to distribute

n

same sized cakes between

k

people equally by cutting every cake at most once. If the number of positive divisors of

n

d(n)

, show that the number of different values of

k

which makes such distribution possible is

n+d(n)

combinatorics proposedcombinatorics

Turkey NMO 2001 Problem 6, Coloring Problem

Source: Turkey NMO 2001 Problem 6

We wish to color the cells of a

n \times n

chessboard with

k

different colors such that for every

i\in \{1,2,...,n\}

2n-1

i

i

. column have all different colors.
a) Prove that for

n=2001

k=4001

, such coloring is not possible.
b) Show that for

n=2^{m}-1

k=2^{m+1}-1

, such coloring is possible.

combinatorics unsolvedcombinatorics