2

Part of 2002 Italy TST

Problems(2)

Divisiblity with a binomial

Source: Italy TST 2002

Prove that for each prime number

p

and positive integer

n

p^n

\binom{p^n}{p}-p^{n-1}.

modular arithmeticnumber theory unsolvednumber theory

soccer tournament

Source: Italy TST 2002

On a soccer tournament with

n\ge 3

teams taking part, several matches are played in such a way that among any three teams, some two play a match.

(a)

n=7

, find the smallest number of matches that must be played.

(b)

Find the smallest number of matches in terms of

n

floor functioncombinatorics unsolvedcombinatorics